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Online Scheduling: Understanding the Impact of Uncertainty ⁎
12th IFAC Symposium on Dynamics and Control of 12th IFAC Symposium on and 12th IFACSystems, Symposium on Dynamics Dynamics and Control Control of of Process including Biosystems 12th IFAC Symposium on Dynamics and Control of Process including Biosystems Process Systems, Systems, including Biosystems Available online at www.sciencedirect.com Florianópolis SC, Brazil, April 23-26, 2019 Process including Biosystems 12th IFACSystems, Symposium on Dynamics Control of Florianópolis -- SC, April 2019 Florianópolis SC, Brazil, Brazil, April 23-26, 23-26,and 2019 Florianópolis SC, Brazil, April 23-26, 2019 Process Systems, including Biosystems Florianópolis - SC, Brazil, April 23-26, 2019
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IFAC PapersOnLine 52-1 (2019) 727–732
Online Scheduling: Understanding Online Online Scheduling: Scheduling: Understanding Understanding Impact of Uncertainty of Uncertainty Online Impact Scheduling: Understanding Impact of Uncertainty Impact ofChristos Uncertainty Dhruv Gupta, T. Maravelias
the the the the
Dhruv Gupta, Christos T. Maravelias Dhruv Dhruv Gupta, Gupta, Christos Christos T. T. Maravelias Maravelias Dhruv Gupta, ChristosMadison, T. Maravelias University of Wisconsin-Madison, University of of Wisconsin-Madison, Wisconsin-Madison, Madison, Madison, WI WI 53706 53706 USA USA University WI 53706 USA [email protected]). University(e-mail: of Wisconsin-Madison, Madison, WI 53706 USA (e-mail: [email protected]). (e-mail: [email protected]). [email protected]). University(e-mail: of Wisconsin-Madison, Madison, WI 53706 USA (e-mail: [email protected]). Abstract: We to quality schedules obtained and Abstract: We We present present aa a framework framework to to study study quality quality of of schedules schedules obtained obtained iteratively iteratively and and Abstract: Abstract: We present present a framework framework to study study quality of ofUsing schedules obtained iteratively iteratively anda online (real-time) in the presence of demand uncertainty. this framework, we carry out online (real-time) (real-time) in in the the presence presence of of demand demand uncertainty. uncertainty. Using Using this this framework, framework, we we carry carry out out aa online computational and interesting observations. First, we find that uncertainty Abstract: Westudy, present a make framework to study quality of schedules obtained iteratively andaa online (real-time) in the of demand uncertainty. Using this framework, we carryplays out computational study, andpresence make interesting observations. First, we find that uncertainty uncertainty plays computational study, and make interesting observations. First, we find that plays a less important role as a manufacturing facility is operated close to capacity. Second, the choice online (real-time) in the presence of demand uncertainty. Using this framework, we carry out computational study, and make interesting observations. First, we find that uncertainty plays less important important role role as as aa manufacturing manufacturing facility facility is is operated operated close close to to capacity. capacity. Second, Second, the the choice choicea less of the horizon for the is dependent on the mean load, independent of computational study, makeiterations, interesting we that but uncertainty a less important as and aonline manufacturing facility is operated close to find capacity. Second, theplays choice of the the horizon role for the the online iterations, is observations. dependent onFirst, the mean mean load, but independent of of horizon for online iterations, is on the load, independent of of the horizonofrole for the online iterations, is dependent dependent onform the of mean load, but but independent of the accuracy the forecasts. Finally, feedback, in the re-optimization, plays a very less important as a manufacturing facility is operated close to capacity. Second, the choice the accuracy accuracy of of the the forecasts. forecasts. Finally, Finally, feedback, feedback, in in the form form of re-optimization, plays aa very the of re-optimization, plays important role in mitigating impact of uncertainty. the analysis presented in of the horizon for the onlinethe iterations, dependent onform thethrough mean load, but independent of the accuracy the forecasts. Finally, in the theThus, of re-optimization, plays a very very important roleofin in mitigating the impact feedback, ofisuncertainty. uncertainty. Thus, through the analysis analysis presented in important role mitigating the impact of Thus, through the presented in this work, we gain insights that are applicable to all general rescheduling approaches. the accuracy of the forecasts. Finally, feedback, in the form of re-optimization, plays a very important role in mitigating the impact of uncertainty. Thus, through the analysis presented in this work, we we gain gain insights insights that are are applicable applicable to to all general general rescheduling rescheduling approaches. this important rolegain in mitigating the are impact of uncertainty. Thus,rescheduling through the approaches. analysis presented in this work, work, we insights that that applicable to all all general approaches. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. this work, we gain insights that are applicable to all general rescheduling approaches. Keywords: Uncertainty, Rescheduling, Operations, MPC, Mixed-integer programming Keywords: Uncertainty, Uncertainty, Rescheduling, Rescheduling, Operations, MPC, Mixed-integer programming Keywords: Operations, MPC, Mixed-integer programming Keywords: Uncertainty, Rescheduling, Operations, MPC, Mixed-integer programming Keywords: Uncertainty, Rescheduling, Operations, MPC, Mixed-integer programming 1. INTRODUCTION In this work, we outline aa framework for studying qual1. INTRODUCTION INTRODUCTION In this work, we outline for studying the the qual1. In this work, we outline aa framework framework for the quality of closed-loop schedules, when uncertainty present. 1. INTRODUCTION In this work, we outline framework for studying studyingis the quality of closed-loop schedules, when uncertainty is present. ity of closed-loop schedules, when uncertainty is present. 1. INTRODUCTION Although, this framework can be used to study several In this work, we outline a framework for studying the quality of closed-loop schedules, when uncertainty is present. Scheduling of operations is one of the many planning this framework can be used to study several Scheduling of of operations operations is is one one of of the the many many planning planning Although, Although, this framework can be used to study several Scheduling types of uncertainties, such as, demand, yield, processing ity of closed-loop schedules, when uncertainty is present. Although, this framework can be used to study several functions in the manufacturing sector (Pinedo and Chao, Scheduling of operations is one of the many planning types of of uncertainties, uncertainties, such such as, as, demand, demand, yield, yield, processing processing functions in in the the manufacturing manufacturing sector sector (Pinedo (Pinedo and and Chao, Chao, types functions types of uncertainties, such as, demand, yield, processing time, unit breakdowns, etc., here to limited space, we Although, this framework can bedue used to study several and Chao, 1999). Finding a schedule once, however, is only part of Scheduling is one of the many planning functions in of theoperations manufacturing sector (Pinedo time, unit breakdowns, etc., here due to limited space, we 1999). Finding a schedule once, however, is only part of time, unit breakdowns, etc., here due to limited space, we 1999). Finding a schedule once, however, is only part of specifically explore the role demand uncertainty plays in types of uncertainties, such as, demand, yield, processing time, unit breakdowns, etc., here due to limited space, we 1999). Finding a schedule once, however, is only part of the whole scheduling process (Harjunkoski et al., 2014). functions in the manufacturing sector (Pinedo and Chao, specifically explore the role demand uncertainty plays in the whole whole scheduling scheduling process process (Harjunkoski (Harjunkoski et et al., al., 2014). 