3708
Ind. Eng. Chem. Res. 1997, 36, 3708-3717
Dynamic Optimization for Batch Design and Scheduling with Process Model Uncertainty Tarun K. Bhatia and Lorenz T. Biegler* Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Design and planning for multiproduct batch plants involves decisions from facility design, product scheduling, inventory planning, etc. These have been addressed for recipe-based processes with fixed stage processing times, although there is significant benefit in incorporating detailed dynamic process models while planning and addressing the overall problem simultaneously (Bhatia, T.; Biegler, L. T. Ind. Eng. Chem. Res. 1996, 35 (7), 2234-2246). However, incorporating processing decisions adequately requires sufficiently accurate nonlinear process models and methods that deal with process model uncertainty. This work simultaneously addresses dynamic processing and planning decisions for multiproduct batch plants, with uncertainty in process parameters. Instances of uncertain parameters are used to construct planning scenarios that are addressed through a multiperiod formulation. In addition, dynamic processing under uncertainty is addressed via a closed loop state feedback correction strategy. In addition, to provide an implementation advantage, parameters in the feedback law are determined in an open loop manner, as design variables that remain invariant across the scenarios in the multiperiod formulation. Introduction Batch processing is intrinsically dynamic and provides transient operating freedom. As a result, incorporating dynamic processing decisions during planning improves overall profitability. Planning decisions include facility design, batch scheduling, production, and inventory control. However, when process parameters vary or are not precisely known, an integration of processing and planning decisions must consider this uncertainty. For the overall processing and planning problem, uncertainty in the processing subproblem affects planning as well. Approaches for addressing these problems under uncertainty differ. For instance, uncertainty in process operation is better addressed through closed loop methods, which rely on monitoring a process and taking remedial measures on-line. One approach is the updating of an input profile via state feedback and correction. In contrast, planning involves broader decisions that are resolved once every planning period. In this sense, planning under uncertainty is usually performed in an open loop manner. This makes it challenging to address dynamic processing and planning decisions simultaneously, when process parameters are uncertain. This paper proposes one such approach. The main idea is to discretize and include closed loop dynamic process models for stages with uncertainty along with the planning constraints and to address each scenario through a multiperiod planning problem. This allows closed loop processing decisions to be resolved simultaneously with open loop planning decisions. The paper is organized as follows. Process integration in planning and approaches for addressing uncertainty are discussed in this section. The batch problem with uncertainty is presented in detail in section 2, followed by the proposed approach. Problem formulation and the model for feedback correction are developed in section 3. These include a novel treatment for addressing bounds on the control profiles. An example with a * To whom correspondence may be addressed: e-mail,
[email protected]; tel, (412) 268-2232; fax, (412) 268-7139. S0888-5885(96)00752-X CCC: $14.00
dynamic reaction stage is analyzed for different levels of uncertainty, in section 4. A larger example, involving dynamic reaction and distillation is solved next in section 5. Finally, in section 6 directions for future work are identified. 1.1. Dynamic Processing and Planning. Market driven specialty chemical production is often addressed in the flexible batch operating mode. At the heart of batch operations lie dynamic processing units, providing transient processing flexibility that is often ignored while planning. Instead, recipe based processes with fixed processing times and operating strategies are usually considered, e.g., Voudouris and Grossmann (1993) and Birewar and Grossmann (1989a,b). This practice is attributed to (i) a lack of accurate process models, (ii) large development time for process models of sufficient detail, and (iii) complexity of the resulting overall problem. There is also little incentive in pursuing detailed process models in view of short product lives. However, dynamic process interactions in planning can be significant for these units. Processing decisions such as input or control variable profiles, stage processing times, and final batch states at processing stages directly influence the planning problem. Processing times affect scheduling and sequencing of products. Final batch states of intermediate stages affect the initial condition of the batch at the next stage, and at the last stage, final batch state affects batch size, which in turn determines the number of product batches to be scheduled. As a result, input variable (or control) profiles influence batch state transitions and affect planning decisions. In a recent work (Bhatia and Biegler, 1996), dynamic processing tradeoffs are explored in design, scheduling, and inventory planning, for a special class of batch operations, i.e., flowshop plants with zero-wait (ZW) and unlimited intermediate storage (UIS) transfer policies and one equipment unit per stage. Processing decisions are resolved by discretizing the dynamic process models through collocation on finite elements. This transforms the infinite dimensional optimal control problem for processing, which requires solution of a system of differential and algebraic equations (DAE), to a finite © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3709
dimensional nonlinear program (NLP). Also, combinatorial scheduling issues are simplified by the class of problems considered, using the formulations of Birewar and Grossmann (1989b) and Voudouris and Grossmann (1993). These studies model the performance of a simplified scheduling problem in the planning objective, through a closed form expression. The overall NLP formulation for ZW policy, developed by Bhatia and Biegler (1996), is shown in model (M1) and is briefly explained below.
