Proactive Scheduling under Uncertainty: A Parametric Optimization

8044

Ind. Eng. Chem. Res. 2007, 46, 8044-8049

Proactive Scheduling under Uncertainty: A Parametric Optimization Approach Jun-hyung Ryu Department of Chemical Engineering, POSTECH, Pohang, Korea

Vivek Dua Centre for Process Systems Engineering, Department of Chemical Engineering, UniVersity College London, London WC1E 7JE, United Kingdom

Efstratios N. Pistikopoulos* Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, London, SW7 2BY, United Kingdom

This paper presents a novel methodology using parametric programming techniques to solve scheduling problems under uncertainty. The uncertainty present in processing times and equipment availabilities is incorporated into scheduling models, which are then transformed to multiparametric mixed-integer linear programming (mp-MILP) problems. A solution procedure that is based on recently proposed state-of-the-art mp-MILP algorithms is then discussed. A key advantage of the proposed methodology is that the complete map of optimal schedules can be obtained as a function of various parameters; rescheduling can thus be performed via simple function evaluations without any further optimization. Therefore, the proposed methodology contributes to the construction of a proactiVe scheduling system. Numerical examples are presented to illustrate the potential of the proposed methodology.

1. Introduction Considerable effort has been exerted over the last two decades in the area of scheduling of the chemical processes. The effort is motivated by the significant impact of scheduling on the on-time delivery of products, as well as an efficient utilization of resources. Therefore, it is a subject that has been extensively studied in the process systems engineering community (see, for example, reviews by Reklaitis,1 Pantelides,2 and Shah3). Most previous research has focused on the “deterministic” scheduling problem, i.e., the case where process-, model-, or market-related parameters are assumed to be known and fixed. The resulting schedule is an off-line deterministic solution, because it is based on fixed parameters that are disconnected from the actual dynamic condition. In reality, variations typically exist (fluctuations in the quality of raw materials, uncertainty in product demands, equipment break-downs, etc.). In the presence of uncertainty, any “optimal” deterministic schedule may not be robust or even feasible. In the literature, three major methodologies have been proposed to address uncertainty in process scheduling. One simple and direct methodology to overcome the effect of uncertainty is to install additional equipment that may be used in the case of uncertainty.4,5 However, this is not practical, because it requires additional high investment costs, which result in low availability during normal conditions. Besides, the uncertainty involved in the operation of the additional units cannot be addressed. Another way is to manipulate the process condition in response to uncertainty based on a fixed operating strategy.6,7 * To whom correspondence should be addressed. Tel.: +44 (0)20 7594 6620. Fax: +44 (0)20 7594 6606. E-mail: [email protected].

An example of this type of manipulation is the adjustment of the operation parameters to use a fixed schedule. However, it cannot be applied generally, because there is not much room for modification in actual complex processes. The third is reactive scheduling, which is a way to address uncertainty issues in (typically on-line) scheduling applications in response to the uncertainty. The main idea of the reactive scheduling is to solve deterministic scheduling problems repeatedly whenever a variation occurs: new schedules are computed and implemented, based on newly realized parameters (for example, from on-line measurements). Studies on the reactive scheduling in the literature can be mainly summarized in the following two ways. First, most studies focus on how to minimize the effect of disturbances after their occurrences. Little attention has been given to the more-positive “proactive” approach, in the form of taking the initiative by acting rather than reacting to events such as predicting new optimal schedules in response to potential variations. Ishii and Muraki8 noticed the importance of predicting the process state in the reactive scheduling; however, their work leaves the question of how the prediction can be made and realized in the framework of the reactive scheduling system unanswered. Second, the computational issue is the major obstacle to improving the performance of reactive scheduling. Because of the need for constant recomputations, reactive scheduling approaches may become computationally expensive. Thus, some of the previous studies have attempted to accelerate the computational performance of solving the underlying deterministic scheduling problem, for instance, by shifting the starting time of the new schedule,9 limiting the search area of scheduling optimization,10 relaxing the constraints,11 or resolving the scheduling problem in a hierarchical way.12