2014). specifically explore the role demand uncertainty plays in the the context of online scheduling. We attempt to answer time, unit breakdowns, etc., here due to limited space, we specifically explore the role demand uncertainty plays in the whole scheduling process (Harjunkoski etonly al., part 2014). Due to disruptions or arrival of new information, the 1999). Finding a schedule once, however, is of the context of online scheduling. We attempt to answer Due to disruptions or arrival of new information, the the context of online scheduling. We attempt to answer Due to disruptions or arrival of new information, the the context of online scheduling. We attempt to answer several questions: (i) how demand uncertainty affects the specifically explore the role demand uncertainty plays in orprocess arrival(Harjunkoski of new information, incumbent schedule can become suboptimal or even infeathe whole scheduling et al., 2014). Due to disruptions the several questions: questions: (i) (i) how how demand demand uncertainty uncertainty affects affects the the incumbent schedule schedule can can become become suboptimal suboptimal or or even even infeainfea- several incumbent several questions: (i) how demand uncertainty affects the quality of the implemented schedules, (ii) how important the context of online scheduling. We attempt to answer sible, thus motivating the need for online (re)scheduling Due to disruptions or arrival of new information, the incumbent schedule can become suboptimal or even infeaquality of of the the implemented implemented schedules, schedules, (ii) (ii) how how important important sible, thus thus motivating motivating the the need need for for online online (re)scheduling (re)scheduling quality sible, is horizon length to this uncertainty, (iii) to several questions: (i) address how demand uncertainty affects the quality of the implemented schedules, (ii) howand important sible, thus schedule motivating the need for important online (re)scheduling (Subramanian et al., 2012). Several have incumbent become suboptimal or works even infeahorizon length to address this uncertainty, and (iii) to (Subramanian et al., al.,can 2012). Several important works have is is horizon length to address this uncertainty, and (iii) to (Subramanian et 2012). Several important works have what extent, feedback, as a function of the frequency of quality of the implemented schedules, (ii) how important is horizon length to address this uncertainty, and (iii) to suggested rescheduling strategies in different contexts (Li sible, thus motivating the need for online (re)scheduling (Subramanian et al., 2012). Several important works have what extent, feedback, as a function of the frequency of suggested rescheduling strategies in different contexts (Li what extent, feedback, as a function of the frequency of suggested rescheduling strategies in different contexts (Li what extent, feedback, as a function of the frequency of re-optimization, can mitigate the effect of this uncertainty horizon length to address this uncertainty, and (iii) to contexts (Li is and Ierapetritou, 2008). For example: Janak et al. (2006) (Subramanian et al., 2012). Several important works have suggested rescheduling strategies in different re-optimization, can mitigate the effect of this uncertainty and Ierapetritou, 2008). For example: Janak et al. (2006) re-optimization, can mitigate the effect of this uncertainty and Ierapetritou, 2008). For example: Janak et al. (2006) re-optimization, can mitigate the effect of this uncertainty which is treated as a disturbance in the nominal model what extent, feedback, as a function of the frequency of proposed partial rescheduling by identifying and fixing suggested rescheduling strategies different contexts (Li which and Ierapetritou, 2008). For example: Janak et al. (2006) is treated as aa disturbance in the nominal model proposed partial rescheduling byinidentifying identifying and fixing which is treated as disturbance in the nominal model proposed partial rescheduling by and fixing that we employ. Rather than proposing a new online re-optimization, can mitigate the effect of this uncertainty which is treated as a disturbance in the nominal model proposed partial rescheduling by identifying and fixing tasks not directly affected an observed disturbance; and For by example: Janak et al. (2006) that we employ. Rather than proposing aa new online tasksIerapetritou, not directly directly2008). affected by an observed observed disturbance; that we employ. than proposing online tasks not affected by an disturbance; scheduling strategy, we focus analysis – to gain which is treated asRather ahere, disturbance inon the nominal model that we employ. Rather than proposing a new new online Pattison et al. (2016) updated schedules respond to proposed partial rescheduling by identifying and fixing tasks not directly affected by an observedto disturbance; scheduling strategy, here, we focus on analysis – to gain Pattison et al. (2016) updated schedules to respond to scheduling strategy, here, we focus on analysis – to gain Pattison et al. (2016) updated schedules to respond to scheduling strategy, here, we focus on analysis – to gain insights that are broadly applicable to general rescheduling that we employ. Rather than proposing a new online process and market disturbances; Chu and You (2014) tasks not directly affected by an observed disturbance; Pattison et al. (2016) updated schedules to respond to insights that are broadly applicable to general rescheduling process and market disturbances; Chu and You (2014) insights that are broadly applicable to general rescheduling process and market disturbances; Chu and You (2014) approaches. scheduling strategy, here, we focus on analysis – to gain insights that are broadly applicable to general rescheduling solved reduced problem online, achieving computational Pattison et al. (2016) updated to You respond to approaches. process market disturbances; Chu and (2014) solved aa a and reduced problem online, schedules achieving computational approaches. solved reduced problem online, achieving computational insights that are broadly applicable to general rescheduling approaches. solved a reduced problem online, achieving computational tractability schedule stability; Novas process and and market disturbances; andand YouHenning (2014) tractability and schedule stability;Chu Novas and Henning tractability and schedule stability; Novas and Henning approaches. 2. (2010) constraint programming for rescheduling; Lapsolved aused reduced problem online, achieving computational tractability and schedule stability; Novas and Henning 2. ONLINE ONLINE FRAMEWORK FRAMEWORK 2. (2010) used constraint programming for rescheduling; Lap(2010) used constraint programming for rescheduling; Lap2. ONLINE ONLINE FRAMEWORK FRAMEWORK pas and Gounaris (2016) used adjustable robust optimizatractability and schedule stability; Novas and Henning (2010) used constraint programming for rescheduling; Lappas and Gounaris (2016) used adjustable robust optimizapas and Gounaris (2016) used adjustable robust optimizaGiven: (i) production facility data, (ii) production recipe, 2. ONLINE FRAMEWORK optimization to generate schedules with a recourse strategy; Cui (2010) constraint programming for rescheduling; pas Gounaris (2016) used adjustable robust (i) production facility data, (ii) production recipe, Given: (i) production facility data, (ii) production recipe, tionand toused generate schedules with a recourse recourse strategy;LapCui Given: tion to generate schedules with a strategy; Cui (iii) resource availability, (iv) feed availability, and (v) Given: (i) production facility data, (ii) production recipe, tion to generate schedules with a recourse strategy; Cui and Engell (2010) applied a two-stage stochastic model in a pas and Gounaris (2016) used adjustable robust optimiza(iii) resource availability, (iv) feed availability, and (v) (iii) resource availability, (iv) feed availability, and (v) and Engell Engell (2010) (2010) applied applied aa two-stage two-stage stochastic stochastic model model in in aa (iii) and resource availability, (iv) feed availability, and (v) production targets; scheduling seeks to find (i) selection Given: (i) production facility data, (ii) production recipe, moving horizon formulation to an expandable polystyrene tion to generate schedules with a recourse strategy; Cui and Engell (2010) applied a two-stage stochastic model in a production targets; scheduling seeks to find (i) selection production targets; scheduling seeks to find (i) selection moving horizon formulation to an expandable polystyrene moving horizon formulation to an expandable polystyrene and sizes of batches (batching), (ii) batch-unit assignment, (iii) resource availability, (iv) feed availability, and (v) production targets; scheduling seeks to find (i) selection manufacturing plant. and Engell (2010) applied a two-stage stochasticpolystyrene model in a and moving horizon formulation to an expandable sizes of batches (batching), (ii) batch-unit assignment, and sizes of batches (batching), (ii) batch-unit assignment, manufacturing plant. manufacturing plant. (iii) timing of these batches. When these decisions are production targets; scheduling seeks to find (i) selection and sizes of batches (batching), (ii) batch-unit assignment, manufacturing plant. moving horizon formulation to an expandable polystyrene and (iii) timing of these batches. When these decisions are and (iii) timing of these batches. When these decisions are However, not enough attention has been paid to undermade repeatedly, an online setting, this decision-making sizes of batches (batching), (ii) batch-unit assignment, (iii) timing ofin these batches. When these decisions are However, not not enough enough attention has has been been paid paid to to underunder- and manufacturing plant. attention However, made repeatedly, in an online setting, this decision-making made repeatedly, in an online setting, this decision-making toschedulunder- made standing how the design of the online (open-loop) However, not enough attention has been paid repeatedly, in an online setting, this decision-making process is termed as online scheduling (Gupta et al., 2016). and (iii) timing of these batches. When these decisions are standing how the design of the online (open-loop) schedulstanding how the design of the the online online (open-loop) schedulis termed as online scheduling (Gupta et al., 2016). process is as online scheduling (Gupta et standing how the design of (open-loop) ing problem the quality of the actual implemented However, notaffects enough paid toschedulunder- process made repeatedly, online setting, this decision-making process is termed termed in as an online scheduling (Gupta et al., al., 2016). 