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Robust Process Scheduling under Uncertainty with Regret
Anton Friedl, Jiří J. Klemeš, Stefan Radl, Petar S. Varbanov, Thomas Wallek (Eds.) Proceedings of the 28th European Symposium on Computer Aided Process Engineering June 10th to 13th, 2018, Graz, Austria. © 2018 Elsevier B.V. All rights reserved. https://doi.org/10.1016/B978-0-444-64235-6.50161-3
Robust Process Scheduling under Uncertainty with Regret Chao Ning, Fengqi You Cornell University, Ithaca, New York, 14853, USA
Abstract In this paper, we address multipurpose batch process scheduling under uncertainty problems. A novel adaptive robust process scheduling model is proposed that effectively incorporates the minimax regret criterion. In batch process scheduling, regret is defined as the deviation of profit obtained over the scheduling horizon from the perfect-information schedule. The regret arises due to the limited knowledge on realized uncertainty, and serves as an important evaluation metric for decision making on process scheduling. In addition to the conventional robustness criterion, the proposed multi-objective scheduling model also simultaneously optimizes the worst-case regret to push the performance of the schedule towards the utopia one under perfect information. Two case studies on multipurpose batch process scheduling under product demand uncertainty are presented. Keywords: batch process scheduling, uncertainty, robust optimization, regret
1. Introduction Multipurpose batch plants play a critical role in producing high-value-added products, such as food, semiconductors and pharmaceuticals (Wassick et al., 2012). Batch process scheduling allocates limited recourses to different tasks over the scheduling horizon for a higher production efficiency and more profit. Therefore, scheduling of multipurpose batch process has attracted tremendous attention from both academia and industry (Chu and You, 2015). A large amount of literature is devoted to deterministic scheduling models which assume all the involved parameters are known for certainty (Méndez et al., 2006). However, in real practice, operators in batch plants need to make a schedule in the face of various uncertainties. As an inherent characteristic of batch processes, uncertainty could render the deterministic schedule suboptimal or even nonimplementable. To this end, extensive research efforts have been made towards the development of systematic methodologies for handling uncertainty in batch process scheduling. Balasubramanian and Grossmann (2004) proposed a multistage stochastic optimization approach to address batch scheduling problems. To hedge against uncertainty, a proactive scheduling strategy was developed based on robust optimization and multiparametric programming (Wittmann-Hohlbein and Pistikopoulos, 2013). Adaptive robust optimization (ARO) was applied for process scheduling such that some decisions can be made in a “wait-and-see” mode based upon uncertainty realizations through a data-driven robust optimization framework (Ning and You, 2017a). Although conventional ARO based scheduling model has a number of attractive features, it does not account for an important evaluation metric, known as regret (Dunning, 2016), in decision-making theory (Bell, 1982). In this work, we propose a novel adaptive robust batch process scheduling model explicitly accounting for regret.
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2. Bi-Criterion Adaptive Robust Scheduling Model In this section, we propose a bi-objective adaptive robust scheduling model with regret based on a deterministic scheduling model (Méndez et al., 2006; Yue and You, 2013): max min s.t.
Profit of selling products minus the production cost Regret over the scheduling horizon Mass balance constraints Batch size constraints Inventory constraints Demand satisfaction constraints Timing constraints
(1) (2) (3) (4) (5) (6) (7)
The resulting batch process scheduling model is formulated as a bi-objective adaptive robust mixed-integer program. The task assignment decision is made “here-and-now”, while other decisions, including continuous decisions on batch sizes, inventory levels, and timings of tasks, are made in a “wait-and-see” manner (Shi and You, 2016). The compact form of the scheduling model, denoted as (BCARO), is presented in (8).