10.1021/ie070018j CCC: $37.00 © 2007 American Chemical Society Published on Web 11/02/2007

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007 8045 Table 1. Parametric Programming at a Glance subjecta mp-MILP mp-MILP mp-MILP mp-MILP mp-QP mp-QP mp-QP mp-MIQP mp-MIQP mp-MIGOP p-MINLP p-MINLP p-MINLP mp-MINLP mp-MINLP mp-MINLP

reference Acevedo and Pistikopoulos and Dua16 Dua and Pistikopoulos17,18 Pistikopoulos et al.19 Bemporad et al.20 Pistikopoulos et al.19 Sakizlis et al.21 Dua et al.22 Sakizlis et al.21 Dua et al.23 Acevedo and Pistikopoulos24 Papalexandri and Dimkou25 Pertsinidis et al.26 Acevedo and Pistikopoulos24 Dua and Pistikopoulos27 Hene´ et al.28

Abbreviations: mp-MILP, multiparametric mixed-integer linear programming; mp-QP, multiparametric quadratic programming; mp-MIGOP, multiparametric global optimization programming; mp-MIQP, multiparametric mixed-integer quadratic programming; p-MINLP, parametric mixedinteger nonlinear programming; and mp-MINLP, multiparametric mixedinteger nonlinear programming. Table 2. Relevant Applications of Parametric Programming application area

reference

multi-objective optimization

Pistikopoulos and Grossmann,29 Pertsinidis et al.,26 and Papalexandri and Dimkou25 Pistikopoulos et al.19 Acevedo and Pistikopoulos,14 Dua and Pistikopoulos17 Pistikopoulos and Dua16 Ryu et al.30 Ryu and Pistikopouls31 Ryu et al.30

process planning bilevel programming zero wait scheduling supply chain management

parameter

Pistikopoulos14,15

a

on-line optimization process synthesis

Table 3. Nomenclature

Considering the scale of conventional scheduling problems, the heavy computation issue still poses a new challenge for reactive scheduling. From the aforementioned discussion of the previous studies, it can be concluded that there is a need for an alternative, proactive approach that would provide optimal reactive schedules in an inexpensive and fast way. Recently, Pistikopoulos and co-workers presented a series of parametric programming techniques and softwares (see Table 1). For example, Dua and Pistikopoulos18 described a state-of-the-art solution procedure for a mixed-integer linear programming (MILP) problem that includes binary variables and multiple right-hand side parameters that are bounded within prespecified intervals. The techniques have been applied to a variety of applications (some of which are listed in Table 2) for the process systems engineering problems under uncertainty. There is an interesting recent report by Ryu and Pistikopoulos.31 They addressed a zero-wait batch process scheduling problem, considering only processing time variation using the parametric programming techniques. In this paper, the scheduling problem under uncertainty will be revisited using parametric programming techniques motivated by their methodology. The remainder of this paper is organized as follows. Key issues of the scheduling problem are briefly addressed. The scheduling problem under uncertainty then is formulated as a parametric programming problem. Two numerical examples are presented to illustrate the potential of the proposed approach. The discussion and remarks are presented at the end of the paper.

Indices time slot (1, ..., N) product (1, ..., N) stage (1, ..., M)

i k j n(j) Pk,j θ1k,j θ2i,j L L θ1k,j , θ2i,j U U θ1k,j, θ2i,j A, E, F b, c, d

Ci,j yi,k wi,k,j x

definition

Parameters number of units in stage j processing time of product k in stage j uncertain processing time of product k in stage j uncertain availability parameter of ith time slot in stage j lower bound upper bound constant matrices constant vectors Variables completion time of ith product in stage j binary variable, if product k is made at ith time slot; otherwise, is 0 auxiliary variable for bilinear term yi,kθ1k,j continuous variables

2. Scheduling under Uncertainty via Parametric Programming In this section, we will present the key idea of using parametric programming techniques to address scheduling problems under uncertainty. A multiproduct batch scheduling model with unlimited intermediate storage (UIS) policy32 will be used, and uncertainty in processing time and equipment availability, which are the most frequently and widely recognized types, will be considered. 2.1. Scheduling Model. A scheduling model generally involves two types of constraints: sequencing and assignment. Sequencing constraints typically denote which products are produced in different time instances (slots) and in what sequence, whereas assignment constraints normally determine the completion times of various products at different stages, based on the selected sequence. Based on the notation in Table 3, the deterministic scheduling model applied in this study corresponds to the following MILP problem:

z ) min CN,M

(1)

N

s.t.