2016). ing problem problem affects theattention quality of ofhas thebeen actual implemented ing affects the quality the actual implemented Online scheduling methodology is analogous to economic ing problem the which quality ofa the actual implemented (closed-loop) schedule result of multiple revistanding howaffects the design of theis (open-loop) schedulOnline scheduling is analogous to economic is termed asmethodology online scheduling (Gupta et 2016). Online scheduling methodology is to economic (closed-loop) schedule which isonline result of multiple multiple revi- process (closed-loop) schedule which is aa the result of reviOnline predictive scheduling methodology is analogous analogous to al., economic model control (Rawlings and Risbeck, 2017). In sions. Open-loop and closed-loop scheduling are two difing problem affects the quality of actual implemented (closed-loop) schedule which is a result of multiple revimodel predictive control (Rawlings and Risbeck, 2017). In model predictive control (Rawlings and Risbeck, 2017). In sions. Open-loop and closed-loop scheduling are two difsions. Open-loop and closed-loop scheduling are two difonline scheduling, as shown in Fig. 1A, a new schedule, for Online scheduling methodology is analogous to economic model predictive control (Rawlings and Risbeck, 2017). In ferent problems, even in the deterministic no (closed-loop) schedule which is a scheduling result of case multiple sions. Open-loop and closed-loop are when tworevidifonline scheduling, as shown in Fig. 1A, a new schedule, for online scheduling, as shown in Fig. 1A, a new schedule, for ferent problems, even in the deterministic case when no ferent problems, even in the deterministic case when no next H hours, is computed every ∆ h, with all decisions model predictive control (Rawlings and Risbeck, 2017). In online scheduling, as shown in Fig. 1A, a new schedule, for ferent problems, even in the deterministic case when no uncertainty is and Maravelias, 2016). sions. Open-loop and (Gupta closed-loop two Furdif- next H hours, every ∆ h, all next Hscheduling, hours, is isa computed computed every ∆ h,a with with allisdecisions decisions uncertainty is present present (Gupta and scheduling Maravelias,are 2016). Furuncertainty is (Gupta and Maravelias, 2016). Furnext H hours, is computed every ∆ h, with all decisions revised through complete re-optimization. H referred online as shown in Fig. 1A, new schedule, for uncertainty is present present (Gupta and Maravelias, 2016). Further, online scheduling methods can rely solely on recourse, ferent problems, even in the deterministic case when no through a complete re-optimization. H is referred revised through complete re-optimization. H referred ther, online online scheduling scheduling methods methods can can rely rely solely solely on on recourse, recourse, revised ther, revised through complete re-optimization. Hallis isdecisions referred as the horizon, and ∆ is termed re-optimization next H hours, isaa computed every ∆as h,the with through feedback, to mitigate the effect of disturbances uncertainty is present (Gupta Maravelias, 2016). Fur- to ther, online scheduling methodsand can rely solely on recourse, to as the horizon, and ∆ is termed as the re-optimization to as the horizon, and ∆ is termed as the re-optimization through feedback, to mitigate the effect of disturbances through feedback, to mitigate the effect of disturbances time-step. We assume that true demand is known to the revised through a complete re-optimization. H is referred to as the horizon, and ∆ is termed as the re-optimization (Gupta et al., 2016). ther, online scheduling methods can solely on recourse, time-step. through to mitigate the rely effect of disturbances We assume that true demand is known to the time-step. We assume that true demand is known to the (Gupta etfeedback, al., 2016). (Gupta et al., 2016). scheduler for next η h, and beyond that, from η to H hours, to as the horizon, and ∆ is termed as the re-optimization time-step. We assume that true demand is known to the (Gupta et al., 2016). through feedback, to mitigate the effect of disturbances scheduler for next η h, and beyond that, from η to H hours, scheduler for next η h, and beyond that, from η to H hours, scheduler for next η h, and beyond that, from η to H hours, in the horizon, a forecast pegged at the mean/expected time-step. We assume that true demand is known to the (Gupta et al., 2016). The authors would like to acknowledge support from the Nain the horizon, aa forecast pegged at the mean/expected in pegged at mean/expected The authors would to support the NaThe Science authorsFoundation would like likeunder to acknowledge acknowledge support from from Nain the the horizon, horizon, aEach forecast pegged at the the mean/expected demand is used. online optimization is referred to as scheduler for next ηforecast h, and beyond that, from η to H hours, tional grants CMMI-1334933 andthe CBETThe authors would like to acknowledge support from the Nademand is used. Each online optimization is referred to as demand is used. Each online optimization is referred to as tional Foundation under CMMI-1334933 and tional Science Science Foundation under grants grants CMMI-1334933 and CBETCBETan (open-loop) iteration. The implemented (closed-loop) in the horizon, a forecast pegged at the mean/expected demand is used. Each online optimization is referred to as 1264096, as well as the Petroleum Research Fund under grant 53313tional Science Foundation under grants CMMI-1334933 and CBETThe authors would like to acknowledge support from the Naan (open-loop) iteration. The implemented (closed-loop) 1264096, as well as the Petroleum Research Fund under grant 53313an (open-loop) iteration. The implemented (closed-loop) 1264096, as well as the Petroleum Research Fund under grant 53313schedule is the result of executing the most up-to-date ND9. Science demand is used. Each online optimization is referred to as an (open-loop) iteration. The implemented (closed-loop) 1264096, as well as the Petroleum Research Fund under grant 53313tional Foundation under grants CMMI-1334933 and CBETschedule is is the the result result of of executing executing the the most most up-to-date up-to-date ND9. schedule ND9. schedule is the iteration. result of The executing the most up-to-date ND9. an (open-loop) implemented (closed-loop) 1264096, as well as the Petroleum Research Fund under grant 53313schedule is the Ltd. result of executing ND9. 2405-8963 © 2019, IFAC (International Federation of Automatic Control) by Elsevier All rights reserved. the most up-to-date Copyright 2019 IFAC 727 Hosting Copyright 2019 727 Copyright ©under 2019 IFAC IFAC 727Control. Peer review© of International Federation of Automatic Copyright © 2019 responsibility IFAC 727 10.1016/j.ifacol.2019.06.149 Copyright © 2019 IFAC 727
2019 IFAC DYCOPS 728 Florianópolis - SC, Brazil, April 23-26, 2019 Dhruv Gupta et al. / IFAC PapersOnLine 52-1 (2019) 727–732
Fig. 1. (A) Online scheduling iterations, with horizon H and with true demand known for next η h, are performed every ∆ h. The implemented schedule is a result of executing the first ∆ h of each iteration. (B) State task network (STN) representation of the process network used in all simulations. M0-M4 are the different materials and T1-T4 are the different tasks. Task-unit mapping, task batch-size bounds (β M IN /β M AX ), and material prices (π) are shown. (recent) decisions at any given time, which corresponds to the first ∆ hours of each online (open-loop) iteration. Within each iteration, any applicable scheduling model can be used. Although, here we use the discrete-time state space model of Gupta and Maravelias (2017), with timegrid granularity δ, the framework and analysis are general. 3. SCHEDULING MODEL The state-space model of Gupta and Maravelias (2017) which builds upon the state-space idea introduced by Subramanian et al. (2012) is used for all numerical studies in this paper (Eqs. 1–9). This is because as outlined in Gupta and Maravelias (2017), state-space is a natural reformulation of models used for online scheduling.
(5)
Binary variable Wijt , when 1, implies task i is starting on ¯n unit j at time t, variable Bijt denotes its batch-size. W ijt ¯ n are the lifted task-states, which enable writing and B ijt the model in the state-space form. Eqs. 1-4 are the lifting equations for the state-space reformulation. Eq. 5, is the assignment constraint, that ensures that only one task can execute at a time on a production unit. Eq. 6 enforces lower and upper bounds on the batch-size of the tasks. Eq. 7 is an inventory balance for all the materials. Eq. 8 enforces the correct domain ranges for the variables. We chose our scheduling objective (Eq. 9) to be minimization of cost which is composed of inventory costs, backlog costs, and unit startup costs.
∀ i, j, t (6) τij ¯ = Skt − BOkt + ρ¯ik Bijt
Although, the above specific discrete-time model is used for the numerical results in this work, the results are general and are applicable to both discrete-time and continuous-time models, in STN as well as resource task network (RTN) representations (Pantelides, 1994).
¯0 ∀j, i ∈ Ij , t W ij(t+1) = Wijt 0 ¯ B ∀j, i ∈ Ij , t ij(t+1) = Bijt n−1 n ¯ ¯ =W ∀j, i ∈ Ij , t, n ∈ {1, 2, ....τij } W ij(t+1) ¯n B ij(t+1) =
ijt ¯ n−1 ∀j, i B ijt τij −1
0 ¯ ijt + W
M IN βij Wijt
≤ Bijt ≤
Sk(t+1) − BOk(t+1) +
j
i∈Ij ∩I− k
∈ Ij , t, n ∈ {1, 2, ....τij }
n ¯ ijt ≤1 W
i∈Ij n=1
i∈Ij
∀ j, t
j
+
j
i∈Ij
(2) (3) (4)
i∈Ij ∩I+ k
(7)
¯ n ∈ {0, 1}; Bijt , B ¯ n , Skt , BOkt ≥ 0 Wijt , W ijt ijt IN V zcost = min πk Skt + πkBO BOkt k
(1)
M AX Wijt βij
ρik Bijt − ξkt ∀ k, t
task i consumes/produces material k equivalent to ρik /¯ ρik mass fraction of its batch-size (ρik < 0 for consumption and ρ¯ik > 0 for production). The subset of tasks that can be carried out on unit j are denoted by Ij ; The processing time of task i, when executed on unit j, is denoted by τij . On any given unit, only one task can be performed at a M IN time with its batch-size between lower (βij ) and upper M AX ); the associated production costs of carcapacities (βij F . ξkt , πk , and πkIN V /πkBO rying out task i on unit j is αij denote net of material deliveries and orders due, material price, and inventory/backlog cost, respectively.