min Worst-case cost = C w ( x ) x
min Worst-case regret = R w ( x ) x
s.t. Ax ≥ d, x ∈ R+n1 × Z n2 C w ( x ) max min cT x + bT y = u∈U y∈Ω ( x , u )
R w ( x ) max min cT x + bT y − Π ( u ) = u∈U y∈Ω ( x , u ) Ω ( x, u ) = y ∈ R+n3 : Wy ≥ h − Tx − Mu
{
min c xˆ + b yˆ Π (u ) = T
(8)
}
T
xˆ , yˆ
s.t. Axˆ ≥ d, xˆ ∈ R+n1 × Z n2 Wyˆ + Txˆ ≥ h − Mu, yˆ ∈ R+n3 where x represents the task assignment decision, and y includes other decisions, such as timing and batch size decisions. Notice that x is made “here-and-now”, whereas y is made in a “wait-and-see” manner. Π(u) is the best objective value that could be achieved by assuming that all the decisions are made in a “wait-and-see” mode. We could further reformulate the worst-case regret to reduce the number of levels, shown in Eq. (9). = R w ( x ) max min ( cT x + bT y ) − Π ( u ) u∈U y∈Ω ( x , u ) = max min ( cT x + bT y ) − min ( cT xˆ + bT yˆ ) xˆ , yˆ ∈Ξ ( u ) u∈U y∈Ω ( x , u ) T T T T =c x + max min b y − ( c xˆ + b yˆ )
(9)
xˆ , yˆ ∈Ξ ( u ) , u∈U y∈Ω ( x , u )
{
}
where Ξ ( u ) = xˆ ∈ R+n1 × Z n2 , y ∈ R+n3 : Axˆ ≥ d, Wyˆ + Txˆ ≥ h − Mu . By explicitly incorporating the minimax regret criterion, it pushes the performance of the resulting solution towards the utopia schedule under perfect information. We further propose a unified optimization model (BCARO-λ) shown in (10).
Robust Process Planning and Scheduling under Uncertainty with Regret min (1 − λ ) ⋅ C w ( x ) + λ ⋅ R w ( x ) x s.t. Ax ≥ d, x ∈ R+n1 × Z n2 C w ( x ) max min cT x + bT y = u∈U y∈Ω ( x , u )
min bT y − ( cT xˆ + bT yˆ ) xˆ , yˆ ∈Ξ ( u ) , u∈U y∈Ω ( x , u ) y ∈ R+n3 : Wy ≥ h − Tx − Mu
R w ( x ) =cT x + Ω ( x, u ) =
{
915
(10)
max
}
where λ is a parameter ranging from 0 to 1. (BCARO-λ) could provide a wide spectrum of solutions by changing the value of λ from 0 to 1. Specifically, it reduces to the conventional ARO model when λ equals to 0, and it becomes the ARO with minimax regret criterion when λ equals to 1. If the value of λ is chosen between 0 and 1, the proposed model makes a trade-off between the conventional robustness and minimax regret criteria. We develop (BCARO-ε) in (11) by transforming the objective on the worst-case regret into an extra constraint that specifies the upper bound of Rw(x). Similarly, (BCARO-ε) can reveal the trade-offs between the conventional robustness and minimax regret criteria in the scheduling problem. min C w ( x ) x
s.t. R w ( x ) ≤ ε Ax ≥ d, x ∈ R+n1 × Z n2 = C w ( x ) max min cT x + bT y
(11)
u∈U y∈Ω ( x , u )
min bT y − ( cT xˆ + bT yˆ ) y ∈ R+n3 : Wy ≥ h − Tx − Mu
R w ( x ) =cT x + Ω ( x, u ) =
{
max
xˆ , yˆ ∈Ξ ( u ) , u∈U y∈Ω ( x , u )
}
where ε is a parameter which bounds the worst-case regret from above.
3. Applications We consider two batch scheduling problems in section to illustrate the proposed methodology. The state-task-network of the first batch process is shown in Figure 1. This batch process involves one heating task, three reaction tasks, and one separation task (Kondili et al., 1993). This network involves three raw materials (A-C), four intermediates (Hot A, Impure E, and I1-I2) and two products (P1-P2). There are four equipment units: one heater, two reactors, and one separator. The scheduling results are shown in Figure 2. By comparing the two Gantt charts, we can observe that the assignment decisions determined by the two approaches are different. Specifically, Reaction 3 starts at a unique time point in the scheduling solution determined by the conventional robustness criterion, while Reaction 3 shares the same starting time point with Reaction 1 in the scheduling solution determined by the minimax regret criterion.
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Figure 1. State-task network of the multipurpose batch process in case study 1.
The values of worst-case profit and regret are listed in Figure 2. The scheduling solution determined by the minimax regret criterion has the lowest worst-case regret of $13.6, and only decreases the worst-case profit by 0.3% compared with solution under the conventional robustness criterion.
Figure 2. Gantt charts determined by (a) the conventional robustness criterion, and (b) the minimax regret criterion in case study 1.