∑i yi,k ) 1

∀k

(1.1)

N

∑k yi,k ) 1

∀i

(1.2)

N

Ci,j g Ci,j-1 +

∑k yi,kPk,j

j > 1, ∀ i

(1.3)

i > n(j), ∀ j

(1.4)

N

Ci,j g Ci-n(j),j +

∑k yi,kPk,j

Equation 1.1 ensures that each product is assigned only one time slot in a sequence, whereas eq 1.2 ensures that only one product is assigned in every time slot. Equations 1.3 and 1.4 indicate that a product in a stage can only be processed if the product and the corresponding unit are available at the same time. The objective of problem 1 is to minimize a makespan CN,M, which is the completion time of the last product in the last stage.

8046

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007

2.2. Uncertainty in Processing Times. By considering the processing times in problem 1 (Pk,j) as varying parameters, problem 1 can be reformulated as the following multiparametric mixed-integer linear programming (mp-MILP) problem (after linearizations):

z ) min CN,M

(4)

N

∑i yi,k ) 1

s.t.

∀k

(4.1)

N

z ) min CN,M

(2)

∑k yi,k ) 1

∀i

(4.2)

N

∑i yi,k ) 1

s.t.

∀k

N

(2.1)

Ci,j > Ci,j-1 +

(2.2)

Ci,j g Ci-n(j),j +

∑k wi,k,j + θ2i,j

j > 1, ∀ i

(4.3)

N

∑k yi,k ) 1

∀i

N

Ci,j g Ci,j-1 +

∑k wi,k,j

j > 1, ∀ i

(2.3)

N

∑k wi,k,j

N

∑k wi,k,j + θ2i,j

i > n(j), ∀ j (4.4)

U (1 - yi,k) e wi,k,j θ1k,j - θ1k,j

∀ i, j, k

(4.5)

L (1 - yi,k) g wi,k,j θ1k,j - θ1k,j

∀ i, j, k

(4.6)

i > n(j), ∀ j

(2.4)

U (1 - yi,k) e wi,k,j θ1k,j - θ1k,j

∀ i, j, k

(2.5)

L U e θ1k,j e θ1k,j θ1k,j

∀ j, k

(4.8)

L θ1k,j - θ1k,j (1 - yi,k) g wi,k,j

∀ i, j, k

(2.6)

L U e θ2i,j e θ2i,j θ2i,j

∀ i, j

(4.9)

Ci,j g Ci-n(j),j +

∀ i, j, k

L U e wi,k,j e yi,kθ1k,j yi,kθ1k,j L U θ1k,j e θ1k,j e θ1k,j

∀ i, j, k

(2.7) (2.8)

where θ1k,j are the varying processing time parameters of product k at stage j; θ1Lk,j and θ1Uk,j are fixed (known) lower and upper bounds. 2.3. Uncertainty in Equipment Availabilities. By considering equipment availabilities in problem 1 as uncertain, problem 1 can be reformulated as the following mp-MILP problem:

(3)

min CN,M N

∑i

s.t.

yi,k ) 1

∀k

(3.1)

N

∑k yi,k ) 1

∀i

(3.2)

∑k yi,kPk,j + θ2i,j)

j > 1, ∀ i (3.3)

N

Ci,j > Ci,j-1 + (

∀ i, j, k

L U e wi,k,j e yi,kθ1k,j yi,kθ1k,j

(4.7)

where θ1k,j denotes the varying processing time parameters of product k at stage j and θ2i,j denotes the varying time parameters U are for the unavailability of time slot i in stage j; θ1Lk,j and θ2i,j U U their fixed (known) lower bounds and θ1k,j and θ2i,j are upper bounds, respectively. Problems 2, 3, and 4 correspond to the following general class of mp-MILP problems:

z(θ) ) min cTx + dTy

(5)

s.t. Ax + Ey e b + Fθ

(5.1)