t
F αij Wijt
k
(8)
t
(9)
t
This model uses the state task network (STN) representation (Kondili et al., 1993) which primarily comprises of tasks i ∈ I, units j ∈ J, and materials k ∈ K. Time is represented by index t ∈ T. The set of tasks − producing/consuming material k are denoted by I+ k /Ik ; 728
4. SIMULATIONS For the purpose of our simulations, we chose to model demand, in the form of orders, with two aspects contributing to the uncertainty: (i) the time ahead for which the actual (true) order sizes are known (η), and (ii) size of each order sampled from uniform distribution [L-DU%, L+DU%]. While generating an instance, each order (l ∈ 1, 2, 3...) was chosen to be due at F × l ± λ h, where λ ∈ {−1, 0, 1} uniformly at random. Thus, L and F represent, respectively, the average order size and the average time between two orders. All due-times are assumed to be known to the scheduler, thus not uncertain. For the illustrative network shown in Fig. 1B, choosing F = 6, and L = 4, 6, and
2019 IFAC DYCOPS Florianópolis - SC, Brazil, April 23-26, 2019 Dhruv Gupta et al. / IFAC PapersOnLine 52-1 (2019) 727–732
A
B
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F6 L8 DU=0% DU=10% DU=20% DU=30%
1.5 1
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729
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8
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Fig. 2. Effect of demand uncertainty (η, DU) on closed-loop cost for fixed H = 48 and ∆ = 1. Each data point is an average cost of closed-loop schedules for 10 demand samples, and is scaled by mean closed-loop cost corresponding to η = 48, DU=0% within each plot. A
B
2.5
F6 L4
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1.5 1
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"=1 "=4 "=8
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2
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2
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Fig. 3. Effect of η and ∆ on closed-loop cost for fixed H = 48 and DU = 30%. Each data point is an average cost of closed-loop schedules for 10 demand samples, and is scaled by mean closed-loop cost corresponding to η = 48, ∆ = 1 within each plot. A
B
2.5
F6 L4
2.5
F6 L6
1.5 1
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20
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30
"=1 "=4 "=8
1.5 1
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2
Cost
2
Cost
2
Cost
C
2.5
0.5 0
10
20
DU (%)
30
0
10
20
30
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Fig. 4. Effect of DU and ∆ on closed-loop cost for fixed H = 48 and η = 8. Each data point is an average cost of closed-loop schedules for 10 demand samples, and is scaled by mean closed-loop cost corresponding to DU=0%, ∆ = 1 within each plot. 8, resulted in an average production capacity utilization (load) of approximately 50%, 75%, and 90%, respectively, to meet the orders. The model is kept identical for each open-loop problem, except for the updated parameter values (demand realizations). For each different “simulation setting”– which comprises of the chosen F, L, η, DU, H, and ∆ – we simulated an implemented schedule of duration 1 week. We performed multiple replications of the online schedules for different samples of the demand, from distributions characterized by F, L, and DU, and discerned statistical significance by carrying out two-way analysis of variance (two-way ANOVA) on the computational results. A total 729
of 10,440 (closed-loop) simulations were carried out in parallel in a cluster of 24 Intel Xeon machines running CentOS 7 operating system. The needed 814,320 (openloop) optimizations were solved to within 0.1% optimality using default solver options in CPLEX 12.6.1 via GAMS 24.4.3. In every plot, in the next section, we fix four out of the six simulation settings as constant; for the two settings that we vary, one is shown on the x-axis, and the other in the form of isolines. F and L are marked in the upper right hand corner in each plot, and the remaining two settings are described in the caption. The (closed-loop) cost is shown on the y-axis. The costs are an average over all the
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Fig. 7. Effect of H and ∆ on closed-loop cost for fixed η = 8 and DU = 30%. Each data point is an average cost of closed-loop schedules for 10 demand samples, and is scaled by mean closed-loop cost corresponding to H = 48, ∆ = 1 within each plot. replications and are scaled by a single chosen mean value within each plot. We also mark one standard deviation intervals resulting from the replications. 5. RESULTS First and foremost, if we could use a very long horizon (H = 48) and could re-optimize as fast as we could (∆ = 1), it is natural to ascertain the additional cost attributable to the presence of uncertainty in demand (η and DU) as compared to if there was no uncertainty. We see, in Fig. 2, as expected, knowing the true demand ahead in time (larger η) is better, and more so if we have larger magnitude of uncertainty (DU). This is true 730
for both lighter and heavier loads. Note that, for a given η, the schedule cost for higher DU need not always be higher. Different DU results in different order-sizes. Given different order-sizes, the resulting backlog and inventory costs can be lower for higher DU. This can be seen, for example, in Fig. 2B, for η = 48 and DU=0% vs. DU = 30%. If the information about true demand comes almost as a surprise (here η = 8), we see, interestingly, that it matters less for heavier loads than for lighter loads. The aforementioned counterintuitive trend is clearly visible, in Figs. 3 and 4, specifically for η = 8, even more so for ∆ = 8, and is further magnified by larger DU. This is
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explainable – a larger mean load requires us to operate at near production capacity, thus, any order much larger than the mean size, will lead to a backlog, irrespective of whether we know about it 8 hours ahead or not. There is not enough slack in the “packed” schedule to adjust production at the last moment. Although, a larger DU also implies that many orders can also be much smaller than the mean, since backlogs, here, are 10 times as expensive as holding inventory (i.e., πkBO = 10πkIN V ), backlog costs influence closed-loop quality more than inventory costs do. In contrast, for smaller mean loads, we have more idle time in our schedules which can be utilized to meet an order with a larger deviation from its mean. Thus, if we know about the actual order, even a short time ahead, we have units available to start new batches. Please note that in all figures, it is incorrect to compare the absolute schedule costs across different loads (keeping other parameters the same). The costs are scaled within each subplot and, hence, a cost of 1 in, for example, Fig. 3A is different in absolute terms than a cost of 1 in Fig. 3C. Next, we see how changing the re-optimization time-step helps us mitigate the effect of demand uncertainty. In Figs. 3 and 4, it is evident that shorter re-optimization time-steps (equivalently– more frequent re-optimizations) lead to better closed-loop quality for any given demand uncertainty (described by both η and DU). This is so, inspite of using a long horizon (H = 48). Since, the horizon is long enough, the consequent role of re-optimization then is to tackle disturbances (due to uncertainty), and not new information. This, as hypothesized in Gupta and Maravelias (2016), is in contrast to the no uncertainty case, wherein a long horizon can completely compensate for infrequent re-optimizations. In addition, given that uncertainty is less important for heavier loads, the cost advantage from shorter re-optimization time-step, under heavier loads, reduces. When we are trying to design an online scheduling algorithm, an important aspect of it is the horizon length that we need to choose. In Figs. 5 and 6, we present how closedloop quality is affected by H, DU, and η. We see that the importance of employing any longer H than a threshold gradually diminishes. This threshold is smaller for lighter loads, and larger for heavier loads. This is because for heavier loads, multiple batches need to be run which need to be scheduled ahead – on the basis of available forecast. Looking closer in Figs. 5 and 6, we see the isolines do not cross-over each other. This leads to an interesting corollary: a longer H is not able to compensate for a higher DU or a smaller η! This makes sense, as a longer horizon is unlikely to help us with a surprise that we get at the start of the horizon. Finally, in Fig.7, we show how H and ∆ influence the cost of the implemented schedule in the presence of demand uncertainty (DU=30%, η = 8). A longer H, dependent on load, and a shorter re-optimization time-step, are beneficial for achieving better closed-loop quality. These general findings, confirm that the recommendations for choosing a good re-optimization algorithm presented by Gupta and Maravelias (2016) on the basis of their results for the no uncertainty case, are also valid when uncertainty is present.
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6. CONCLUSIONS First, the results presented in this work demonstrate that demand uncertainty is an important consideration in online scheduling. We validate, that it is indeed beneficial to invest in obtaining good forecasts (shortening η) as it greatly influences the closed-loop schedule. Second, we find that this, counter-intuitively, is more important for lightly loaded facilities than heavily loaded facilities, since heavily loaded facilities will run close to capacity irrespective of fluctuations in order sizes. Third, we find that a good choice of the horizon is influenced by the mean load at which we are operating, and not by the accuracy of the demand forecast, as has been sometimes covertly assumed by practitioners. Fourth, if the forecast is not perfect, feedback, acted upon in the form of re-optimization, can mitigate the effect of uncertainty, which is observed, in hindsight, as a disturbance. Re-optimization is very effective, especially, for underloaded facilities, as it can help exploit uncommitted resources, which are not necessarily present under heavy loads. Finally, we also see how certain recommendations based on studies assuming no uncertainty, can also be useful, even when uncertainty is present. The analysis, presented in this work, yields several insights, that are applicable to general rescheduling approaches, thus taking us one step forward towards synthesis of general strategies to obtain high-quality closedloop schedules. REFERENCES Chu, Y. and You, F. (2014). Moving horizon approach of integrating scheduling and control for sequential batch processes. AIChE Journal, 60(5), 1654–1671. doi: 10.1002/aic.14359. Cui, J. and Engell, S. (2010). Medium-term planning of a multiproduct batch plant under evolving multiperiod multi-uncertainty by means of a moving horizon strategy. Computers and Chemical Engineering, 34(5), 598–619. doi:10.1016/j.compchemeng.2010.01.013. Gupta, D. and Maravelias, C.T. (2016). On deterministic online scheduling: Major considerations, paradoxes and remedies. Computers & Chemical Engineering, 94, 312– 330. doi:10.1016/j.compchemeng.2016.08.006. Gupta, D. and Maravelias, C.T. (2017). A General StateSpace Formulation for Online Scheduling. Processes, 5(4), 69. Gupta, D., Maravelias, C.T., and Wassick, J.M. (2016). From rescheduling to online scheduling. Chemical Engineering Research and Design, 116, 83–97. doi: 10.1016/j.cherd.2016.10.035. Harjunkoski, I., Maravelias, C.T., Bongers, P., Castro, P.M., Engell, S., Grossmann, I.E., Hooker, J., M´endez, C.A., Sand, G., and Wassick, J.M. (2014). Scope for industrial applications of production scheduling models and solution methods. Computers and Chemical Engineering, 62, 161–193. doi: 10.1016/j.compchemeng.2013.12.001. Janak, S.L., Floudas, C.A., Kallrath, J., and Vormbrock, N. (2006). Production scheduling of a large-scale industrial batch plant. II. Reactive scheduling. Industrial and Engineering Chemistry Research, 45(25), 8253– 8269. doi:10.1021/ie0600590.
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Kondili, E., Pantelides, C.C., and Sargent, R.W.H. (1993). A general algorithm for short-term scheduling of batch operations-I. MILP formulation. Computers and Chemical Engineering, 17(2), 211–227. doi:10.1016/00981354(93)80015-F. Lappas, N.H. and Gounaris, C.E. (2016). Multi-stage adjustable robust optimization for process scheduling under uncertainty. AIChE Journal, 62(5), 1646–1667. doi:10.1002/aic.15183. Li, Z. and Ierapetritou, M.G. (2008). Process scheduling under uncertainty: Review and challenges. Computers and Chemical Engineering, 32(4-5), 715–727. doi: 10.1016/j.compchemeng.2007.03.001. Novas, J.M. and Henning, G.P. (2010). Reactive scheduling framework based on domain knowledge and constraint programming. Computers and Chemical Engineering, 34(12), 2129–2148. doi: 10.1016/j.compchemeng.2010.07.011. Pantelides, C.C. (1994). Unified frameworks for optimal process planning and scheduling. In Proceedings on the second conference on foundations of computer aided operations, 253—-274. Publications, Cache, New York. Pattison, R.C., Touretzky, C.R., Harjunkoski, I., and Baldea, M. (2016). Moving horizon closed-loop production scheduling using dynamic process models. AIChE Journal. Pinedo, M. and Chao, X. (1999). Operations scheduling with applications in manufacturing and services. McGraw Hill, Boston, USA. Rawlings, J.B. and Risbeck, M.J. (2017). Model predictive control with discrete actuators: Theory and application. Automatica, 78, 258–265. doi: 10.1016/j.automatica.2016.12.024. Subramanian, K., Maravelias, C.T., and Rawlings, J.B. (2012). A state-space model for chemical production scheduling. Computers & Chemical Engineering, 47, 97– 110. doi:10.1016/j.compchemeng.2012.06.025.
