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

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Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Short-Term Scheduling of Continuous Plants Based on Conditional Sequencing for Planned Maintenance Rohit Omar and Munawar A. Shaik* Department of Chemical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India S Supporting Information *

ABSTRACT: In this paper we present an extension of an earlier model for short-term scheduling of continuous/semicontinuous plants based on conditional sequencing, where a consumption task is conditionally aligned to a production task only if it receives material from production task; otherwise, if material is received from storage then sequencing is relaxed. Four-index binary and continuous variables are used to handle conditional sequencing by modifying the material balances and sequencing constraints in the earlier models. Two models with different timing variables are presented, and appropriate lower and upper bounds are postulated for different scenarios involving planned maintenance. The proposed models are demonstrated on different instances of a motivating example and on literature examples related to scheduling of continuous plants under planned maintenance operation, resulting in either improved objective function value and/or the converged solution is found using less number of events.

1. INTRODUCTION Scheduling of continuous and semicontinuous plants has been an important research problem with many applications such as in refineries, petrochemicals, and other fast-moving consumer goods (FMCG) manufacturing plants. Several works have been done in the literature for short-term and cyclic scheduling of continuous plants using continuous time representations such as slot based, global-event based, unit-specific event based, and precedence based, having different process representations, i.e., state-task, resource-task networks. Important features such as unit task allocation, sequencing of tasks, sequence-dependent changeovers, different storage policies, i.e., unlimited, finite, no intermediate storage, zero-wait, dedicated, and flexible storage are the essential elements in mathematical models. Different objective functions such as minimization of makespan, minimization of total cost, and maximization of profit have been considered for solving the realistic problems. Extensive reviews are available (Floudas and Lin,1 Mendez et al.,2 Shaik et al.,3 Maravelias,4 Harjunkoski et al.5) that present comparisons of different approaches for scheduling. Sahindis and Grossmann6 formulated an MINLP model for periodic scheduling of continuous parallel lines by deploying the decomposition method and solved the realistic problem using Bender’s decomposition and/or outer approximation. This work was extended by Pinto and Grossmann7 to multistage continuous plants. Munawar et al.8 extended the model of Pinto and Grossmann7 for cyclic scheduling of hybrid flowshops where they modified time slots for handling feed losses during changeovers. Alle et al.9 proposed cyclic scheduling model accounting for cleaning operation of a unit © XXXX American Chemical Society

whose performance continually degrades with time. Lamba and Karimi10,11 presented a slot-based model and a decomposition algorithm for solving the short-term scheduling problem of semicontinuous processes with resource constraints. Castro et al.12 proposed a short-term scheduling model for batch and continuous plants using a global event-based time representation. Their model can effectively handle continuous tasks through a new set of timing constraints, and the efficacy of their model can be attributed to an improved finite intermediate storage scenario. Zhang and Sargent13 proposed an MINLP model for solving the optimal operation of mixed production facilities using a global event-based time representation and RTN process representation. Schilling and Pantelides14 formulated the general mathematical model for solving scheduling problems using a continuous-time representation and RTN process representation and also proposed a branch and bound algorithm to reduce the computation size of the problem. McDonald and Karimi15 and Karimi and McDonald16 developed two slot-based models: process slots and unit slots for parallel semicontinuous processes for production planning and short-term scheduling. Their model includes several realistic features such as sequence-dependent changeovers, inventory costs, safety stock penalties, and multiple due dates. However, the resource constraints are not defined explicitly. Received: Revised: Accepted: Published: A

October 16, 2017 February 7, 2018 February 7, 2018 February 7, 2018 DOI: 10.1021/acs.iecr.7b04294 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

problem with flexible maintenance and nonresumable jobs29−32 for the objective of minimizing makespan. The single machine scheduling problem with periodic maintenance and nonresumable jobs has been well studied.33−39 Periodic maintenance includes several maintenance periods each scheduled after a fixed periodic time interval. The periodic maintenance scheduling problem for parallel machines has been studied by Xu et al.40 Low et al.41 presented the single machine scheduling problem with flexible and periodic maintenance simultaneously. Lee and Lin42 considered the single machine scheduling problem for surface-mount technology of electronic assembly lines. They assumed maintenance and repair of the unit simultaneously, which is termed as rate modifying activities in their model. Schmidt43 presented a comprehensive review for scheduling of single and multiple machines with limited available time. Cheung et al.44 proposed the MILP model for a short-term maintenance scheduling problem that includes interaction between production and utility systems. Yu and Seif45 solved the conventional flow shop scheduling problem using a genetic algorithm for the objective of minimizing tardiness and maintenance costs. For handling complex plant structures network-based models such as state-task (STN) and resource-task networks (RTN) are widely used. Dedopoulos and Shah46 developed a scheduling model for a lubricant process based on discretetime STN process representation. Their objective was to maximize the plant profitability for multipurpose batch/ semicontinuous plants by considering production and maintenance. Sanmarti et al.47 addressed the problem for batch scheduling with preventive maintenance, where they considered a reliability index for each unit and scheduled tasks in order to avoid equipment failure. Hazaras et al.48 extended the globalevent based model of Maravelias and Grossmann49 for maintenance scheduling incorporating fixed, flexible, and recurring maintenance activities. Production planning and scheduling problems together with preventive maintenance have been gaining importance recently.50−53 Jain and Grossmann 54 presented the MINLP model for cyclic scheduling of ethylene plants under performance decay. Liu et al.55 proposed the MILP model for production and maintenance planning of biopharmaceutical process under performance decay. The issues related to unavailability of units can be better handled using a conditional sequencing model. The earlier proposed models assumed unconditional sequencing where production and consumption tasks were always aligned, irrespective of actual usage. An interesting case is shown in a motivating example given in Figure 1. Here a production task “m1” is taking place in “mixer A” and a consumption task “p1” is occurring in “line 1”. If line 1 is blocked for some duration, say due to planned maintenance, then mixer A also remains idle for the same duration due to unconditional alignment, which is undesirable. Avoiding unnecessary machine downtime, which is