θmin e θ e θmax

(5.2)

x ∈ X; y ∈ {0,1}m

(5.3)

x,y

where x denotes the continuous variables that correspond to the completion times and y are the binary variables that represent the selection of a specific schedule. The superscript “T” denotes a transpose. For the solution of problem 5, algorithms and software have been recently proposed by Pistikopoulos and co-workers. This paper takes advantage of the algorithm described in the work by Ryu and Pistkopoulos,31 which was based on the work of Dua and Pistikopoulos.18 For reference, they are described in the Appendix.

N

∑k yi,kPk,j + θ2i,j)

Ci,j g Ci-n(j),j + (

L U θ2i,j e θ2i,j e θ2i,j

i > n(j), ∀ j

∀ i, j

(3.4) (3.5)

where θ2i,j denotes the varying time parameters for the unavailL U and θ2i,j are the lower and ability of time slot i in stage j; θ2i,j upper bounds. A similar formulation can be derived and used in the following mp-MILP problem for the case when both the processing times and equipment availabilities are considered to be uncertain:

3. Illustrative Examples Based on the proposed approach with the aforementioned algorithm, two numerical examples are solved to illustrate the potential of the proposed approach. The first example is a scheduling problem that considers the processing time uncertainty, and the second considers the processing time and equipment availability uncertainty simultaneously. 3.1. Example 1. Consider a manufacturing process that consists of two stages (stage 1 and stage 2) for the production of three products (A, B, and C). The process has one unit for each stage. Table 4 shows the processing times of the products at each stage.

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007 8047

Figure 2. Process configuration for example 2. Figure 1. Result of example 1: (a) optimal sequences and (b) final parametric solutions. Table 4. Processing Time Data for Example 1

Table 6. Processing Time Data for Example 2 Processing Time (h/batch) product

mixing

reaction

separation

A B C D E

6 9 12 14 15

25 14 θ1 11 20

11 17 5 16 8

Processing Time, Pkj (h/batch) product, k

stage 1 (j ) 1)

stage 2 (j ) 2)

A B C

3 θ1 6

5 θ2 3

Table 7. Parametric Solutions of Example 2 Table 5. Parametric Solutions of Example 1 notation

critical region (CR)

CR1

4 e θ1 e 5, 5 e θ2 e 8; θ1 + 1 e θ2 5 e θ1 e 7, 6 e θ2 e 8 4 e θ1 e 7, 5 e θ2 e 8; θ1 + 1 g θ2 {4 e θ1 e 5, 3 e θ2 e 4}; {5 < θ1 < 7, 3 < θ2 e 4}, 5 e θ1 e 7, θ2 ) 3

CR2 CR3 CR4 CR5

optimal sequence

makespan (h)

notation

critical region (CR)

CR1 CR2

8 e θ1 e 15, 0 e θ2 e 17 8 e θ1 e 19, 17 e θ2 e 21; θ 1 + 2 e θ2 8 e θ1 e 20, 21 e θ2 e 30; θ1 + 2 e θ2, θ1 + θ2 e 42 12 e θ1 e 20, 22 e θ2 e 30; 42 e θ1 + θ2, θ1 + θ2 e 46 16 e θ1 e 20, 26 e θ2 e 30; 46 e θ1 + θ2 15 e θ1 e 19, 0 eθ21 e 21; θ2 e θ1 + 2 19 e θ1 e 20, 0 e θ2 e 22; θ2 e θ1 + 2

A-B-C

θ2 + 11

A-B-C A-B-C

θ 1 + θ2 + 6 θ1 + 12

A-B-C, B-A-C A-B-C, B-A-C, A-C-B

θ1 + 12

CR5

θ1 + 12

CR6

The processing times of product B at both stages (PB1 and PB2) are assumed to be varying parameters, denoted as θ1 and θ2, respectively, which are bounded as follows:

CR3 CR4

CR7

optimal sequence

makespan (h)