We then apply the proposed scheduling approach to another process scheduling problem (Ning and You, 2017b). This case study is originated from an industrial multipurpose
Robust Process Planning and Scheduling under Uncertainty with Regret
917
batch process in The Dow Chemical Company (Chu et al., 2013), and the network is shown in Figure 3. This multipurpose batch process has one preparation task, three reaction tasks, two packing tasks and two drumming tasks. The equipment units include one mixer, two reactors, one finishing system and one drumming line (Chu et al., 2013). The scheduling horizon is one week, i.e. 168 hours, and there are 11 time points.
Figure 3. The state-task network of the multipurpose batch process in case study 2.
We solve the resulting scheduling problems with different values of parameter λ. The Gantt charts of the ARO solutions under the conventional robustness criterion and the minimax regret criterion are shown in Figure 4 (a) and (b), respectively. From the Gantt chart, we can observe that the assignment decisions are different, but the worst-case profit and regret remain unchanged for the two solutions. This is because of a different problem setup from Case study 1.
Figure 4. Gantt charts determined by (a) the conventional robustness criterion, and (b) the minimax regret criterion in case study 2.
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4. Conclusions In this paper, a novel multi-objective robust batch process scheduling model was proposed to account for the minimax regret criterion. Apart from the conventional robustness criterion, the worst-case regret was introduced as another objective to minimize the deviation of profit from the perfect-information schedule. A set of Paretooptimal schedules was generated to reveal the systematic trade-offs between the conventional robustness and minimax regret criteria. The results of two case studies showed that the proposed scheduling decision could make a systematic trade-off between the conventional robustness criterion and minimax regret criterion.
References J. Balasubramanian, I. E. Grossmann, 2004, Approximation to multistage stochastic optimization in multiperiod batch plant scheduling under demand uncertainty, Industrial & Engineering Chemistry Research, 43, 3695–3713. D. Bell, 1982, Regret in decision making under uncertainty, Operations Research, 30, 961-981. A. Ben-Tal A. Nemirovski, 2002, Robust optimization, Mathematical Programming, 92, 453-480. D. Bertsimas, D. B. Brown, C. Caramanis, 2011, Theory and applications of robust optimization, SIAM Review, 53, 464-501. Y. Chu, J.M. Wassick, F. You, 2013, Efficient scheduling method of complex batch processes with general network structure via agent-based modeling, AIChE Journal, 59, 2884-2906. Y. Chu, F. You, 2013, Integration of Scheduling and Dynamic Optimization of Batch Processes under Uncertainty: Two-stage Stochastic Programming Approach. Industrial & Engineering Chemistry Research, 52, 16851-16869. Y. Chu, F. You, 2015, Model-based integration of control and operations: Overview, challenges, advances, and opportunities, Computers & Chemical Engineering, 83, 2-20. I. R. Dunning, 2016, Advances in robust and adaptive optimization: algorithms, software, and insights, Massachusetts Institute of Technology, Massachusetts, USA. E. Kondili, C. C. Pantelides, and R. W. H. Sargent, 1993, A general algorithm for short-term scheduling of batch operations," Computers & Chemical Engineering, 17, 211-227. C.A. Méndez, J. Cerdá, I.E. Grossmann, I. Harjunkoski, 2006, State-of-the-art review of optimization methods for short-term scheduling of batch processes, Computers & Chemical Engineering, 30, 913-946. C. Ning, F. You, 2017a, A data-driven multistage adaptive robust optimization framework for planning and scheduling under uncertainty. AIChE Journal, 63, 4343-4369. C. Ning, F. You, 2017b, Data-driven adaptive nested robust optimization: General modeling framework and efficient computational algorithm for decision making under uncertainty. AIChE Journal, 63, 3790-3817. C. Shang, X. Huang, X., F. You, 2017, Data-Driven Robust Optimization Based on Kernel Learning. Computers & Chemical Engineering, 106, 464–479. H. Shi, F. You, 2016, A computational framework and solution algorithms for two-stage adaptive robust scheduling of batch manufacturing processes under uncertainty, AIChE Journal, 62, 687-703. J.M. Wassick, A. Agarwal, N. Akiya, J. Ferrio, S. Bury, F. You, 2012, Addressing the operational challenges in the development, manufacture, and supply of advanced materials and performance products, Computers & Chemical Engineering, 47, 157-169. M. Wittmann-Hohlbein, E.N. Pistikopoulos, 2013, Proactive scheduling of batch processes by a combined robust optimization and multiparametric programming approach, AIChE Journal, 59, 4184-211. D. Yue, F. You, 2013, Sustainable Scheduling of Batch Processes under Economic and Environmental Criteria with MINLP Models and Algorithms. Computers & Chemical Engineering, 54, 44-59.