Ierapetritou and Floudas17 proposed a unit-specific event based model for continuous plants which is applicable only for unlimited intermediate storage (UIS) policy. The limitations of their model were addressed by Shaik and Floudas18 by considering storage as a separate task for rigorously handling different storage policies and for different cases of storage bypassing allowed and not allowed, which gave better computational performance compared to other models from the literature. Shaik et al.19 solved a medium-term scheduling problem for a large-scale industrial continuous plant based on a rolling-horizon approach. Li et al.20 demonstrated the different advancements in unit-specific event-based models for batch and continuous/semicontinuous plants using more benchmark examples and highlighted the need for using explicit storage tasks for continuous processes for dedicated finite intermediate storage cases as well. Yadav and Shaik21 formulated the scheduling model for refinery crude oil operations based on a unit-specific event-based time representation. Their model effectively handles material flow in tanks based on whether mixing is allowed or not, whether simultaneous input and output is allowed or not, and whether bypassing is allowed or not. Seifert et al.22 presented design aspects of multiproduct continuous plants in two phases, with the first phase involving selection of suitable modular setups and in the second phase the setups are evaluated using real option analysis. However, in all of the above unit-specific event-based models, the sequencing constraints were written such that the production and consumption tasks were always aligned (unconditional alignment), irrespective of whether the consumption task receives material from production task or not, an issue which was recently addressed in batch process scheduling by Seid and Majozi23 and Vooradi and Shaik.24 This unconditional alignment often leads to the requirement of a higher number of events to find the optimal solution or may give suboptimal solution especially when planned maintenance is considered. Moreover, these models assume that all processing units operate without any intervention throughout the scheduling horizon. However, in reality the performance of a unit gradually decreases with time due to excessive usage of resources and requires regular maintenance to improve unit efficiency. Therefore, the importance for maintenance of the unit has been recognized by decision makers to optimize the plant profitability. The problem can be classified into two categories, namely, stochastic and deterministic. Stochastic problems are usually considered when there is an abrupt breakdown of a unit leading to unplanned maintenance and repair, and deterministic problems are handled with preventive and planned maintenance. The scheduled maintenance problem for a single machine has been addressed by Adiri et al.25 with the objective of minimizing makespan. They considered a single breakdown situation which was solved using both stochastic and deterministic approaches. The same problem has been extended by Lee and Liman26 for the deterministic case. Lee27 presented the fixed timing scheduled maintenance problem for single and multiple machines. Their algorithm could handle a single maintenance event on both preemptive and nonpreemptive production jobs using different performance measures such as makespan minimization and minimization of maximum tardy jobs. Xu et al.28 addressed the single machine scheduling problem with fixed duration of maintenance activity. The researchers have also addressed the

Figure 1. Motivating example. B

DOI: 10.1021/acs.iecr.7b04294 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research resulting merely due to some assumptions made in the literature models, is certainly desirable and is a significant development in resource management; given the global competition in the manufacturing sector it can lead to significant economic advantage. This is the motivation behind the proposed study so as to extend the earlier unconditional sequencing-based models to avoid such unnecessary nonproductive time.

Figure 2. Sample Gantt chart for planned maintenance: (a) ideal case and (b) SF model.

2. MATHEMATICAL FORMULATION In this work we present an extension of the earlier model of Shaik and Floudas18 (referred to as the SF model here) for short-term scheduling of continuous/semicontinuous plants based on conditional sequencing, where a consumption task is conditionally aligned to a production task only if it receives material from that production task; otherwise, if material is received from storage then sequencing is relaxed. Conditional sequencing in batch process scheduling was recently addressed by Seid and Majozi23 and Vooradi and Shaik.24 Additional fourindex binary and continuous variables are used to handle conditional sequencing by modifying the material balances and sequencing constraints in the earlier models. The four-index variables are used to monitor if the amount produced by a production task is actually consumed by a consumption task. We propose two models where the timing variables are modeled differently. In model M1 we use task-specific timing variables Ts(i,n) and Tf(i,n), and in model M2 we use unitspecific timing variables Ts(j,n) and Tf(j,n) to denote start and finish times at event n. Constraints 1−4 are common for both models and are similar as discussed in Shaik and Floudas18 model, which are repeated here for completeness.

duration due to maintenance taking place at the end of scheduling horizon then in Figure 2b the producing unit jp also gets blocked for the SF model as already discussed due to unconditional alignment of enforcing the finish time of consumption task which should be equal to the finish time production task at event n + 1, whereas in Figure 2a the ideal case is shown where the model does not have any effect to continue processing the material in the production unit. 2.1. Mathematical Model Based on Task-Specific Timing Variables (Model M1). 2.1.1. Duration, Capacity and Sequencing Constraints. Equation 5 describes the duration constraint in which task duration would be enforced to zero if the task is not activated at event n. Equation 6a defines the capacity constraint for tasks with variable processing rates and eq 6b for a fixed processing rate. Equations 5, 6a, and 6b also enforce the amount processed to be zero if the task is not active at event n. Equations 7 and 8−10 describe the sequencing constraints for the same task in the same unit and different tasks in the same unit, respectively. Equation 8 is applicable for sequence-independent changeovers, eq 9 for sequence-dependent changeovers, and eq 10 for handling of no changeover time.

∑ w(i , n) ≤ 1 ∀ j ∈ J , n ∈ N i ∈ Ij

T f (i , n) − T s(i , n) ≤ Hw(i , n) ∀ i ∈ I , n ∈ N

(1)

ST0(s , n) +

∑ ρsic b(i , n) = 0 ∀ s ∈ S R , n ∈ N i ∈ Isc

R imin(T f (i , n) − T s(i , n))

(2)

ST (s , n) = ST (s , n − 1) +

≤ b(i , n)

∑ ρsip b(i , n)

i ∈ Isp

+

∑ ρsic b(i , n) ∀ s ∈ Sin , n ∈ N , n > 1

i ∈ Isc

(3a)

ST (s , n) = ST0(s) +

∑ i ∈ Isp

ρsip b(i ,

n) +

∑ i ∈ Isc

ρsic b(i ,

s ∈ Sin , n = 1

Dsmin ≤

≤ R imax(T f (i , n) − T s(i , n)) ∀ i ∈ I , n ∈ N

(6a)

b(i , n) = R i(T f (i , n) − T s(i , n)) ∀ i ∈ I , n ∈ N

(6b)

T s(i , n + 1) ≥ T f (i , n) ∀ i ∈ I , n ∈ N , n ≠ N

n) ∀

(7)

T s(i , n + 1) ≥ T f (i′, n) + τjw(i′, n) ∀ j ∈ J cl1 , i , i′ ∈ Ij , i ≠ i′, n ∈ N , n ≠ N

(3b)

(8)

T s(i , n) ≥ T f (i′, n′) + ticl iw(i , n) − H(1 − w(i′, n′)) ′ − H ∑ ∑ w(i″ , n″)

∑ ∑ ρsip b(i , n) ≤ Dsmax ∀ s ∈ Sp

n ∈ N i ∈ Isp

(5)

(4)

i ″∈ Ij n ′< n ″< n

Equation 1 describes the allocation constraint in which at most one task can occur in each unit at each event. Equation 2 describes material balance for raw material states and eqs 3a and 3b for intermediate states. Equation 4 describes demand constraint for final products, where total production is bounded between minimum and maximum demand. Consider a sample Gantt chart for planned maintenance as shown in Figure 2. Here there are two production tasks ip1 and ip2 producing 10 and 30 ton of material in production unit (jp) at events n and n + 1. There are two consumption tasks ic1 and ic2 each consuming 10 ton of material in consumption unit (jc) at events n and n + 1. If consuming unit jc is blocked for some

(9)

T s(i , n + 1) ≥ T f (i′, n) ∀ j ∉ J cl 2 , i , i′ ∈ Ij , i ≠ i′, n ∈ N, n ≠ N

(10)

2.1.2. Sequencing Constraints Based on Usage of Storage Material. The constraint for different tasks occurring in different units is modified to take care of conditional sequencing and to avoid unconditional sequencing done in earlier unit-specific event-based models. For this purpose a four-index continuous variable b1(i′,i,s,n) is activated, in eqs 11a, 11b, 12a, and 12b, if there is not sufficient material in the C