B-A-D-C-E B-A-D-C-E

80 63 + θ2

A-B-D-C-E

84

A-B-D-C-E

42 + θ1 + θ2

A-B-D-E-C

88

B-D-A-E-C

80

B-D-A-E-C

61 + θ1

4 h e θ1 e 7 h

have one unit (see Figure 2). The processing times of the products in the three stages are shown in Table 6. Uncertainty is involved in the processing time of product C at the reaction stage,

3 h e θ2 e 8 h

8 h e θ1 e 20 h

The application of our proposed algorithm in section 2.4 yields the results that are summarized in Table 5 and are graphically depicted in Figure 1. As can be seen in Table 5 and Figure 1, we can obtain information on how the optimal schedule changes as the parameters change. For some parameters, we can also determine that multiple optimal solutions exist (CR4 and CR5). The advantage of the proposed methodology is that these multiple optimal solutions for each parameter values are obtained without solving them multiple times. Furthermore, without the proposed methodology, it would be quite difficult to determine when solutions change as the parameters change. Note that, in this example, [A-B-C] is a robust optimal schedule, because it is an optimal solution for the entire range of variations. The proposed approach provides proactive information on the process operation, because a robust schedule is obtained before the start of the operation. From the computational viewpoint, the results were achieved by solving eight MILPs (via GAMS/CPLEX (from the work of Brooke et al.33) in 0.25 s) and nine mp-LP problems (in 0.25 s) on a SUN Ultrasparc workstation. 3.2. Example 2. Consider a manufacturing process that involves five products (A, B, C, D, and E) in the three stages (mixing, reaction, and separation). The reaction stage consists of two units (reactor1 and reactor2), whereas the other stages

and the availability of reactor1, which may be unavailable after the first task is finished and before the second task is started, for as long as 30 h:

0 e θ2 e 30 h The application of our proposed algorithm in section 2.4 yields the results that are summarized in Table 7 and are graphically depicted in Figure 3. The results were obtained by solving 33 MILPs (via GAMS/CPLEX (from the work of Brooke et al.33) in 1.32 s) and 12 multiparametric linear programming (mp-LP) problems (in 5.25 s) on a SUN Ultrasparc workstation, based on the software implementation that was developed by Bozinis et al.34 3.3. Remarks. Many important observations and remarks can be made from the results of the previously mentioned two examples. (i) The proposed methodology can derive a complete map of optimal schedules explicitly as a function of the parameters in corresponding critical regions. For the case of example 2, if no uncertainty is involved (i.e., the processing time of product C at the reaction stage is 8 h and all equipment operated normally), the optimal schedule is [B-A-D-C-E], with the corresponding makespan of 80 h (see point A in Figure 3, CR1). Then, assume that the processing time of product C at the reaction

8048

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007

that enforce its optimal schedule are changed in multiple modes, which is inevitable in practice, advanced methodologies such as those proposed allow us to prepare proactively against them. The proposed framework can be applied for the large-size problems. (v) If we know the exact parameter before the start of a process, we can always operate any process optimally. However, it is almost impossible to know it exactly before the start, from the author’s industrial experience. On the other hand, if we try to guess their values as a boundary, we can get fairly exact values. Our idea then is to determine how the schedule will be affected within the boundary in the least-expensive way. This has been the major motive of this manuscript, and we think that the proactive scheduling framework that we have proposed can be a good alternative. Figure 3. Results of example 2: (a) optimal sequences in the different critical regions and (b) the corresponding optimal makespans.