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Online Scheduling: Understanding Online Online Scheduling: Scheduling: Understanding Understanding Impact of Uncertainty of Uncertainty Online Impact Scheduling: Understanding Impact of Uncertainty Impact ofChristos Uncertainty Dhruv Gupta, T. Maravelias
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Dhruv Gupta, Christos T. Maravelias Dhruv Dhruv Gupta, Gupta, Christos Christos T. T. Maravelias Maravelias Dhruv Gupta, ChristosMadison, T. Maravelias University of Wisconsin-Madison, University of of Wisconsin-Madison, Wisconsin-Madison, Madison, Madison, WI WI 53706 53706 USA USA University WI 53706 USA [email protected]). University(e-mail: of Wisconsin-Madison, Madison, WI 53706 USA (e-mail: [email protected]). (e-mail: [email protected]). [email protected]). University(e-mail: of Wisconsin-Madison, Madison, WI 53706 USA (e-mail: [email protected]). Abstract: We to quality schedules obtained and Abstract: We We present present aa a framework framework to to study study quality quality of of schedules schedules obtained obtained iteratively iteratively and and Abstract: Abstract: We present present a framework framework to study study quality of ofUsing schedules obtained iteratively iteratively anda online (real-time) in the presence of demand uncertainty. this framework, we carry out online (real-time) (real-time) in in the the presence presence of of demand demand uncertainty. uncertainty. Using Using this this framework, framework, we we carry carry out out aa online computational and interesting observations. First, we find that uncertainty Abstract: Westudy, present a make framework to study quality of schedules obtained iteratively andaa online (real-time) in the of demand uncertainty. Using this framework, we carryplays out computational study, andpresence make interesting observations. First, we find that uncertainty uncertainty plays computational study, and make interesting observations. First, we find that plays a less important role as a manufacturing facility is operated close to capacity. Second, the choice online (real-time) in the presence of demand uncertainty. Using this framework, we carry out computational study, and make interesting observations. First, we find that uncertainty plays less important important role role as as aa manufacturing manufacturing facility facility is is operated operated close close to to capacity. capacity. Second, Second, the the choice choicea less of the horizon for the is dependent on the mean load, independent of computational study, makeiterations, interesting we that but uncertainty a less important as and aonline manufacturing facility is operated close to find capacity. Second, theplays choice of the the horizon role for the the online iterations, is observations. dependent onFirst, the mean mean load, but independent of of horizon for online iterations, is on the load, independent of of the horizonofrole for the online iterations, is dependent dependent onform the of mean load, but but independent of the accuracy the forecasts. Finally, feedback, in the re-optimization, plays a very less important as a manufacturing facility is operated close to capacity. Second, the choice the accuracy accuracy of of the the forecasts. forecasts. Finally, Finally, feedback, feedback, in in the form form of re-optimization, plays aa very the of re-optimization, plays important role in mitigating impact of uncertainty. the analysis presented in of the horizon for the onlinethe iterations, dependent onform thethrough mean load, but independent of the accuracy the forecasts. Finally, in the theThus, of re-optimization, plays a very very important roleofin in mitigating the impact feedback, ofisuncertainty. uncertainty. Thus, through the analysis analysis presented in important role mitigating the impact of Thus, through the presented in this work, we gain insights that are applicable to all general rescheduling approaches. the accuracy of the forecasts. Finally, feedback, in the form of re-optimization, plays a very important role in mitigating the impact of uncertainty. Thus, through the analysis presented in this work, we we gain gain insights insights that are are applicable applicable to to all general general rescheduling rescheduling approaches. this important rolegain in mitigating the are impact of uncertainty. Thus,rescheduling through the approaches. analysis presented in this work, work, we insights that that applicable to all all general approaches. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. this work, we gain insights that are applicable to all general rescheduling approaches. Keywords: Uncertainty, Rescheduling, Operations, MPC, Mixed-integer programming Keywords: Uncertainty, Uncertainty, Rescheduling, Rescheduling, Operations, MPC, Mixed-integer programming Keywords: Operations, MPC, Mixed-integer programming Keywords: Uncertainty, Rescheduling, Operations, MPC, Mixed-integer programming Keywords: Uncertainty, Rescheduling, Operations, MPC, Mixed-integer programming 1. INTRODUCTION In this work, we outline aa framework for studying qual1. INTRODUCTION INTRODUCTION In this work, we outline for studying the the qual1. In this work, we outline aa framework framework for the quality of closed-loop schedules, when uncertainty present. 1. INTRODUCTION In this work, we outline framework for studying studyingis the quality of closed-loop schedules, when uncertainty is present. ity of closed-loop schedules, when uncertainty is present. 1. INTRODUCTION Although, this framework can be used to study several In this work, we outline a framework for studying the quality of closed-loop schedules, when uncertainty is present. Scheduling of operations is one of the many planning this framework can be used to study several Scheduling of of operations operations is is one one of of the the many many planning planning Although, Although, this framework can be used to study several Scheduling types of uncertainties, such as, demand, yield, processing ity of closed-loop schedules, when uncertainty is present. Although, this framework can be used to study several functions in the manufacturing sector (Pinedo and Chao, Scheduling of operations is one of the many planning types of of uncertainties, uncertainties, such such as, as, demand, demand, yield, yield, processing processing functions in in the the manufacturing manufacturing sector sector (Pinedo (Pinedo and and Chao, Chao, types functions types of uncertainties, such as, demand, yield, processing time, unit breakdowns, etc., here to limited space, we Although, this framework can bedue used to study several and Chao, 1999). Finding a schedule once, however, is only part of Scheduling is one of the many planning functions in of theoperations manufacturing sector (Pinedo time, unit breakdowns, etc., here due to limited space, we 1999). Finding a schedule once, however, is only part of time, unit breakdowns, etc., here due to limited space, we 1999). Finding a schedule once, however, is only part of specifically explore the role demand uncertainty plays in types of uncertainties, such as, demand, yield, processing time, unit breakdowns, etc., here due to limited space, we 1999). Finding a schedule once, however, is only part of the whole scheduling process (Harjunkoski et al., 2014). functions in the manufacturing sector (Pinedo and Chao, specifically explore the role demand uncertainty plays in the whole whole scheduling scheduling process process (Harjunkoski (Harjunkoski et et al., al., 2014). 2014). specifically explore the role demand uncertainty plays in the the context of online scheduling. We attempt to answer time, unit breakdowns, etc., here due to limited space, we specifically explore the role demand uncertainty plays in the whole scheduling process (Harjunkoski etonly al., part 2014). Due to disruptions or arrival of new information, the 1999). Finding a schedule once, however, is of the context of online scheduling. We attempt to answer Due to disruptions or arrival of new information, the the context of online scheduling. We attempt to answer Due to disruptions or arrival of new information, the the context of online scheduling. We attempt to answer several questions: (i) how demand uncertainty affects the specifically explore the role demand uncertainty plays in orprocess arrival(Harjunkoski of new information, incumbent schedule can become suboptimal or even infeathe whole scheduling et al., 2014). Due to disruptions the several questions: questions: (i) (i) how how demand demand uncertainty uncertainty affects affects the the incumbent schedule schedule can can become become suboptimal suboptimal or or even even infeainfea- several incumbent several questions: (i) how demand uncertainty affects the quality of the implemented schedules, (ii) how important the context of online scheduling. We attempt to answer sible, thus motivating the need for online (re)scheduling Due to disruptions or arrival of new information, the incumbent schedule can become suboptimal or even infeaquality of of the the implemented implemented schedules, schedules, (ii) (ii) how how important important sible, thus thus motivating motivating the the need need for for online online (re)scheduling (re)scheduling quality sible, is horizon length to this uncertainty, (iii) to several questions: (i) address how demand uncertainty affects the quality of the implemented schedules, (ii) howand important sible, thus schedule motivating the need for important online (re)scheduling (Subramanian et al., 2012). Several have incumbent become suboptimal or works even infeahorizon length to address this uncertainty, and (iii) to (Subramanian et al., al.,can 2012). Several important works have is is horizon length to address this uncertainty, and (iii) to (Subramanian et 2012). Several important works have what extent, feedback, as a function of the frequency of quality of the implemented schedules, (ii) how important is horizon length to address this uncertainty, and (iii) to suggested rescheduling strategies in different contexts (Li sible, thus motivating the need for online (re)scheduling (Subramanian et al., 2012). Several important works have what extent, feedback, as a function of the frequency of suggested rescheduling strategies in different contexts (Li what extent, feedback, as a function of the frequency of suggested rescheduling strategies in different contexts (Li what extent, feedback, as a function of the frequency of re-optimization, can