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remaining in storage at the end of event n. Equations 12a and 12b allow for partial consumption from the production task. Thus, the concept of conditional sequencing allows for storage of fresh material in the storage tank, which is useful for perishable products such as fast-moving consumer goods (FMCG). 2.1.3. Alignment of Different Tasks in Different Units. A four-index binary variable z1(i′, i, s, n) is activated when the corresponding continuous variable b1(i′, i, s, n) is nonzero as shown in eq 13.

storage and based on whether consumption task i uses the amount of material from production task i′ at event n. − ∑ ρsic b(i , n) ≤ ST0, s + i ∈ Isc

∑ ∑ b1(i′, i , s , n) ∀ i ′∈ Isp i ∈ Isc

s ∈ Sin , n = 1

(11a)

− ∑ ρsic b(i , n) ≤ ST (s , n − 1) + i ∈ Isc

∑ ∑ b1(i′, i , s , n) ∀ i ′∈ Isp i ∈ Isc

s ∈ Sin , n ∈ N , n > 1

ρsip′b(i′, n) ≥

(11b)

in b1(i′, i , s , n) ≤ z1(i′, i , s , n)ρsip R imax ′ H ∀ s ∈ S , i′ ′ ∈ Isp , i ∈ Isc , j , j′ ∈ J , j

∑ b1(i′, i , s , n) ∀ s ∈ Sin , i′ ∈ Isp , n ∈ N i ∈ Isc

(12a)

−ρsic b(i ,

n) ≥

∑

in

b1(i′, i , s , n) ∀ s ∈ S , i ∈

Isc ,

≠ j′, i ∈ Ij , i′ ∈ Ij ′ , n ∈ N

n∈N

(13)

Accordingly, the constraint for different tasks occurring in different units is modified to model conditional sequencing using the four-index variable z1, so that a consumption task is conditionally aligned to a production task, only if it receives material from it; otherwise, it is relaxed, as shown in eqs 14a and 14b. On the other hand, if the consumption task consumes the entire required amount from storage then these constraints are relaxed, unlike in earlier models where the consumption tasks were unconditionally always aligned to production tasks.

i ′∈ Isp

(12b)

Consider a sample case shown in Figure 3 with 10 ton of material available in the storage tank at event n − 1 and task ip

T s(i , n) ≥ T s(i′, n) − H(1 − z1(i′, i , s , n)) ∀ s ∈ Sin , i′ ∈ Isp , i ∈ Isc , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′ ∈ Ij ′ , n ∈ N

(14a)

T f (i , n) ≥ T f (i′, n) − H(1 − z1(i′, i , s , n))

Figure 3. Motivating example: (a) no activation of variable b1(i′p,ic,s,n) and (b) activation of variable b1(i′p,ic,s,n).

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

producing 50 ton of material at event n. In case a, task ic consumes only 10 ton of material at event n, and since there is a large enough amount available in storage, variable b1(i′,i,s,n) is not activated here. Therefore, the consumption task receives the required amount of material (10 ton) from storage instead of receiving from the producing task. On the other hand, in case b, task ic consumes 60 ton of material at event n, with 10 ton coming from storage and 50 ton of material from production task ip at event n, thus activating the variable b1(i′,i,s,n). Indirectly we are also enforcing usage of the stored amount first with relative priority compared to the fresh amount produced. Consider a sample Gantt chart shown in Figure 4 in which task ic has a choice to either consume 10 ton of material from storage at event n − 1 followed by 40 ton of material from production task ip at event n or consume the entire 50 ton of amount produced at event n. Constraints 11a and 11b will enforce the former case, leading to 10 ton of fresh material

(14b)

Consider a sample Gantt chart shown in Figure 5 with multiple production and consumption tasks in which two production

Figure 5. Sample Gantt chart for multiple production and consumption tasks.

tasks, ip1 and ip2, produce 30 and 40 ton of material, respectively, and two consumption tasks, ic1 and ic2, consume 30 and 40 ton of material, respectively. Consumption task ic1 may consume material from any of the production tasks, and the same applies to task ic2. In case a, with unconditional sequencing, three events are necessary for the consumption task to consume material from the production task as shown in Figure 5a, due to unconditional enforcement of the same start (or end) times for all production and consumption tasks.

Figure 4. Sample Gantt chart for partial consumption from production. D


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the start time of the production task at event n. Two production tasks ip1 and ip2 each produce 10 ton of material in different units at event n; hence, storage continues until the finish time of the producing tasks, and the mixed material is still in the storage until the start time of the consumption task ic2 at event n. Although it appears that the entire amount of consumption task ic2 at event n is taken from storage at event n, eq 15 still enforces alignment with the production task at event n − 1, because it partially draws material stored at event n − 1, and it is not possible to distinguish fresh material here. Thus, there is still unconditional alignment allowed in eq 15 due to mixing of material that is allowed in storage. Equation 16 is a tightening constraint which limits the total time available in each unit by subtracting the minimum changeover time in each unit and maintenance time, if any, in the given scheduling horizon.

However, conditional sequencing allows a consumption task to align only with that production task from which material is actually consumed, without any alignment issues with other unrelated tasks, as shown in Figure 5b. Thus, only one event is necessary for conditional sequencing as shown in Figure 5b. Additionally, for the unlimited storage (UIS) case, only one event is sufficient even if the production and consumption tasks are of different durations as shown in Figure 5b. Equation 15 prevents real-time storage violation if the consumption task uses material from storage, because material balance constraints in eq 4 assume that the producing task, leading to storage at previous event n − 1, would have completed before starting the consumption task at event n. T s(i , n) ≥ T f (i′, n − 1) − H(1 − w(i′, n − 1)) ∀ s ∈ Sin , i′ ∈ Isp , i ∈ Isc , j , j′ ∈ J , j ≠ j′, i ∈ Ij , i′ ∈ Ij ′ , n ∈ N , n > 1

∑ ∑ (T f (i , n) − T s(i , n)) ≤ Hj − τjmin ∀ j ∈ J

(15)

n ∈ N i ∈ Ij

Consider a sample case as shown in Figure 6, in which 30 ton is to be consumed from storage, ST(s, n − 1), by the

(16)

2.1.4. Objective: Maximization of Profit. The objective function includes maximization of profit based on revenue generated from product sales. The binary variables are penalized to avoid inconsistencies so that they are not activated unnecessarily in eqs 5 and 13. If the left-hand term is zero in these equations then the binary variables can still take a value of 1, which is referred to as inconsistency here. The study on the effect of penalty factors (C1−C3) was already reported by Shaik and Floudas18 and Li et al.20

Figure 6. Real-time storage violation for UIS and FIS cases.