stage turns out to be 14 h and reactor1 is unavailable for 24 h. This new condition is substituted into the constraints that define the critical regions in Table 7, and it only satisfies CR3 (see point B in Figure 3, CR3). In this case, the optimal schedule corresponds to [A-B-D-C-E], with a corresponding makespan of 84 h, which is the same as that obtained by resolving the entire scheduling problem. (ii) The proposed methodology presents a complete map of all optimal schedules against potential occurrences of uncertainty before the start of the process, thereby constructing a proactive scheduling system. The reason for using the term “proactive” is that we can have full information on the optimal process schedules in response to the varying process conditions. We are not forced to wait for the specific values of varying parameters to be realized to obtain the solution of the corresponding scheduling problem (i.e., we are not forced to solve the deterministic scheduling problem exhaustively for different combinations of varying parameters). We solve corresponding multiparametric problems, which are solved in a finite number of iterations, using parametric programming. The advantage of having all information is quite clear, in that we can have more initiative by preparing for the variations. (iii) The change in the scheduling policy can be obtained without any need for further computations (i.e., re-solving the scheduling problem); it is achieved simply by function evaluations (see Table 5 for example 1 and Table 7 for example 2). It would be then asked if the proposed framework directly provides information on shifting from one schedule to the other, in response to the variation. The proposed methodology does not directly provide that information. On the other hand, we can still utilize the schedule with the condition that the same initial sequences are used. For example, we can shift from a schedule of [A-B-D-C-E] to that of [A-B-C-D-E] if we identify a variation after operating A and B. If an optimal schedule does not change entirely from one to the other (for instance, [A-B-D-C-E] and [B-D-A-E-C] in Example 2), the proposed framework can allow us to know the shift of one optimal sequence to the other in advance. (iv) This paper has illustrated the application of the proposed framework using relatively small-sized problems. It is thought that the contribution of this paper is properly highlighted using relatively small-sized problems. Although the sizes of the presented examples are relatively small, the insight conveyed in this paper can be significant in solving other types of scheduling problems, considering the presence of uncertainty. That is to say, when we accept the fact that parameter variations

4. Conclusion This paper has proposed a novel methodology for the construction of the proactive scheduling systems under uncertainty, using parametric programming techniques. The uncertainty present in processing times and equipment availabilities is incorporated into scheduling models, which are then transformed to multiparametric mixed-integer linear programming (mp-MILP) problems. A key advantage of the proposed methodology is that the complete map of optimal schedules can be obtained as a function of various parameters; thus, rescheduling can be performed via simple function evaluations without any further optimization. Therefore, a proactive scheduling system can be constructed using the proposed methodology. Numerical examples have been presented to illustrate the potential of the proposed approach. Appendix. A Solution Procedure of the mp-MILP Algorithm Step 1: Initialization. Define an initial region of θ, with the best upper bound zˆ*(θ) ) ∞, and obtain an initial integer structure yj by solving the initial MILP, whose θ are treated as variables.

z ) min cTx + dTy x,y,θ

(6)

s.t. Ax + Ey e b + Fθ

(6.1)

θmin e θ e θmax

(6.2)

x ∈ X; y ∈ {0,1}

m

(6.3)

where θ is a free variable. The solution of problem 6 is given by y ) yj. Step 2: Multiparametric LP Problem. For each region with a new integer structure, yj: (a) Solve the multiparametric LP subproblem (problem 7) to obtain a set of parametric upper bounds zˆ(θ) and corresponding valid regions, which will be called critical regions (denoted as CRs).

zˆ(θ) ) min cTx + dTyj

(7)

s.t. Ax + Eyj e b + Fθ

(7.1)

θmin e θ e θmax

(7.2)

x∈X

(7.3)

x

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007 8049

(b) If zˆ(yj, θ) e zˆ*(θ) for some region of θ, update the best upper bound function, zˆ*(θ), and the corresponding integer solutions, y*. (c) If infeasibility is found in some region CR, go to Step 3. Step 3: Master Problem. For each region CR, formulate and solve the deterministic MILP master problem by (i) treating θ as a variable bounded in the region CR, (ii) introducing an integer cut, and (iii) introducing a parametric cut z e zˆ*(θ). Return to Step 2 with new integer solutions and corresponding CRs.

z ) min cTx + dTy x,y,θ

(8)

s.t. Ax + Ey e b + Fθ

(8.1)

d y + c x e zˆ(θ)

(8.2)

T

T

i

yj - ∑ yj e |J| - 1 ∑ j∈ J j∈L

(8.3)