Robust Process Scheduling under Uncertainty with Regret Chao Ning, Fengqi You Cornell University, Ithaca, New York, 14853, USA
Abstract In this paper, we address multipurpose batch process scheduling under uncertainty problems. A novel adaptive robust process scheduling model is proposed that effectively incorporates the minimax regret criterion. In batch process scheduling, regret is defined as the deviation of profit obtained over the scheduling horizon from the perfect-information schedule. The regret arises due to the limited knowledge on realized uncertainty, and serves as an important evaluation metric for decision making on process scheduling. In addition to the conventional robustness criterion, the proposed multi-objective scheduling model also simultaneously optimizes the worst-case regret to push the performance of the schedule towards the utopia one under perfect information. Two case studies on multipurpose batch process scheduling under product demand uncertainty are presented. Keywords: batch process scheduling, uncertainty, robust optimization, regret
1. Introduction Multipurpose batch plants play a critical role in producing high-value-added products, such as food, semiconductors and pharmaceuticals (Wassick et al., 2012). Batch process scheduling allocates limited recourses to different tasks over the scheduling horizon for a higher production efficiency and more profit. Therefore, scheduling of multipurpose batch process has attracted tremendous attention from both academia and industry (Chu and You, 2015). A large amount of literature is devoted to deterministic scheduling models which assume all the involved parameters are known for certainty (Méndez et al., 2006). However, in real practice, operators in batch plants need to make a schedule in the face of various uncertainties. As an inherent characteristic of batch processes, uncertainty could render the deterministic schedule suboptimal or even nonimplementable. To this end, extensive research efforts have been made towards the development of systematic methodologies for handling uncertainty in batch process scheduling. Balasubramanian and Grossmann (2004) proposed a multistage stochastic optimization approach to address batch scheduling problems. To hedge against uncertainty, a proactive scheduling strategy was developed based on robust optimization and multiparametric programming (Wittmann-Hohlbein and Pistikopoulos, 2013). Adaptive robust optimization (ARO) was applied for process scheduling such that some decisions can be made in a “wait-and-see” mode based upon uncertainty realizations through a data-driven robust optimization framework (Ning and You, 2017a). Although conventional ARO based scheduling model has a number of attractive features, it does not account for an important evaluation metric, known as regret (Dunning, 2016), in decision-making theory (Bell, 1982). In this work, we propose a novel adaptive robust batch process scheduling model explicitly accounting for regret.
914
C. Ning and .F. You
2. Bi-Criterion Adaptive Robust Scheduling Model In this section, we propose a bi-objective adaptive robust scheduling model with regret based on a deterministic scheduling model (Méndez et al., 2006; Yue and You, 2013): max min s.t.
Profit of selling products minus the production cost Regret over the scheduling horizon Mass balance constraints Batch size constraints Inventory constraints Demand satisfaction constraints Timing constraints
(1) (2) (3) (4) (5) (6) (7)
The resulting batch process scheduling model is formulated as a bi-objective adaptive robust mixed-integer program. The task assignment decision is made “here-and-now”, while other decisions, including continuous decisions on batch sizes, inventory levels, and timings of tasks, are made in a “wait-and-see” manner (Shi and You, 2016). The compact form of the scheduling model, denoted as (BCARO), is presented in (8).
min Worst-case cost = C w ( x ) x
min Worst-case regret = R w ( x ) x
s.t. Ax ≥ d, x ∈ R+n1 × Z n2 C w ( x ) max min cT x + bT y = u∈U y∈Ω ( x , u )
R w ( x ) max min cT x + bT y − Π ( u ) = u∈U y∈Ω ( x , u ) Ω ( x, u ) = y ∈ R+n3 : Wy ≥ h − Tx − Mu
{
min c xˆ + b yˆ Π (u ) = T
(8)
}
T
xˆ , yˆ
s.t. Axˆ ≥ d, xˆ ∈ R+n1 × Z n2 Wyˆ + Txˆ ≥ h − Mu, yˆ ∈ R+n3 where x represents the task assignment decision, and y includes other decisions, such as timing and batch size decisions. Notice that x is made “here-and-now”, whereas y is made in a “wait-and-see” manner. Π(u) is the best objective value that could be achieved by assuming that all the decisions are made in a “wait-and-see” mode. We could further reformulate the worst-case regret to reduce the number of levels, shown in Eq. (9). = R w ( x ) max min ( cT x + bT y ) − Π ( u ) u∈U y∈Ω ( x , u ) = max min ( cT x + bT y ) − min ( cT xˆ + bT yˆ ) xˆ , yˆ ∈Ξ ( u ) u∈U y∈Ω ( x , u ) T T T T =c x + max min b y − ( c xˆ + b yˆ )
(9)
xˆ , yˆ ∈Ξ ( u ) , u∈U y∈Ω ( x , u )
{
}
where Ξ ( u ) = xˆ ∈ R+n1 × Z n2 , y ∈ R+n3 : Axˆ ≥ d, Wyˆ + Txˆ ≥ h − Mu . By explicitly incorporating the minimax regret criterion, it pushes the performance of the resulting solution towards the utopia schedule under perfect information. We further propose a unified optimization model (BCARO-λ) shown in (10).