C1 ∑ p

∑ Ps ∑ ρsip b(i , n)

s∈S n∈N

consumption task at event n. However, in real time it seems that 30 ton of material is consumed from the current production task, as shown in Figure 6a. Therefore, constraint 15 is added to avoid the real-time storage violation, which leads to case b shown in Figure 6b, which states that if the production task is active at event n − 1 then the start time of the consumption task at event n should be greater than or equal to the finish time of the production task at event n − 1, and if the production task is not active, the above constraint will be relaxed. We still assume unconditional alignment in eq 15 to avoid further complications. Due to mixing of material taking place in storage, as shown in Figure 7, sometimes it is not possible to distinguish fresh material from stored material. In the example shown in Figure 7, an initial amount of 100 ton is available in storage from which 60 ton of material is consumed by task ic1 at event n − 1 and 40 ton of material remains in the storage until

− C2

i ∈ Isp

∑ ∑ ∑ ∑ z1(i′, i , s , n) i ′∈ Isp i ∈ Isc s ∈ Sin n ∈ N

− C3 ∑

∑ w(i , n) (17)

i∈I n∈N

We consider the same objective function of maximization of profit as reported in the literature so that we have the same basis for comparison and to demonstrate the proposed benefits. Other terms such as inventory costs and penalties for underproduction also can be included easily as reported in Shaik et al.19 2.2. Mathematical Model Based on Unit-Specific Timing Variables (Model M2). Now we consider the next model where the timing variables are modeled in unit-specific fashion. The following constraints are applicable in addition to constraints 1, 2, 3a, 3b, 4, 11a, 11b, 12a, 12b, 13, and 17.

Figure 7. Motivating example for mixing of material from storage tank at a previous event and production task at a current event. E


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Industrial & Engineering Chemistry Research Equation 18 describes the capacity constraint for tasks with variable processing rates and eq 19 for a fixed processing rate. Equation 20 enforces the amount processed by task i at event n to be zero if task i is not active at event n, and together with eqs 18 and 19 it enforces the duration to be zero if task i is not active at event n.

−ρsic b(i , n) ≥

j ∈ Ji

(18) s

b(i , n) = R i ∑ (T (j , n) − T (j , n))∀i ∈ I , n ∈ N j ∈ Ji

(19)

b(i , n) ≤ R imaxHw(i , n) ∀ i ∈ I , n ∈ N

T s(j , n) ≥ T s(j′, n) − H(1 − z1(j′, j , s , n)) ∀ s ∈ Sin , j , j

(20)

′ ∈ J , j ≠ j′, j ∈ Jsc , j′ ∈ Jsp , n ∈ N

Equation 21 states that processing unit j would start at the next event only after it finishes an earlier event. Equation 22 incorporates handling of sequence-independent changeovers, and eq 23 addresses sequence-dependent changeovers, which states that if task i′ is active at event n′ and another task i is active at a later event n, with no other task active in between, then a changeover time is added before starting next task. T s(j , n + 1) ≥ T f (j , n) ∀ j ∈ J , n ∈ N , n ≠ N

, j′ ∈ J , j ≠ j′, j ∈ Jsc , j′ ∈ Jsp , n ∈ N T s(j , n) ≥ T f (j′, n − 1) − H(1 −

(21)

∑ w(i′, n) ∀

(31)

Equation 32 describes the modified tightening constraint, and eq 33 defines the modified objective function.

(22)

∑ (T f (j , n) − T s(j , n)) ≤ Hj − τjmin ∀ j ∈ J

n) − H(1 − w(i′, n′)) w(i″ , n″)

n∈N

C1

∀ j ∈ J cl 2 , i , i′ ∈ Ij , i ≠ i′, n , n′ ∈ N , n > n′

Constraints 24−28 are used to take care of conditional sequencing where the four-index variables b1(j′,j,s,n) and z1(j′,j,s,n) are also modeled in unit-specific fashion. Binary variable z1(j′,j,s,n) is activated when the corresponding continuous variable is active as shown in eq 28.

∑ ∑ ∑ ∑ z1(j′, j , s , n) ∑ w(i , n) (33)

i∈I n∈N

2.3. Bounds on Variables. Equation 34a is used for model M1 and eq 34b for model M2. 0 ≤ T s(i , n) ≤ H , 0 ≤ T f (i , n) ≤ H ∀ i ∈ I , n ∈ N (34a)

(24)

0 ≤ T s(j , n) ≤ H , 0 ≤ T f (j , n) ≤ H ∀ j ∈ J , n ∈ N (34b)

− ∑ ρsic b(i , n) ≤ ST (s , n − 1)

2.3.1. Handling of Planned Maintenance for Model M1. Figure 8 depicts different cases considered for handling of planned maintenance. The time horizon is considered to be H, Tjms, and Tjmf represent the start and finish times of maintenance in the unit j, respectively. In case 1 we consider unit j1 that requires planned maintenance at the end of time mf horizon, with unit blocked for production between Tms j1 and Tj1 . For this case, the upper bound for start and finish times of all suitable tasks (i1) in eq 34a will correspond to Tms j1 as shown in eq 35.

i ∈ Isc

∑ ∑ b1(j′, j , s , n) ∀ s ∈ Sin , n ∈ N , n > 1 j ′∈ Jsp j ∈ Jsc

(25)

ρsip′b(i′, n) ≥

i ∈ Isp

j ′∈ Jsp j ∈ Jsc s ∈ Sin n ∈ N

− C3 ∑

∑ ∑ b1(j′, j , s , n) ∀

s∈S ,n=1

+

s∈S p n∈N

j ′∈ Jsp j ∈ Jsc

in

∑ b1(j′, j , s , n) ∀ s ∈ Sin , i′ ∈ Isp , j′ j ∈ Jsc

∈ Jsp , i′ ∈ Ij ′ , n ∈ N

(32)

∑ ∑ Ps ∑ ρsip b(i , n)

− C2

(23)

i ∈ Isc

∑ w(i′, n − 1)) ∀

s ∈ Sin , j , j′ ∈ J , j ≠ j′, j ∈ Jsc , j′ ∈ Jsp , n ∈ N , n > 1

i ″∈ Ij n ′< n ″< n

− ∑ ρsic b(i , n) ≤ ST0(s) +

(30)

i ′∈ Ij ′

ticl′ iw(i ,

T (j , n) ≥ T (j , n′) + −H∑ ∑

(29)

T f (j , n) ≥ T f (j′, n) − H(1 − z1(j′, j , s , n)) ∀ s ∈ Sin , j

i ′∈ Ij

j ∈ J cl1 , n ∈ N , n ≠ N

(28)

Equations 29 and 30 enforce conditional alignment that start and finish times of consumption unit j at event n should be later than or equal to start and finish times of production unit j′ at the same event n, only if the consumption unit uses the material produced from the current production unit. Constraint 31 is used for avoiding the real-time storage violation as discussed in model M1.