θ ∈ CRi ; x ∈ X; y ∈ {0,1}m

(8.4)

where J ) (y|y ) 1) and L ) (y|y ) 0), and |J| is the cardinality of J. Note that the inequality dTy + cTx e zˆ(θ)i represents a parametric cut for the identification of a new integer solution which has not been explored previously; the inequality ∑j∈J yj - ∑j∈L yj e |J| - 1 corresponds to integer cuts prohibiting previous integer solutions from appearing again. Step 4: Convergence. The algorithm terminates when the solution of the deterministic MILP problem is infeasible for each region CR. The final solution is given by the current upper bounds zˆ*(θ) in corresponding CRs. For details of the algorithm, the readers are referred to the work of Dua and Pistikopoulos.18 Literature Cited (1) Reklaitis, G. V. Perspectives of scheduling and planning of process operations. In Proceedings of the Process Systems Engineering Conference, PSE’91, 4th International Symposium on Process Systems Engineering, Montebello, Quebec, Canada, June 1991. (2) Pantelides, C. C. Unified Framework for the Optimal Process Planning Scheduling. In Proceedings of the Second International Conference on Foundations of Computer Aided Process Operations; Rippin, D. W. T., Hale, J., Davis, J. F., Eds.; CACHE Corp: Ausstin, TX, 1994. (3) Shah, N. Single and Multisite Planning and Scheduling: Current Status and Future Challenges. In Proceedings of Foundations of Computer Aided Process Operations (FOCAPO), Snowbird, UT, 1998; CACHE Corp.: Austin, TX, 1998. (4) Karimi, I. A.; Reklaitis, G. V. Deterministic Variability Analysis for Intermediate Storage in Noncontinuous Processes. Part I: Allowability Conditions. AIChE J. 1985, 31, 1516-1527. (5) Karimi, I. A.; Reklaitis, G. V. Deterministic Variability Analysis for Intermediate Storage in Noncontinuous Processes. Part II: Storage Sizing for Serial Systems. AIChE J. 1985, 31, 1528-1537. (6) Onogi, K.; Nishimura, Y.; Nakata, Y.; Inomata, T. An On-line Operating Control System for a Class of Combined Batch/Semi Continuous Processes. J. Chem. Eng. Jpn. 1986, 19, 542-548. (7) Hvala, N.; Strmcˇnik, S.; C ˇ erneticˇ, J. Scheduling of Batch Digesters According to Different Control Targets and Servicing Limitations. Comput. Chem. Eng. 1993, 17, 739-750. (8) Cott, B. J.; Macchietto, S. Minimizing the effects of batch process variability using online schedule modification. Comput. Chem. Eng. 1989, 13, 105-113. (9) Kanakamedala, K. B.; Reklaitis, G. V.; Venkatasubramanian, V. Reactive Schedule Modification in Multipurpose Batch Chemical Plants. Ind. Eng. Chem. Res. 1994, 33, 77-90. (10) Huercio, A.; Espun˜a, A.; Puigjaner, L. Incorporating on-line scheduling strategies in integrated batch production control. Comput. Chem. Eng. 1995, 19, S609-S614. (11) Ishii, N.; Muraki, M. A process-variability-based online scheduling system in multiproduct batch process. Comput. Chem. Eng. 1996, 20, 217234.