Robust Process Planning and Scheduling under Uncertainty with Regret min (1 − λ ) ⋅ C w ( x ) + λ ⋅ R w ( x ) x s.t. Ax ≥ d, x ∈ R+n1 × Z n2 C w ( x ) max min cT x + bT y = u∈U y∈Ω ( x , u )
min bT y − ( cT xˆ + bT yˆ ) xˆ , yˆ ∈Ξ ( u ) , u∈U y∈Ω ( x , u ) y ∈ R+n3 : Wy ≥ h − Tx − Mu
R w ( x ) =cT x + Ω ( x, u ) =
{
915
(10)
max
}
where λ is a parameter ranging from 0 to 1. (BCARO-λ) could provide a wide spectrum of solutions by changing the value of λ from 0 to 1. Specifically, it reduces to the conventional ARO model when λ equals to 0, and it becomes the ARO with minimax regret criterion when λ equals to 1. If the value of λ is chosen between 0 and 1, the proposed model makes a trade-off between the conventional robustness and minimax regret criteria. We develop (BCARO-ε) in (11) by transforming the objective on the worst-case regret into an extra constraint that specifies the upper bound of Rw(x). Similarly, (BCARO-ε) can reveal the trade-offs between the conventional robustness and minimax regret criteria in the scheduling problem. min C w ( x ) x
s.t. R w ( x ) ≤ ε Ax ≥ d, x ∈ R+n1 × Z n2 = C w ( x ) max min cT x + bT y
(11)
u∈U y∈Ω ( x , u )
min bT y − ( cT xˆ + bT yˆ ) y ∈ R+n3 : Wy ≥ h − Tx − Mu
R w ( x ) =cT x + Ω ( x, u ) =
{
max
xˆ , yˆ ∈Ξ ( u ) , u∈U y∈Ω ( x , u )
}
where ε is a parameter which bounds the worst-case regret from above.
3. Applications We consider two batch scheduling problems in section to illustrate the proposed methodology. The state-task-network of the first batch process is shown in Figure 1. This batch process involves one heating task, three reaction tasks, and one separation task (Kondili et al., 1993). This network involves three raw materials (A-C), four intermediates (Hot A, Impure E, and I1-I2) and two products (P1-P2). There are four equipment units: one heater, two reactors, and one separator. The scheduling results are shown in Figure 2. By comparing the two Gantt charts, we can observe that the assignment decisions determined by the two approaches are different. Specifically, Reaction 3 starts at a unique time point in the scheduling solution determined by the conventional robustness criterion, while Reaction 3 shares the same starting time point with Reaction 1 in the scheduling solution determined by the minimax regret criterion.
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C. Ning and .F. You
Figure 1. State-task network of the multipurpose batch process in case study 1.
The values of worst-case profit and regret are listed in Figure 2. The scheduling solution determined by the minimax regret criterion has the lowest worst-case regret of $13.6, and only decreases the worst-case profit by 0.3% compared with solution under the conventional robustness criterion.
Figure 2. Gantt charts determined by (a) the conventional robustness criterion, and (b) the minimax regret criterion in case study 1.
We then apply the proposed scheduling approach to another process scheduling problem (Ning and You, 2017b). This case study is originated from an industrial multipurpose
Robust Process Planning and Scheduling under Uncertainty with Regret
917
batch process in The Dow Chemical Company (Chu et al., 2013), and the network is shown in Figure 3. This multipurpose batch process has one preparation task, three reaction tasks, two packing tasks and two drumming tasks. The equipment units include one mixer, two reactors, one finishing system and one drumming line (Chu et al., 2013). The scheduling horizon is one week, i.e. 168 hours, and there are 11 time points.