≤ R imax ∑ (T f (j , n) − T s(j , n)) ∀ i ∈ I , n ∈ N

f

(27)

∀ s ∈ Sin , j , j′ ∈ J , j ≠ j′, j ∈ Jsc , j′ ∈ Jsp , i′ ∈ Isp , i′ ∈ Ij ′ , n ∈ N

≤ b(i , n)

s

b1(j′, j , s , n) ∀ s ∈ Sin , i ∈ Isc , j ∈ Jsc , i

b1(j′, j , s , n) ≤ z1(j′, j , s , n)ρsip′R imax ′ H

j ∈ Ji

T s(j , n + 1) ≥ T f (j , n) + τj

j ′∈ Jsp

∈ Ij , n ∈ N

R imin ∑ (T f (j , n) − T s(j , n))

f

∑

(26) F


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0 ≤ T s(j3 , n) ≤ T jms , 0 ≤ T f (j3 , n) ≤ T jms ∀ n ∈ N1 3

3

(40a)

T jmf ≤ T s(j3 , n′) ≤ H , T jmf ≤ T f (j3 , n′) ≤ H ∀ n′ ∈ N2 3

3

(40b)

Although a single maintenance task is depicted in Figure 8 for illustration purposes, the proposed approach is applicable for multiple maintenance tasks per unit, where for each segment of time horizon (for instance in case 3 of Figure 8 there are two segments, before and after maintenance) we need to postulate the number of events required per segment in an iterative fashion by trial and error, as discussed above. 2.4. Important Enhancements of the Proposed Conditional Sequencing Model. The novelty and important contributions of the proposed models are in terms of modeling conditional sequencing of production and consumption tasks for continuous plants using four-index binary and continuous variables based on availability of initial inventory for intermediates and unavailability of unit/planned maintenance. This is the first time that a conditional sequencing concept has ever been identified and modeled for continuous or semicontinuous plants, which is an important milestone in itself in the scheduling literature of continuous processes. More importantly, we identify the limitation/drawback of earlier established models and present ways to overcome the same. Two different models with task-specific and unit-specific timings are presented, which is also novel in scheduling of continuous plants. The consumption task is conditionally aligned to a production task only if it receives material from that production task; otherwise, if material is received from storage then sequencing is relaxed. The constraint for real-time storage violation is included; sequencing constraints and material balances are reformulated to enforce conditional sequencing. The importance of the proposed development in the context of planned maintenance has been highlighted. We also presented how to extend the proposed scheduling models for handling of planned maintenance as explained in sections 2.3.1 and 2.3.2, which is also not addressed in the literature in the context of unit-specific event-based models. Three different scenarios are presented covering different aspects of planned maintenance, and appropriate lower and upper bounds are postulated for each case. The computational results presented in the next section further establish the advantages of the proposed approach both at the conceptual level as well as from the optimality viewpoint.

Figure 8. (a) Unit blocked at the end of time horizon [case 1], (b) unit blocked at the beginning of time horizon [case 2], and c) unit blocked in between the time horizon [case 3].

0 ≤ T s(i1 , n) ≤ T jms , 0 ≤ T f (i1 , n) ≤ T jms ∀ i1 ∈ j1 , n 1

1

(35)

∈N

In case 2 we consider unit j2 that requires maintenance at the beginning of scheduling horizon, with unit blocked for mf production between Tms j2 and Tj2 . For this case, the lower bound for start and finish times of all suitable tasks (i2) will correspond to Tmf j2 and the upper bound will be H as shown in eq 36. T jmf ≤ T s(i2 , n) ≤ H , T jmf ≤ T f (i2 , n) ≤ H ∀ i2 ∈ j2 , n 2

2

(36)

∈N

In case 3 we consider unit j3 that requires maintenance in between the scheduling horizon with a unit blocked for mf production between Tms j3 and Tj3 . For this case, two set of events are necessary: one before the blocked period (n ∈ N1) with eq 37a and other after the blocked period (n′ ∈ N2) with eq 37b. 0 ≤ T s(i3 , n) ≤ T jms , 0 ≤ T f (i3 , n) ≤ T jms ∀ i3 ∈ j3 , n 3

3

∈ N1

(37a)

T jmf ≤ T s(i3 , n′) ≤ H , T jmf ≤ T f (i3 , n′) ≤ H ∀ i3 ∈ j3 , n′ 3

3

∈ N2

(37b)

For a given total number of events (N), we consider an iterative procedure to determine the number of subevents N1 and N2 required by trial and error, with all possible combinations considered in which N1 and N2 events are used before and after the blocked time period, respectively. Equation 37a is valid for events n = 1 to N1, with N1 representing the total number of subevents before the blocked time period. Equation 37b is valid for events n′ = N1 + 1 to N which is denoted by N2. 2.3.2. Handling of Planned Maintenance for Model M2. The corresponding equations for all three cases using unitspecific timing variables are given in eqs 38−40b. 0 ≤ T s(j1 , n) ≤ T jms , 0 ≤ T f (j1 , n) ≤ T jms ∀ n ∈ N 1

1

3. COMPUTATIONAL RESULTS Three literature examples along with a motivating example are solved for a fast-moving consumer goods (FMCG) manufacturing plant which involves mixing and packing tasks. The computational performance of the proposed models is compared with the Shaik and Floudas18 model (SF model), which is considered as the base case. In section 3.1, a motivating example with four different case studies is solved under planned maintenance. In sections 3.2 and 3.3, literature examples are solved under planned maintenance for case 1 and combined cases 1 and 3 (of Figure 8), respectively. In section 3.4, the same examples are solved without planned maintenance. All of the examples are solved using GAMS 23.9/ CPLEX 12.2 on a 3.10 GHz Intel Xeon processor with 32 GB RAM running on a Linux operating system.

(38)

T jmf ≤ T s(j2 , n) ≤ H , T jmf ≤ T f (j2 , n) ≤ H ∀ n ∈ N 2

2

(39) G


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Industrial & Engineering Chemistry Research 3.1. Maximization of Profit for Motivating Example under Planned Maintenance Operation. 3.1.1. Motivating Example. The motivating example consists of one raw material (base A), one intermediate (int1), and a final product (S1). The processing task “m1” consumes the raw material state ́ “base A” and produces the intermediate state “int1” that is processed in the “Mixer A” unit. The intermediate state “int1” is consumed by processing task “p1” that is processed in the “Line1” unit, and it produces the final product “S1”. The production recipe in terms of a state-task network (STN) is shown in Figure 9, and Table S1 (in Supporting Information)

Figure 10. Gantt chart of case study 1 for the proposed models (M1 and M2).

Figure 9. State-task network (STN) representation for motivating example.

Figure 11. Gantt chart of case study 1 for the SF model.

provides data for four cases with different availability of initial inventory (ST0,s). Also, we consider different down times for these units due to scheduled maintenance, where conditional sequencing becomes more pertinent as demonstrated in the results shown in Table 1. For the case studies which are solved without the penalty terms in the objective function (eqs 17 and 33), profit and MILP values are same. The data in parentheses represents the objective function value with penalty. The price for any intermediate left in storage by the last event is considered to be one-half of the final product price. All cases are solved to zero integrality gap. Case Study 1. This case is solved considering a scheduling horizon of 5 h with the objective of maximization of profit. It is assumed that the consumption unit Line1 is down for 1 h at the end (case 1 of Figure 8) and hence is blocked for 4−5 h. There is 40 ton of initial inventory available for intermediate state int1, and the storage capacity is unlimited for that state. The Gantt charts for proposed conditional sequencing models and the SF model are presented in Figures 10 and 11. In the proposed models, the production task produces 25 ton of material within the scheduling horizon of 5 h. However,

in the SF model, the production task produces only 20 ton as the production unit also gets blocked for a 4−5 h period due to unconditional alignment of the finish time of production and consumption tasks, even though the unit is idle. In Table 1 the objective value for both conditional sequencing models (M1 and M2) is 523.0, which correspond to a profit of $52.5, which is better as compared to the SF model, which could not find the optimal solution at higher events as well due to unconditional alignment. The proposed model found the converged optimal solution using one event. Case Study 2. In this case the scheduling horizon is the same (5 h) as case study 1 but with different initial inventory available for the intermediate, which is assumed to be 50 ton. It is assumed that the production unit Mixer A is down for 1 h at the beginning (case 2 of Figure 8) and hence is blocked for 0−1 h and no down time for the consumption unit. The Gantt charts for the proposed conditional sequencing models and the SF model are presented in Figures 12 and 13. The results are given in Table 1, where it can be observed that although there is enough initial inventory (50 ton) available to process the material without depending on the production task, due to the