mitigate the effect of this uncertainty horizon length to address this uncertainty, and (iii) to contexts (Li is and Ierapetritou, 2008). For example: Janak et al. (2006) (Subramanian et al., 2012). Several important works have suggested rescheduling strategies in different re-optimization, can mitigate the effect of this uncertainty and Ierapetritou, 2008). For example: Janak et al. (2006) re-optimization, can mitigate the effect of this uncertainty and Ierapetritou, 2008). For example: Janak et al. (2006) re-optimization, can mitigate the effect of this uncertainty which is treated as a disturbance in the nominal model what extent, feedback, as a function of the frequency of proposed partial rescheduling by identifying and fixing suggested rescheduling strategies different contexts (Li which and Ierapetritou, 2008). For example: Janak et al. (2006) is treated as aa disturbance in the nominal model proposed partial rescheduling byinidentifying identifying and fixing which is treated as disturbance in the nominal model proposed partial rescheduling by and fixing that we employ. Rather than proposing a new online re-optimization, can mitigate the effect of this uncertainty which is treated as a disturbance in the nominal model proposed partial rescheduling by identifying and fixing tasks not directly affected an observed disturbance; and For by example: Janak et al. (2006) that we employ. Rather than proposing aa new online tasksIerapetritou, not directly directly2008). affected by an observed observed disturbance; that we employ. than proposing online tasks not affected by an disturbance; scheduling strategy, we focus analysis – to gain which is treated asRather ahere, disturbance inon the nominal model that we employ. Rather than proposing a new new online Pattison et al. (2016) updated schedules respond to proposed partial rescheduling by identifying and fixing tasks not directly affected by an observedto disturbance; scheduling strategy, here, we focus on analysis – to gain Pattison et al. (2016) updated schedules to respond to scheduling strategy, here, we focus on analysis – to gain Pattison et al. (2016) updated schedules to respond to scheduling strategy, here, we focus on analysis – to gain insights that are broadly applicable to general rescheduling that we employ. Rather than proposing a new online process and market disturbances; Chu and You (2014) tasks not directly affected by an observed disturbance; Pattison et al. (2016) updated schedules to respond to insights that are broadly applicable to general rescheduling process and market disturbances; Chu and You (2014) insights that are broadly applicable to general rescheduling process and market disturbances; Chu and You (2014) approaches. scheduling strategy, here, we focus on analysis – to gain insights that are broadly applicable to general rescheduling solved reduced problem online, achieving computational Pattison et al. (2016) updated to You respond to approaches. process market disturbances; Chu and (2014) solved aa a and reduced problem online, schedules achieving computational approaches. solved reduced problem online, achieving computational insights that are broadly applicable to general rescheduling approaches. solved a reduced problem online, achieving computational tractability schedule stability; Novas process and and market disturbances; andand YouHenning (2014) tractability and schedule stability;Chu Novas and Henning tractability and schedule stability; Novas and Henning approaches. 2. (2010) constraint programming for rescheduling; Lapsolved aused reduced problem online, achieving computational tractability and schedule stability; Novas and Henning 2. ONLINE ONLINE FRAMEWORK FRAMEWORK 2. (2010) used constraint programming for rescheduling; Lap(2010) used constraint programming for rescheduling; Lap2. ONLINE ONLINE FRAMEWORK FRAMEWORK pas and Gounaris (2016) used adjustable robust optimizatractability and schedule stability; Novas and Henning (2010) used constraint programming for rescheduling; Lappas and Gounaris (2016) used adjustable robust optimizapas and Gounaris (2016) used adjustable robust optimizaGiven: (i) production facility data, (ii) production recipe, 2. ONLINE FRAMEWORK optimization to generate schedules with a recourse strategy; Cui (2010) constraint programming for rescheduling; pas Gounaris (2016) used adjustable robust (i) production facility data, (ii) production recipe, Given: (i) production facility data, (ii) production recipe, tionand toused generate schedules with a recourse recourse strategy;LapCui Given: tion to generate schedules with a strategy; Cui (iii) resource availability, (iv) feed availability, and (v) Given: (i) production facility data, (ii) production recipe, tion to generate schedules with a recourse strategy; Cui and Engell (2010) applied a two-stage stochastic model in a pas and Gounaris (2016) used adjustable robust optimiza(iii) resource availability, (iv) feed availability, and (v) (iii) resource availability, (iv) feed availability, and (v) and Engell Engell (2010) (2010) applied applied aa two-stage two-stage stochastic stochastic model model in in aa (iii) and resource availability, (iv) feed availability, and (v) production targets; scheduling seeks to find (i) selection Given: (i) production facility data, (ii) production recipe, moving horizon formulation to an expandable polystyrene tion to generate schedules with a recourse strategy; Cui and Engell (2010) applied a two-stage stochastic model in a production targets; scheduling seeks to find (i) selection production targets; scheduling seeks to find (i) selection moving horizon formulation to an expandable polystyrene moving horizon formulation to an expandable polystyrene and sizes of batches (batching), (ii) batch-unit assignment, (iii) resource availability, (iv) feed availability, and (v) production targets; scheduling seeks to find (i) selection manufacturing plant. and Engell (2010) applied a two-stage stochasticpolystyrene model in a and moving horizon formulation to an expandable sizes of batches (batching), (ii) batch-unit assignment, and sizes of batches (batching), (ii) batch-unit assignment, manufacturing plant. manufacturing plant. (iii) timing of these batches. When these decisions are production targets; scheduling seeks to find (i) selection and sizes of batches (batching), (ii) batch-unit assignment, manufacturing plant. moving horizon formulation to an expandable polystyrene and (iii) timing of these batches. When these decisions are and (iii) timing of these batches. When these decisions are However, not enough attention has been paid to undermade repeatedly, an online setting, this decision-making sizes of batches (batching), (ii) batch-unit assignment, (iii) timing ofin these batches. When these decisions are However, not not enough enough attention has has been been paid paid to to underunder- and manufacturing plant. attention However, made repeatedly, in an online setting, this decision-making made repeatedly, in an online setting, this decision-making toschedulunder- made standing how the design of the online (open-loop) However, not enough attention has been paid repeatedly, in an online setting, this decision-making process is termed as online scheduling (Gupta et al., 2016). and (iii) timing of these batches. When these decisions are standing how the design of the online (open-loop) schedulstanding how the design of the the online online (open-loop) schedulis termed as online scheduling (Gupta et al., 2016). process is as online scheduling (Gupta et standing how the design of (open-loop) ing problem the quality of the actual implemented However, notaffects enough paid toschedulunder- process made repeatedly, online setting, this decision-making process is termed termed in as an online scheduling (Gupta et al., al., 2016). 2016). ing problem problem affects theattention quality of ofhas thebeen actual implemented ing affects the quality the actual implemented Online scheduling methodology is analogous to economic ing problem the which quality ofa the actual implemented (closed-loop) schedule result of multiple revistanding howaffects the design of theis (open-loop) schedulOnline scheduling is analogous to economic is termed asmethodology online scheduling (Gupta et 2016). Online scheduling methodology is to economic (closed-loop) schedule which isonline result of multiple multiple revi- process (closed-loop) schedule which is aa the result of reviOnline predictive scheduling methodology is analogous analogous to al., economic model control (Rawlings and Risbeck, 2017). In sions. Open-loop and closed-loop scheduling are two difing problem affects the quality of actual implemented (closed-loop) schedule which is a result of multiple revimodel predictive control (Rawlings and Risbeck, 2017). In model predictive control (Rawlings and Risbeck, 2017). In sions. Open-loop and closed-loop scheduling are two difsions. Open-loop and closed-loop scheduling are two difonline scheduling, as shown in Fig. 1A, a new schedule, for Online scheduling methodology is analogous to economic model predictive control (Rawlings and Risbeck, 2017). In ferent problems, even in the deterministic no (closed-loop) schedule which is a scheduling result of case multiple sions. Open-loop and closed-loop are when tworevidifonline scheduling, as shown in Fig. 1A, a new schedule, for online scheduling, as shown in Fig. 1A, a new schedule, for ferent problems, even in the deterministic case when no ferent problems, even in the deterministic case when no next H hours, is computed every ∆ h, with all decisions model predictive control (Rawlings and Risbeck, 2017). In online scheduling, as shown in Fig. 1A, a new schedule, for ferent problems, even in the deterministic case when no uncertainty is and Maravelias, 2016). sions. Open-loop and (Gupta closed-loop two Furdif- next H hours, every ∆ h, all next Hscheduling, hours, is isa computed computed every ∆ h,a with with allisdecisions decisions uncertainty is present present (Gupta and scheduling Maravelias,are 2016). Furuncertainty is (Gupta and Maravelias, 2016). Furnext H hours, is computed every ∆ h, with all decisions revised through complete re-optimization. H referred online as shown in Fig. 1A, new schedule, for uncertainty is present present (Gupta and Maravelias, 2016). Further, online scheduling methods can rely solely on recourse, ferent problems, even in the deterministic case when no through a complete re-optimization. H is referred revised through