(12) Schilling, G.; Pantelides, C. C. General Algorithms for Reactive Scheduling of Multipurpose Plants. Presented at the AIChE Annual Meeting, Los Angeles, CA, 1997. (13) Vin, J. P.; Ierapetritou, M. G. A New Approach for Efficient Rescheduling of Multiproduct Batch Plants. Ind. Eng. Chem. Res. 2000, 39, 4228-4238. (14) Acevedo, J.; Pistikopoulos, E. N. A multiparametric programming approach for linear process engineering problems under uncertainty. Ind. Eng. Chem. Res. 1997, 36, 717-728. (15) Acevedo, J.; Pistikopoulos, E. N. An algorithm for multiparametric mixed integer linear programming problems under uncertainty. Oper. Res. Lett. 1999, 24, 139-148. (16) Pistikopoulos, E. N.; Dua, V. Planning under uncertainty; a parametric optimization approach. In Proceedings of the Third International Conference on FOCAPO; Pekny, J. F., Blau, G. E., Eds.; CACHE Corp.: Austin, TX, 1998; pp 164-169. (17) Dua, V.; Pistikopoulos, E. N. Optimization techniques for process synthesis and material design under uncertainty. Trans. Inst. Chem. Eng. 1998, 76 (Part A), 408-416. (18) Dua, V.; Pistikopoulos, E. N. An algorithm for the solution of multiparametric mixed integer linear programming problems. Ann. Oper. Res. 2000, 99, 123-139. (19) Pistikopoulos, E. N.; Dua, V.; Bozinis, N. A.; Bemporad, A.; Morari, M. On-line optimization via off-line parametric optimization tools. Comput. Chem. Eng. 2000, 24, 182-188. (20) Bemporad, A.; Morari, M.; Dua, V.; Pistikopoulos, E. N. The explicit linear quadratic regulator for constrained systems. Automatica 2002, 38, 3-20. (21) Sakizlis, V.; Dua, V.; Kakalis, N.; Perkins, J. D.; Pistikopoulos, E. N. The explicit control law for hybrid systems via parametric programming. Proc. Am. Control Conf. 2002, 1, 674-679. (22) Dua, V.; Bozinis, N. A.; Pistikopoulos, E. N. A new multiparametric mixed-integer quadratic programming algorithm. In Proceedings of ESCAPE11, Kolding, Denmark, May 27-30, 2001; Gani, R., Jørgensen, S. B., Eds.; Elsevier B.V. Science Publishers: Amsterdam, 2001. (23) Dua, V.; Papalexandri, K. P.; Pistikopoulos, E. N. A parametric mixed-integer global optimization framework for the solution of process engineering problems under uncertainty. Comput. Chem. Eng. 1999, 23, S19-S22. (24) Acevedo, J., Pistikopoulos, E. N. A parametric MINLP algorithm for process synthesis problem under uncertainty. Ind. Eng. Chem. Res. 1996, 35, 147-158. (25) Papalexandri, K. P.; Dimkou, T. I. A parametric mixed integer optimization algorithm for multi-objective engineering problems involving discrete decisions. Ind. Eng. Chem. Res. 1998, 37, 1866-1882. (26) Pertsinidis, A.; Grossmann, I. E.; McRae, G. J. Parametric optimization of MILP programs and a framework for the parametric optimization of MINLPs. Comput. Chem. Eng. 1998, 22, S205-S210. (27) Dua, V.; Pistikopoulos, E. N. Algorithms for the solutions of multiparametric mixed-integer nonlinear optimization problems. Ind. Eng. Chem. Res. 1999, 38, 3976-3987. (28) Hene´, T. S.; Dua, V.; Pistikopoulos, E. N. A Hybrid Parametric/ Stochastic Programming Approach for Mixed-Integer Nonlinear Problems under Uncertainty. Ind. Eng. Chem. Res. 2001, 41, 66-77. (29) Pistikopoulos, E. N.; Grossmann, I. E. Optimal retrofit design for improving process flexibility in linear systems. Comput. Chem. Eng. 1988, 12, 719-731. (30) Ryu, J.; Dua, V.; Pistikopoulos, E. N. A bilevel programming framework for enterprise-wide process networks under uncertainty. Comput. Chem. Eng. 2004, 28, 1121-1129. (31) Ryu, J.; Pistikopoulos, E. N. A novel approach to scheduling of zero-wait batch processes under processing time variations. Comput. Chem. Eng. 2007, 31, 101-106. (32) Ryu, J.; Lee, I. A new completion time algorithm considering an out-of-phase policy in batch processes. Ind. Eng. Chem. Res. 1997, 36, 5321-5328. (33) Brooke, A.; Kendrick, D.; Meeraus, A. GAMS: A Users Guide; GAMS Development Corp.: Washington, DC, 1996. (34) Bozinis, N. A.; Dua, V.; Pistikopoulos, E. N. A MATLAB Implementation of Multi-parametric Mixed-integer Linear Programming Algorithm; Imperial College: London, 2000.

ReceiVed for reView January 5, 2007 ReVised manuscript receiVed June 5, 2007 Accepted September 7, 2007 IE070018J