Table 1. Results for Motivating Example with Availability of Initial Inventory and Unavailability of Unitsa model

binary variables

continuous variables

SF M1 M2

1 1 1

2 3 3

9 10 10

SF

1 2 1 1

2 4 3 3

9 16 10 10

M1 M2

1 2 1 1

2 4 3 3

9 16 10 10

SF M1 M2

2 2 2

4 6 6

16 18 18

M1 M2 SF

a

events

constraints case study 13 19 19 case study 13 24 19 19 case study 13 24 19 19 case study 24 35 35

RMIP

1 (H = 5 h) 51.3 52.5 52.5 2 (H = 5 h) 60.0 60.0 60.0 60.0 3 (H = 5 h) 50.0 50.0 50.0 50.0 4 (H = 5 h) 50.0 50.0 50.0

MILP

profit ($)

CPU time (s)

(510.8) (523.2) (523.2)

50.0 (498.0) 52.5 (523.0) 52.5 (523.0)

50.0 52.5 52.5

0.010 (0.010) 0.007 (0.007) 0.007 (0.006)

(598.2) (598.2) (598.2) (598.2)

55.0 60.0 60.0 60.0

(548.0) (598.0) (598.0) (598.0)

55.0 60.0 60.0 60.0

0.009 0.015 0.010 0.008

(0.011) (0.015) (0.008) (0.008)

(498.4) (498.4) (498.4) (498.4)

42.5 47.5 50.0 50.0

(423.0) (473.0) (498.0) (498.0)

42.5 47.5 50.0 50.0

0.011 0.030 0.009 0.008

(0.015) (0.015) (0.008) (0.006)

(498.4) (498.4) (498.4)

47.5 (472.0) 50.0 (497.0) 50.0 (497.0)

47.5 50.0 50.0

0.011 (0.016) 0.010 (0.013) 0.018 (0.009)

SF: Shaik and Floudas.18 H


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shown in Figure 16, whereas the conditional sequencing model gives no idle time except for the blocked times as shown in Figure 14.


Figure 16. Gantt chart of case study 3 for the SF model using two events.

Case Study 4. This case is solved considering the same scheduling horizon of 5 h and an initial stock level of 40 ton for the intermediate. Here, it is assumed that production unit Mixer A is down for 1 h in between the scheduling horizon (case 3 of Figure 8) and is blocked for a period of 2−3 h, whereas consumption unit Line 1 is down for 1 h at the end (case 1 of Figure 8) and is blocked for a period of 4−5 h. Therefore, it is a combination of cases 1 and 3 of Figure 8. The Gantt charts for the proposed models and the SF model are presented in Figures 17 and 18. In the proposed models, the


unconditional alignment of the start time of production and consumption tasks in the SF model the consumption task is not able to start immediately using one event and gets blocked for a 0−1 h period, but it overcomes this difficulty using two events for the SF model. The proposed models found the same optimal solution ($60) using one event itself and hence are computationally superior. Case Study 3. This case is solved considering the same scheduling horizon of 5 h and an initial stock level of 40 ton for the intermediate (similar to case study 1), but here we assume a down time of 1 h on both production and consumption units (combination of cases 1 and 2 in Figure 8). The production unit Mixer A is down for 1 h and is blocked for 0−1 h, whereas the consumption unit Line 1 is down for 1 h and is blocked for 4−5 h. Here also the SF model is unable to find the optimal solution even at higher events due to the same reason (unconditional alignment), whereas the proposed conditional sequencing models are able to find the converged optimal solution using one event itself. The Gantt charts for the proposed conditional sequencing models and the SF model are shown in Figures 14



production task produces 10 ton of material for 0−2 and 3−5 h period. However, in the SF model, the production task produces 10 ton for a 0−2 h period and only 5 ton for a 3− 4 h period as the production unit also gets blocked for 4−5 h due to unconditional alignment. Thus, the SF model gives an idle time of 1 h, whereas in the proposed models there is no idle time except for the blocked times. 3.2. Maximization of Profit for Examples 1−3 for Case 1 under Planned Maintenance. 3.2.1. Example 1. This is a simple example adopted from Li et al. (2010). The problem consists of one raw material (base A), two intermediates (int1, int2), and five final products (S1−S5). The sequential tasks (m1 and m2) consume the raw material state and produce intermediates. Both tasks are suitable in the mixer A unit in which the first processing task m1 produces int1 that is to be consumed by processing tasks p1 and p2 in Line1 and Line2 units, respectively. Similarly, processing task m2 produces int2 that is to be consumed by tasks p3, p4, and p5 in Line1, Line2, and Line3 units, respectively. The relevant data is given in Table S2 (in Supporting Information) for all examples, which is modified from the


and 15 using one event. From Figure 15 it can be observed that using one event, although there is an idle time of 1 h each on the production and consumption units, it cannot use the same due to unconditional alignment in the SF model. For two events also the SF model gives an idle time of 1 h on Mixer A as

Figure 15. Gantt chart of case study 3 for the SF model using one event. I


Article

Industrial & Engineering Chemistry Research Table 2. Results for Examples 1−3 for Case 1 with Planned Maintenancea model

binary variables


SF M1 M2

1 1 1

7 12 12

25 30 24

SF M1 M2

1 2 1 1

13 26 21 21

49 93 57 45

SF M1 M2

2 2 2

54 104 100

180 230 150

SF M1

2 2 3 2 3

14 24 36 24 36

48 58 86 46 68

SF M1 M2

3 3 3

39 63 63

137 161 125

SF M1 M2

4 4 4

108 208 200

356 456 296

M2

a

events

constraints

RMIP

(i) without demand constraints example 1 (H = 30 h) 36 3246.2 57 3245.7 57 3345.4 example 2 (H = 12 h) 66 9503.4 149 9653.3 105 9652.5 105 9832.2 example 3 (H = 120 h) 353 45327.1 510 45327.1 433 87863.7 (ii) with demand constraints example 1a (H = 35 h) 84 3721.1 124 3721.1 189 3721.1 118 6767.4 177 6768.1 example 2a (H = 12 h) 258 9088.3 361 9088.3 341 13 588.4 example 3a (H = 120 h) 1040 45 180.0 1213 45 180.0 964 128 893.1

MILP

profit ($)

CPU time (s)