complete re-optimization. H referred ther, online online scheduling scheduling methods methods can can rely rely solely solely on on recourse, recourse, revised ther, revised through complete re-optimization. Hallis isdecisions referred as the horizon, and ∆ is termed re-optimization next H hours, isaa computed every ∆as h,the with through feedback, to mitigate the effect of disturbances uncertainty is present (Gupta Maravelias, 2016). Fur- to ther, online scheduling methodsand can rely solely on recourse, to as the horizon, and ∆ is termed as the re-optimization to as the horizon, and ∆ is termed as the re-optimization through feedback, to mitigate the effect of disturbances through feedback, to mitigate the effect of disturbances time-step. We assume that true demand is known to the revised through a complete re-optimization. H is referred to as the horizon, and ∆ is termed as the re-optimization (Gupta et al., 2016). ther, online scheduling methods can solely on recourse, time-step. through to mitigate the rely effect of disturbances We assume that true demand is known to the time-step. We assume that true demand is known to the (Gupta etfeedback, al., 2016). (Gupta et al., 2016). scheduler for next η h, and beyond that, from η to H hours, to as the horizon, and ∆ is termed as the re-optimization time-step. We assume that true demand is known to the (Gupta et al., 2016). through feedback, to mitigate the effect of disturbances scheduler for next η h, and beyond that, from η to H hours, scheduler for next η h, and beyond that, from η to H hours, scheduler for next η h, and beyond that, from η to H hours, in the horizon, a forecast pegged at the mean/expected time-step. We assume that true demand is known to the (Gupta et al., 2016). The authors would like to acknowledge support from the Nain the horizon, aa forecast pegged at the mean/expected in pegged at mean/expected The authors would to support the NaThe Science authorsFoundation would like likeunder to acknowledge acknowledge support from from Nain the the horizon, horizon, aEach forecast pegged at the the mean/expected demand is used. online optimization is referred to as scheduler for next ηforecast h, and beyond that, from η to H hours, tional grants CMMI-1334933 andthe CBETThe authors would like to acknowledge support from the Nademand is used. Each online optimization is referred to as demand is used. Each online optimization is referred to as tional Foundation under CMMI-1334933 and tional Science Science Foundation under grants grants CMMI-1334933 and CBETCBETan (open-loop) iteration. The implemented (closed-loop) in the horizon, a forecast pegged at the mean/expected demand is used. Each online optimization is referred to as 1264096, as well as the Petroleum Research Fund under grant 53313tional Science Foundation under grants CMMI-1334933 and CBETThe authors would like to acknowledge support from the Naan (open-loop) iteration. The implemented (closed-loop) 1264096, as well as the Petroleum Research Fund under grant 53313an (open-loop) iteration. The implemented (closed-loop) 1264096, as well as the Petroleum Research Fund under grant 53313schedule is the result of executing the most up-to-date ND9. Science demand is used. Each online optimization is referred to as an (open-loop) iteration. The implemented (closed-loop) 1264096, as well as the Petroleum Research Fund under grant 53313tional Foundation under grants CMMI-1334933 and CBETschedule is is the the result result of of executing executing the the most most up-to-date up-to-date ND9. schedule ND9. schedule is the iteration. result of The executing the most up-to-date ND9. an (open-loop) implemented (closed-loop) 1264096, as well as the Petroleum Research Fund under grant 53313schedule is the Ltd. result of executing ND9. 2405-8963 © 2019, IFAC (International Federation of Automatic Control) by Elsevier All rights reserved. the most up-to-date Copyright 2019 IFAC 727 Hosting Copyright 2019 727 Copyright ©under 2019 IFAC IFAC 727Control. Peer review© of International Federation of Automatic Copyright © 2019 responsibility IFAC 727 10.1016/j.ifacol.2019.06.149 Copyright © 2019 IFAC 727
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Fig. 1. (A) Online scheduling iterations, with horizon H and with true demand known for next η h, are performed every ∆ h. The implemented schedule is a result of executing the first ∆ h of each iteration. (B) State task network (STN) representation of the process network used in all simulations. M0-M4 are the different materials and T1-T4 are the different tasks. Task-unit mapping, task batch-size bounds (β M IN /β M AX ), and material prices (π) are shown. (recent) decisions at any given time, which corresponds to the first ∆ hours of each online (open-loop) iteration. Within each iteration, any applicable scheduling model can be used. Although, here we use the discrete-time state space model of Gupta and Maravelias (2017), with timegrid granularity δ, the framework and analysis are general. 3. SCHEDULING MODEL The state-space model of Gupta and Maravelias (2017) which builds upon the state-space idea introduced by Subramanian et al. (2012) is used for all numerical studies in this paper (Eqs. 1–9). This is because as outlined in Gupta and Maravelias (2017), state-space is a natural reformulation of models used for online scheduling.
(5)
Binary variable Wijt , when 1, implies task i is starting on ¯n unit j at time t, variable Bijt denotes its batch-size. W ijt ¯ n are the lifted task-states, which enable writing and B ijt the model in the state-space form. Eqs. 1-4 are the lifting equations for the state-space reformulation. Eq. 5, is the assignment constraint, that ensures that only one task can execute at a time on a production unit. Eq. 6 enforces lower and upper bounds on the batch-size of the tasks. Eq. 7 is an inventory balance for all the materials. Eq. 8 enforces the correct domain ranges for the variables. We chose our scheduling objective (Eq. 9) to be minimization of cost which is composed of inventory costs, backlog costs, and unit startup costs.
∀ i, j, t (6) τij ¯ = Skt − BOkt + ρ¯ik Bijt
Although, the above specific discrete-time model is used for the numerical results in this work, the results are general and are applicable to both discrete-time and continuous-time models, in STN as well as resource task network (RTN) representations (Pantelides, 1994).
¯0 ∀j, i ∈ Ij , t W ij(t+1) = Wijt 0 ¯ B ∀j, i ∈ Ij , t ij(t+1) = Bijt n−1 n ¯ ¯ =W ∀j, i ∈ Ij , t, n ∈ {1, 2, ....τij } W ij(t+1) ¯n B ij(t+1) =
ijt ¯ n−1 ∀j, i B ijt τij −1
0 ¯ ijt + W
M IN βij Wijt
≤ Bijt ≤
Sk(t+1) − BOk(t+1) +
j
i∈Ij ∩I− k
∈ Ij , t, n ∈ {1, 2, ....τij }
n ¯ ijt ≤1 W
i∈Ij n=1
i∈Ij
∀ j, t
j
+
j
i∈Ij
(2) (3) (4)
i∈Ij ∩I+ k
(7)
¯ n ∈ {0, 1}; Bijt , B ¯ n , Skt , BOkt ≥ 0 Wijt , W ijt ijt IN V zcost = min πk Skt + πkBO BOkt k
(1)
M AX Wijt βij
ρik Bijt − ξkt ∀ k, t
task i consumes/produces material k equivalent to ρik /¯ ρik mass fraction of its batch-size (ρik < 0 for consumption and ρ¯ik > 0 for production). The subset of tasks that can be carried out on unit j are denoted by Ij ; The processing time of task i, when executed on unit j, is denoted by τij . On any given unit, only one task can be performed at a M IN time with its batch-size between lower (βij ) and upper M AX ); the associated production costs of carcapacities (βij F . ξkt , πk , and πkIN V /πkBO rying out task i on unit j is αij denote net of material deliveries and orders due, material price, and inventory/backlog cost, respectively.
t
F αij Wijt
k
(8)
t
(9)
t
This model uses the state task network (STN) representation (Kondili et al., 1993) which primarily comprises of tasks i ∈ I, units j ∈ J, and materials k ∈ K. Time is represented by index t ∈ T. The set of tasks − producing/consuming material k are denoted by I+ k /Ik ; 728
4. SIMULATIONS For the purpose of our simulations, we chose to model demand, in the form of orders, with two aspects contributing to the uncertainty: (i) the time ahead for which the actual (true) order sizes are known (η), and (ii) size of each order sampled from uniform distribution [L-DU%, L+DU%]. While generating an instance, each order (l ∈ 1, 2, 3...) was chosen to be due at F × l ± λ h, where λ ∈ {−1, 0, 1} uniformly at random. Thus, L and F represent, respectively, the average order size and the average time between two orders. All due-times are assumed to be known to the scheduler, thus not uncertain. For the illustrative network shown in Fig. 1B, choosing F = 6, and L = 4, 6, and
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A
B
C
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1.5 1
2
F6 L6
1.5
Cost
F6 L4
Cost
Cost
2
1
0.5 16
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40
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F6 L8 DU=0% DU=10% DU=20% DU=30%
1.5 1
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729
0.5 8
16
2 (Hours)
24
32
40
48
8
16
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24
32
40
48
2 (Hours)
Fig. 2. Effect of demand uncertainty (η, DU) on closed-loop cost for fixed H = 48 and ∆ = 1. Each data point is an average cost of closed-loop schedules for 10 demand samples, and is scaled by mean closed-loop cost corresponding to η = 48, DU=0% within each plot. A
B
2.5
F6 L4
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F6 L6
1.5 1
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0.5 16
24
32
40
48
"=1 "=4 "=8
1.5 1
0.5 8
F6 L8
2
Cost
2
Cost
2
Cost
C
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0.5 8
2 (Hours)
16
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16
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40
48
2 (Hours)
Fig. 3. Effect of η and ∆ on closed-loop cost for fixed H = 48 and DU = 30%. Each data point is an average cost of closed-loop schedules for 10 demand samples, and is scaled by mean closed-loop cost corresponding to η = 48, ∆ = 1 within each plot. A
B
2.5
F6 L4
2.5
F6 L6
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0.5 10
20
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30
"=1 "=4 "=8
1.5 1
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F6 L8
2
Cost
2
Cost
2
Cost
C
2.5
0.5 0
10
20
DU (%)
30
0
10
20
30
DU (%)