Figure 3. The state-task network of the multipurpose batch process in case study 2.
We solve the resulting scheduling problems with different values of parameter λ. The Gantt charts of the ARO solutions under the conventional robustness criterion and the minimax regret criterion are shown in Figure 4 (a) and (b), respectively. From the Gantt chart, we can observe that the assignment decisions are different, but the worst-case profit and regret remain unchanged for the two solutions. This is because of a different problem setup from Case study 1.
Figure 4. Gantt charts determined by (a) the conventional robustness criterion, and (b) the minimax regret criterion in case study 2.
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C. Ning and .F. You
4. Conclusions In this paper, a novel multi-objective robust batch process scheduling model was proposed to account for the minimax regret criterion. Apart from the conventional robustness criterion, the worst-case regret was introduced as another objective to minimize the deviation of profit from the perfect-information schedule. A set of Paretooptimal schedules was generated to reveal the systematic trade-offs between the conventional robustness and minimax regret criteria. The results of two case studies showed that the proposed scheduling decision could make a systematic trade-off between the conventional robustness criterion and minimax regret criterion.
References J. Balasubramanian, I. E. Grossmann, 2004, Approximation to multistage stochastic optimization in multiperiod batch plant scheduling under demand uncertainty, Industrial & Engineering Chemistry Research, 43, 3695–3713. D. Bell, 1982, Regret in decision making under uncertainty, Operations Research, 30, 961-981. A. Ben-Tal A. Nemirovski, 2002, Robust optimization, Mathematical Programming, 92, 453-480. D. Bertsimas, D. B. Brown, C. Caramanis, 2011, Theory and applications of robust optimization, SIAM Review, 53, 464-501. Y. Chu, J.M. Wassick, F. You, 2013, Efficient scheduling method of complex batch processes with general network structure via agent-based modeling, AIChE Journal, 59, 2884-2906. Y. Chu, F. You, 2013, Integration of Scheduling and Dynamic Optimization of Batch Processes under Uncertainty: Two-stage Stochastic Programming Approach. Industrial & Engineering Chemistry Research, 52, 16851-16869. Y. Chu, F. You, 2015, Model-based integration of control and operations: Overview, challenges, advances, and opportunities, Computers & Chemical Engineering, 83, 2-20. I. R. Dunning, 2016, Advances in robust and adaptive optimization: algorithms, software, and insights, Massachusetts Institute of Technology, Massachusetts, USA. E. Kondili, C. C. Pantelides, and R. W. H. Sargent, 1993, A general algorithm for short-term scheduling of batch operations," Computers & Chemical Engineering, 17, 211-227. C.A. Méndez, J. Cerdá, I.E. Grossmann, I. Harjunkoski, 2006, State-of-the-art review of optimization methods for short-term scheduling of batch processes, Computers & Chemical Engineering, 30, 913-946. C. Ning, F. You, 2017a, A data-driven multistage adaptive robust optimization framework for planning and scheduling under uncertainty. AIChE Journal, 63, 4343-4369. C. Ning, F. You, 2017b, Data-driven adaptive nested robust optimization: General modeling framework and efficient computational algorithm for decision making under uncertainty. AIChE Journal, 63, 3790-3817. C. Shang, X. Huang, X., F. You, 2017, Data-Driven Robust Optimization Based on Kernel Learning. Computers & Chemical Engineering, 106, 464–479. H. Shi, F. You, 2016, A computational framework and solution algorithms for two-stage adaptive robust scheduling of batch manufacturing processes under uncertainty, AIChE Journal, 62, 687-703. J.M. Wassick, A. Agarwal, N. Akiya, J. Ferrio, S. Bury, F. You, 2012, Addressing the operational challenges in the development, manufacture, and supply of advanced materials and performance products, Computers & Chemical Engineering, 47, 157-169. M. Wittmann-Hohlbein, E.N. Pistikopoulos, 2013, Proactive scheduling of batch processes by a combined robust optimization and multiparametric programming approach, AIChE Journal, 59, 4184-211. D. Yue, F. You, 2013, Sustainable Scheduling of Batch Processes under Economic and Environmental Criteria with MINLP Models and Algorithms. Computers & Chemical Engineering, 54, 44-59.