2996.0 3244.0 3244.0

300.0 325.0 325.0

0.014 0.009 0.018

8693.0 8723.7 9651.0 9651.0

870.0 873.3 966.0 966.0

0.011 0.214 0.017 0.012

44475.0 45322.0 45320.0

4448.5 4533.5 4533.5

0.481 0.359 0.657

3418.0 3414.0 3665.0 3414.0 3665.0

342.5 342.5 367.5 342.5 367.5

0.029 0.028 0.398 0.024 0.410

7561.0 8457.0 8457.0

757.5 847.5 847.5

0.393 0.400 0.550

44 182.3 45 027.3 45 027.3

4420.4 4505.4 4505.4

356.30 145.10 7311.76

SF: Shaik and Floudas.18

final packed products (S1−S5) using five intermediates (int1− int5) stored in five different storage tanks. Example 2 is solved for a scheduling horizon of 12 h with the objective of maximization of profit without any demand constraints. It is assumed that consumption unit Line1 is down for 3 h at the end of scheduling horizon for maintenance, and hence is blocked during the period of 9−12 h. There is 90 ton of initial inventory available for intermediate state int2 with an unlimited storage capacity. In Table 2 model M2 requires less continuous variables as compared to models M1 and SF. The proposed models (M1and M2) found an optimal solution with the objective function of 9651 that corresponds to a profit of $966 using one event, and the SF model requires two events for finding the converged solution with the objective function of 8723.7 that corresponds to a profit of $873.3. Thus, the proposed models give an optimal solution with a better objective function value and less events as compared to the SF model. Example 2a is solved considering the same scheduling horizon of 12 h and with minimum and maximum demand constraints. The proposed models (M1 and M2) found an optimal solution using three events with objective function value of 8457.0 that corresponds to a profit of $847.5. However, SF model could not find an optimal solution using same event points. Model M2 requires less number of continuous variables and constraints as compared to model M1. However, model M2 has higher RMIP value and it takes more CPU time.

original data to take into account initial inventory. This example is solved for a scheduling horizon of 30 h with the objective of maximization of profit without any demand constraints. It is assumed that consumption unit Line1 is down for 5 h at the end of the scheduling horizon for maintenance and hence is blocked for 25−30 h. There is 100 ton of initial inventory available for intermediate state int1 with unlimited storage capacity. The results are given in Table 2, where model M2 requires less continuous variables as compared to model M1 and the SF model. The proposed models (M1 and M2) found an optimal solution using one event with an objective function value of 3244 that corresponds to a profit of $325. However, the SF model could not find the optimal solution at higher events as well due to unconditional alignment. If we use a minimum demand constraint, then example 1 is infeasible for a scheduling horizon of 30 h; hence, we consider a scheduling horizon of 35 h (example 1a). Here, the consumption unit Line1 is blocked for the period of 30−35 h. The proposed models (M1and M2) found the optimal solution using 3 events with an objective function value of 3665 that corresponds to a profit of $367.5. However, the SF model gives suboptimal solution using two (or higher) events. Model M2 requires less continuous variables and constraints as compared to model M1. 3.2.2. Example 2. This example is from Li et al. (2010), which involves five mixing tasks in four mixers (Mixers A−D) and eight packing tasks in three packing lines (Lines 1−3). Four raw materials (Bases A−D) are required to produce five J


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Figure 19. Gantt chart of example 3a (case 1) for the proposed model (M1).

Figure 20. Gantt chart of example 3 (combined cases 1 and 3) for the proposed model (M1).

3.2.3. Example 3. This example is adopted from Shaik and Floudas,18 and it is a relatively large problem as compared to previous examples. The problem consists of 3 raw materials (Base A, B, and C), 7 intermediates (int1−int7), and 15 final products (S1−S15). There are 7 mixing tasks (m1−m7) operating in 3 parallel mixer units (Mixer A, B, and C) and 15 packing tasks (p1−p15) processed in 5 packing lines (Line1− Line5). This problem is solved considering a scheduling horizon of 120 h for the objective of maximization of profit without any demand constraints. Here, we assumed that consumption unit Line1 is down for 10 h at the end of the scheduling horizon for maintenance and is blocked during the period of 110−120 h. There is 650 ton of initial inventory available for intermediate state int2 with unlimited storage capacity. In Table 2 model M2 requires less binary variables, continuous variables, and constraints as compared to model M1. However, model M2 reported more CPU time due to a higher RMIP value. The proposed models (M1 and M2) require two events for finding the optimal solution with an objective function of 45 322 that corresponds to a profit of $4533.5, whereas the SF model could not find the optimal solution using two events (also at higher events). Example 3a is solved considering the same scheduling horizon of 120 h but with minimum demand constraint. The Gantt chart for the proposed model M1 is shown in Figure 19.

Here, the proposed models (M1 and M2) found an optimal solution using four event points with an objective function value of 45 027.3 that corresponds to a profit of $4505.4. However, the SF model could not find the optimal solution using the same event points. Model M1 requires a higher number of binary variables, continuous variables, and constraints as compared to the SF model, but less CPU time is reported. However, model M2 reported less binary variables, continuous variables, and constraints as compared to model M1, but again it takes more CPU time due to a higher RMIP value. Thus, model M1 is computationally better as compared to model M2. 3.3. Maximization of Profit for Examples 1−3 for Combined Cases 1 and 3 under Planned Maintenance. All of the above examples are solved for combined cases 1 and 3 (of Figure 8) under the planned maintenance operation. Example 1 is solved considering a scheduling horizon of 35 h with a down time of 5 h on production unit Mixer A and consumption unit Line 1; hence, production unit Mixer A is blocked for 15−20 h, and consumption unit Line 1 is blocked for 30−35 h. The proposed models (M1 and M2) give the optimal solution with a profit of $344.5 using four events (with two events used before and after the blocked period). The SF model gives a suboptimal solution ($319.5) due to unconditional alignment. For four events, other combinations of subevents are also tested such as one event before the blocked K


Article

Industrial & Engineering Chemistry Research Table 3. Results for Examples 1−3 for Combined Cases 1 and 3 with Demand Constraintsa model

total events

events before (N1)

events after (N2)

SF M1 M2

4 4 4

2 2 2

2 2 2

SF M1 M2

3 3 3

1 1 1

2 2 2

SF M1 M2

4 4 4

2 2 2

2 2 2

a

binary variables


constraints

example 1 (H = 35 h) 94 114 90 example 2 (H = 12 h) 39 137 63 161 63 125 example 3 (H = 120 h) 108 356 208 456 200 296 28 48 40

RMIP

MILP

profit ($)

CPU time (s)

188 257 239

3471.3 3471.3 6268.5

3184.0 3430.0 3430.0

319.5 344.5 344.5

0.352 0.279 0.251

258 361 341

8488.5 8488.5 11 790.1

6960.0 7856.0 7856.0

697.5 787.5 787.5

0.446 0.362 0.283

1040 1213 964

44 330.1 44 330.1 127 193.3

43 331.3 44 176.3 44 176.3

4335.4 4420.4 4420.4

282.03 448.20 6061.95


Table 4. Results for Motivating Example and Examples 1−3 without Planned Maintenancea model

a

events

binary variables


SF M1 M2

1 1 1

2 3 3

9 10 10

SF M1 M2

2 2 2

14 24 24

48 58 46

SF M1 M2

2 2 2

26 42 42

93 109 85

SF M1 M2

4 4 4

108 208 200

356 456 296

constraints

motivating example (H = 5 13 19 19 example 1 (H = 30 h) 84 124 118 example 2 (H = 12 h) 154 227 217 example 3 (H = 120 h) 1040 1213 964