Fig. 4. Effect of DU and ∆ on closed-loop cost for fixed H = 48 and η = 8. Each data point is an average cost of closed-loop schedules for 10 demand samples, and is scaled by mean closed-loop cost corresponding to DU=0%, ∆ = 1 within each plot. 8, resulted in an average production capacity utilization (load) of approximately 50%, 75%, and 90%, respectively, to meet the orders. The model is kept identical for each open-loop problem, except for the updated parameter values (demand realizations). For each different “simulation setting”– which comprises of the chosen F, L, η, DU, H, and ∆ – we simulated an implemented schedule of duration 1 week. We performed multiple replications of the online schedules for different samples of the demand, from distributions characterized by F, L, and DU, and discerned statistical significance by carrying out two-way analysis of variance (two-way ANOVA) on the computational results. A total 729
of 10,440 (closed-loop) simulations were carried out in parallel in a cluster of 24 Intel Xeon machines running CentOS 7 operating system. The needed 814,320 (openloop) optimizations were solved to within 0.1% optimality using default solver options in CPLEX 12.6.1 via GAMS 24.4.3. In every plot, in the next section, we fix four out of the six simulation settings as constant; for the two settings that we vary, one is shown on the x-axis, and the other in the form of isolines. F and L are marked in the upper right hand corner in each plot, and the remaining two settings are described in the caption. The (closed-loop) cost is shown on the y-axis. The costs are an average over all the
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A
B
4
F6 L4
4
F6 L6
2 1
2 1
0 24
30
36
42
48
DU=0% DU=10% DU=20% DU=30%
2 1
0 18
F6 L8
3
Cost
3
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3
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C
4
0 18
24
H (Hours)
30
36
42
48
18
24
H (Hours)
30
36
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H (Hours)
Fig. 5. Effect of H and DU on closed-loop cost for fixed η = 8 and ∆ = 1. Each data point is an average cost of closed-loop schedules for 10 demand samples, and is scaled by mean closed-loop cost corresponding to H = 48, DU=0% within each plot. A
B
C
6
4 2
6
F6 L6
4
Cost
F6 L4
Cost
Cost
6
2
0 24
30
36
42
48
2 2 2 2 2 2
4 2
0 18
F6 L8
=8 =16 =24 =32 =40 =48
0 18
24
H (Hours)
30
36
42
48
18
24
H (Hours)
30
36
42
48
H (Hours)
Fig. 6. Effect of H and η on closed-loop cost for fixed ∆ = 1 and DU=30%. Each data point is an average cost of closed-loop schedules for 10 demand samples, and is scaled by mean closed-loop cost corresponding to η = 48, H = 48 within each plot. A
B
4
F6 L4
4
F6 L6
2 1
2 1
0 24
30
36
42
48
H (Hours)
"=1 "=4 "=8
2 1
0 18
F6 L8
3
Cost
3
Cost
3
Cost
C
4
0 18
24
30
36
H (Hours)
42
48
18
24
30
36
42
48
H (Hours)
Fig. 7. Effect of H and ∆ on closed-loop cost for fixed η = 8 and DU = 30%. Each data point is an average cost of closed-loop schedules for 10 demand samples, and is scaled by mean closed-loop cost corresponding to H = 48, ∆ = 1 within each plot. replications and are scaled by a single chosen mean value within each plot. We also mark one standard deviation intervals resulting from the replications. 5. RESULTS First and foremost, if we could use a very long horizon (H = 48) and could re-optimize as fast as we could (∆ = 1), it is natural to ascertain the additional cost attributable to the presence of uncertainty in demand (η and DU) as compared to if there was no uncertainty. We see, in Fig. 2, as expected, knowing the true demand ahead in time (larger η) is better, and more so if we have larger magnitude of uncertainty (DU). This is true 730
for both lighter and heavier loads. Note that, for a given η, the schedule cost for higher DU need not always be higher. Different DU results in different order-sizes. Given different order-sizes, the resulting backlog and inventory costs can be lower for higher DU. This can be seen, for example, in Fig. 2B, for η = 48 and DU=0% vs. DU = 30%. If the information about true demand comes almost as a surprise (here η = 8), we see, interestingly, that it matters less for heavier loads than for lighter loads. The aforementioned counterintuitive trend is clearly visible, in Figs. 3 and 4, specifically for η = 8, even more so for ∆ = 8, and is further magnified by larger DU. This is
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explainable – a larger mean load requires us to operate at near production capacity, thus, any order much larger than the mean size, will lead to a backlog, irrespective of whether we know about it 8 hours ahead or not. There is not enough slack in the “packed” schedule to adjust production at the last moment. Although, a larger DU also implies that many orders can also be much smaller than the mean, since backlogs, here, are 10 times as expensive as holding inventory (i.e., πkBO = 10πkIN V ), backlog costs influence closed-loop quality more than inventory costs do. In contrast, for smaller mean loads, we have more idle time in our schedules which can be utilized to meet an order with a larger deviation from its mean. Thus, if we know about the actual order, even a short time ahead, we have units available to start new batches. Please note that in all figures, it is incorrect to compare the absolute schedule costs across different loads (keeping other parameters the same). The costs are scaled within each subplot and, hence, a cost of 1 in, for example, Fig. 3A is different in absolute terms than a cost of 1 in Fig. 3C. Next, we see how changing the re-optimization time-step helps us mitigate the effect of demand uncertainty. In Figs. 3 and 4, it is evident that shorter re-optimization time-steps (equivalently– more frequent re-optimizations) lead to better closed-loop quality for any given demand uncertainty (described by both η and DU). This is so, inspite of using a long horizon (H = 48). Since, the horizon is long enough, the consequent role of re-optimization then is to tackle disturbances (due to uncertainty), and not new information. This, as hypothesized in Gupta and Maravelias (2016), is in contrast to the no uncertainty case, wherein a long horizon can completely compensate for infrequent re-optimizations. In addition, given that uncertainty is less important for heavier loads, the cost advantage from shorter re-optimization time-step, under heavier loads, reduces. When we are trying to design an online scheduling algorithm, an important aspect of it is the horizon length that we need to choose. In Figs. 5 and 6, we present how closedloop quality is affected by H, DU, and η. We see that the importance of employing any longer H than a threshold gradually diminishes. This threshold is smaller for lighter loads, and larger for heavier loads. This is because for heavier loads, multiple batches need to be run which need to be scheduled ahead – on the basis of available forecast. Looking closer in Figs. 5 and 6, we see the isolines do not cross-over each other. This leads to an interesting corollary: a longer H is not able to compensate for a higher DU or a smaller η! This makes sense, as a longer horizon is unlikely to help us with a surprise that we get at the start of the horizon. Finally, in Fig.7, we show how H and ∆ influence the cost of the implemented schedule in the presence of demand uncertainty (DU=30%, η = 8). A longer H, dependent on load, and a shorter re-optimization time-step, are beneficial for achieving better closed-loop quality. These general findings, confirm that the recommendations for choosing a good re-optimization algorithm presented by Gupta and Maravelias (2016) on the basis of their results for the no uncertainty case, are also valid when uncertainty is present.
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6. CONCLUSIONS First, the results presented in this work demonstrate that demand uncertainty is an important consideration in online scheduling. We validate, that it is indeed beneficial to invest in obtaining good forecasts (shortening η) as it greatly influences the closed-loop schedule. Second, we find that this, counter-intuitively, is more important for lightly loaded facilities than heavily loaded facilities, since heavily loaded facilities will run close to capacity irrespective of fluctuations in order sizes. Third, we find that a good choice of the horizon is influenced by the mean load at which we are operating, and not by the accuracy of the demand forecast, as has been sometimes covertly assumed by practitioners. Fourth, if the forecast is not perfect, feedback, acted upon in the form of re-optimization, can mitigate the effect of uncertainty, which is observed, in hindsight, as a disturbance. Re-optimization is very effective, especially, for underloaded facilities, as it can help exploit uncommitted resources, which are not necessarily present under heavy loads. Finally, we also see how certain recommendations based on studies assuming no uncertainty, can also be useful, even when uncertainty is present. The analysis, presented in this work, yields several insights, that are applicable to general rescheduling approaches, thus taking us one step forward towards synthesis of general strategies to obtain high-quality closedloop schedules. REFERENCES Chu, Y. and You, F. (2014). Moving horizon approach of integrating scheduling and control for sequential batch processes. AIChE Journal, 60(5), 1654–1671. doi: 10.1002/aic.14359. Cui, J. and Engell, S. (2010). Medium-term planning of a multiproduct batch plant under evolving multiperiod multi-uncertainty by means of a moving horizon strategy. Computers and Chemical Engineering, 34(5), 598–619. doi:10.1016/j.compchemeng.2010.01.013. Gupta, D. and Maravelias, C.T. (2016). On deterministic online scheduling: Major considerations, paradoxes and remedies. Computers & Chemical Engineering, 94, 312– 330. doi:10.1016/j.compchemeng.2016.08.006. Gupta, D. and Maravelias, C.T. (2017). A General StateSpace Formulation for Online Scheduling. Processes, 5(4), 69. Gupta, D., Maravelias, C.T., and Wassick, J.M. (2016). From rescheduling to online scheduling. Chemical Engineering Research and Design, 116, 83–97. doi: 10.1016/j.cherd.2016.10.035. Harjunkoski, I., Maravelias, C.T., Bongers, P., Castro, P.M., Engell, S., Grossmann, I.E., Hooker, J., M´endez, C.A., Sand, G., and Wassick, J.M. (2014). Scope for industrial applications of production scheduling models and solution methods. Computers and Chemical Engineering, 62, 161–193. doi: 10.1016/j.compchemeng.2013.12.001. Janak, S.L., Floudas, C.A., Kallrath, J., and Vormbrock, N. (2006). Production scheduling of a large-scale industrial batch plant. II. Reactive scheduling. Industrial and Engineering Chemistry Research, 45(25), 8253– 8269. doi:10.1021/ie0600590.
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