RMIP

MILP

profit ($)

CPU time (s)

248.5 247.5 247.5

248.0 247.0 247.0

25.0 25.0 25.0

0.016 0.009 0.013

2696.10 2696.02 5091.75

2493.0 2488.0 2488.0

250.0 250.0 250.0

0.017 0.029 0.013

4985.83 4985.83 8930.24

3979.0 3975.0 3975.0

399.0 399.0 399.0

0.020 0.231 0.020

26 931.18 26 925.18 26 925.18

2695.32 2695.32 2695.32

3.845 23.720 5.161

h)

26 946.72 26 946.72 71 084.34


period and three events after the blocked period, and the best case is reported here. Example 2 is solved considering a scheduling horizon of 12 h with a down time of 3 h on production unit Mixer A and consumption unit Line 1; hence, production unit Mixer A is blocked for 3−6 h, and consumption unit Line 1 is blocked for 9−12 h. Here, the proposed models (M1 and M2) give the optimal solution ($787.5) using three events, which remains the same for other combinations of subevents as well. Example 3 is solved considering a scheduling horizon of 120 h with a down time of 10 h on production unit Mixer A and consumption unit Line 1; hence, production unit Mixer A is blocked for 60−70 h, and consumption unit Line 1 is blocked for 110−120 h. The proposed models found the converged optimal solution ($4420.4) using four events, whereas the SF model gives suboptimal solution at same events. The Gantt chart for the proposed model (M1) for example 3 is shown in Figure 20. The results are given in Table 3, where model M2 shows less binary variables, continuous variables, and constraints as compared to model M1, but it requires more CPU time due to a higher RMIP for large examples. Therefore, model M1 is computationally better as compared to model M2 for example 3. However, CPU time is less for model M2 in other examples (1 and 2), even though the RMIP value is large.

In the case studies considered although model M2 requires less binary variables, continuous variables, and constraints compared to model M1, model M2 reported more CPU time due to a higher RMIP value. Thus, model M1 is computationally better as compared to model M2 for medium- to large-scale problems, whereas model M2 is better for small-scale problems. The reason for this behavior could be that unit-specific timing variables provide aggregation (leading to less problem size), so it takes more time to confirm optimality due to loss of accuracy in aggregation. Moreover, The issue is not just about the magnitude of numerical improvements shown in the results (which can be tweaked by assigning higher penalty for machine idle time), but conceptually unconditional sequencing gives suboptimal solutions (as shown in Figures 1, 2, 5, 6, and 8). Moreover it also depends on the cost assigned to the nonproductive time. If there is a significant cost assigned to every hour of nonproductive time, the benefits would still be huge even for small-scale problems. 3.4. Maximization of Profit for Motivating Example and Examples 1−3 without Planned Maintenance. Now we consider the normal case without any down time or requirement of maintenance for comparison. The motivating example and all of the above three examples are solved for the L


Article

Industrial & Engineering Chemistry Research n = events s = states

objective of maximization of profit. The motivating example is solved for scheduling horizon of 5 h, example 1 for horizon of 30 h, example 2 for horizon of 12 h, and example 3 for horizon of 120 h. The model statistics and computational results are given in Table 4, where model M2 requires less continuous variables and constraints but a higher RMIP value compared to model M1 in all of the examples. In spite of higher RMIP values, model M2 still gives better computational performance compared to model M1. However, both models M1 and M2 have higher model statistics due to usage of four-index variables compared to the SF model, which performs better for large examples. Therefore, when machine downtime/maintenance is not considered the SF model performs better, but when planned maintenance is considered it gives suboptimal solutions due to unconditional alignment; thus, it is absolutely necessary to consider conditional sequencing.

Sets

I Ips Ics J Jcl1

tasks tasks which produce state s tasks which consume state s units sequence-independent changeover between consecutive tasks in unit Jcl2 sequence-dependent changeover between consecutive tasks in unit N total number of events N1 total number of subevents before the blocked period N2 total number of subevents after the blocked period S states SR raw material states Sin intermediate states SP final product states

4. CONCLUSION In this work, we proposed an improved unit-specific eventbased model for short-term scheduling of continuous/semicontinuous plants based on conditional sequencing, an issue which has been recently addressed in batch plants. The material balances and sequencing constraints are modified using fourindex binary and continuous variables based on whether a consumption task receives material from a production task or not to overcome the limitations of unconditional sequencing assumed in earlier models. Two models with different timing variables are presented, and appropriate lower and upper bounds are postulated for different scenarios involving planned maintenance. Different cases of a motivating example and some benchmark examples are solved to demonstrate the concept using different availability of initial inventory for intermediates and unavailability of processing units due to downtime/ maintenance. It was observed that the proposed models give better objective values and converged optimal solutions using lesser events, which is not possible for unconditional sequencing-based models. We presented the models for the case of unlimited intermediate storage (UIS) policy, and the future work involves extending the proposed models to handle different storage policies including finite storage.

■

Parameters

Rmin i Rmax i Ri ST0,s H Tms j Tmf j Ps Dmin s Dmax s τj tclii′ ρsi

Binary Variables

w(i,n)

binary variable which takes a value of 1 if task i starts at event n z1(i′,i,s,n) binary variable which takes a value of 1 if consumption task i at event n uses the amount of state s produced by production task i′ at event n (model M1) z1(j′,j,s,n) binary variable which takes a value of 1 if consumption unit j at event n uses the amount of state s produced by production unit j′ at event n (model M2)

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.7b04294. Data of processing rates and state specific data for motivating example and data of processing rates and state specific data for examples 1−3 (PDF)

■

Non-negative Variables

b(i,n)

amount of material processed by task i at event n, ton ST0 (s,n) amount of raw material state s ∈ SR required at event n, ton ST (s,n) excess amount of state s stored in inventory tank at event n, ton 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 (model M1) b1(j′,j,s,n) amount of state s consumed by unit j at event n from the total amount produced by unit j′ at event n (model M2) Ts (i,n) time at which task i starts at event n, h (model M1) Tf (i,n) time at which task i finishes at event n, h (model M1) Ts (j,n) time at which unit j starts at event n, h (model M2)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Munawar A. Shaik: 0000-0002-4364-483X Notes

The authors declare no competing financial interest.

■

minimum processing rate of task i, ton/h maximum processing rate of task i, ton/h constant processing rate of task i, ton/h initial inventory available for state s, ton time horizon, h start time of maintenance in unit j, h finish time of maintenance in unit j, h price of state s, $ minimum demand for state s, ton maximum demand for state s, ton sequence-independent clean up time required in unit j, h sequence-dependent clean up time required between tasks i and i′, h fraction of state s produced (ρpsi ≥ 0) or consumed (ρcsi ≤ 0) by task i

NOMENCLATURE

Indices

i = tasks j = units M


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Industrial & Engineering Chemistry Research Tf (j,n)

■

time at which unit j finishes at event n, h (model M2)

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