max ψprofit(CTj,Vj,Qi)
zij,uij,d,tij
s.t. z0ij - f(zij,uij,d) ) 0 ∀i,j g(zij,uij,d) e 0 ∀i,j ge(zfij,tij,d) e 0 ∀i,j h(zij,uij,d) ) 0 ∀i,j zLij e zij e zU ij uLij e uij e uU ij f z0ij ) f(zi,j-1 ) ∀i, j; j * 1
Bi ) f(zfij) ∀i, j ) J Vj g SijBi ∀i, j; j ∉ Jd Vj g SFijBSi ∀i, j; j ∈ Jd ni )
Qi ∀i Bi
Np
∑ NPRSik ) ni
∀i
k)1 Np
NPRSik ) nk ∑ i)1
∀k
NPRSii e ni - 1 ∀i Np
CTj )
Np
(nitij + ∑ SLikjNPRSik) ∑ i)1 k)1
∀j
CTj e H ∀j SLik,j + tk,j ) ti,j+1 + SLik,j+1 ∀ik, j; j < J Vj g 0, ni, Bi g 0, NPRSik g 0, ti,j g 0, SLikj g 0
(M1)
In model (M1), indices i and k denote a product (1 ... Np) and j denotes a processing stage (1 ... J). Each operation ij requires a processing time tij. Dynamic models contain state and control (or input) variable profiles (z(t) and u(t)) as well as variables d that cover continuous processing or design decisions. Constraints
g and h include inequality and equality path constraints and ge are constraints at the end of an operation. Variables zij0 and zijf denote initial and final conditions for state profile zij(t). As in Bhatia and Biegler (1996) the differential equations and dynamic profiles are discretized to algebraic equations and variables, respectively, by using collocation on finite elements. Further details of this discretization can be found in the work of Bhatia and Biegler (1996) and supporting references. Also, the overall profit objective is a function of stage cycle times (CTj), equipment design parameters (Vj), and production (Qi) and is maximized in (M1) subject to processing and planning constraints. Planning decisions include size and number of product batches Bi and ni, number of pairs of product batches NPRSik where a batch of product i is followed by one of product k, and SLikj, is the slack time enforced at stage j in ZW when a batch of product i is followed by product k. Stages modeled dynamically belong to the set Jd, equipment sizes are related to batch sizes through size factors Sij and SFij, and parameter H is the total planning horizon. Using this formulation from Bhatia and Biegler (1996), improved planning with dynamic input policies for a three-stage (one dynamic reactor) problem provides better and more time efficient product conversions than operating with constant input policies. With the best transient operating policy in the reactor, the overall profit in this example improves by about 16% for ZW and 11% for UIS policy, over operating with the best constant recipe. With more than one dynamic processing stage, processing times change and process loads shift among units to provide efficient schedules with reduced idleness. In a four-stage example, with best transient operating policies at the reaction and distillation stages, close to 7% improvement in the planning objective is realized over operating with the best time invariant recipes. Similar effects improve an inventory planning objective by 2.3%. Considerable savings can thus result from an integration of dynamic processing decisions in planning. 1.2. Uncertainty in Processing. The previous work demonstrates significant motivation to integrate processing and planning decisions for a special class of problems, with totally deterministic process models. However, plant model mismatch (both parametric and structural) could give suboptimal, and perhaps infeasible, performance and wipe out some of the incentive for incorporating processing and planning decisions. For instance, if the three-stage example of Bhatia and Biegler (1996) is subject to, say, a 10% variation in reaction constants for just one product, then profits would be lower since optimal input profiles corresponding to the nominal case are now implemented in a different situation. This lost profit is 3.5% for profit maximization with UIS and 3.9% for ZW. Consequently, this motivates addressing uncertainty in the process models when we consider problem (M1). Design and processing under uncertainty is reviewed by Pistikopoulos (1995). In this area, issues such as feasibility, flexibility, controllability, and operability are important and incorporated at the design stage. In a broad sense the area covers (i) a feasibility test that requires constraint satisfaction over a specified space of uncertain parameters (Halemane and Grossmann, 1983), (ii) a flexibility index associated with a given design that quantifies the portion of the uncertain space that passes the feasibility test (Swaney and Grossmann 1985a,b), and (iii) integration of design and operations
3710 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
where tradeoffs between flexible operations and fixed design costs are explored based on the first two concepts (Grossmann et al. 1983). In performing the flexibility analysis, many instances of the parameters need to be considered to represent the uncertainty space, and this can lead to a very expensive optimization problem. To circumvent this, critical points in this space can be generated to construct new scenarios or periods that are incorporated within multiperiod NLP or MINLP models. These problems can be large due to repeated model instances for each scenario, which are linked via a set of invariant design variables. 1.3. Multiperiod Planning Formulation. The overall dynamic processing and planning formulation (M1) presented by Bhatia and Biegler (1996) is a nonconvex NLP of the general form (M2).
max ψ(z,u,v;θ) ) 0 z,u,v
hz(z,u,v;θ) ) 0
s.t.
hp(z,u,v;θ) ) 0 z, u ∈ Z, v ∈ P, θ ) {θz,θp}
(M2)
The profit objective ψ is maximized subject to constraints hz, for the discretized dynamic process models, and hp, for the planning problem. Variables z,u ∈ Z cover processing decisions and v ∈ P, the planning decisions. Problem parameters include θz for processing and θp for the planning problem. Often these parameters are uncertain or subject to change. Discrete instances of uncertain process parameters combine to generate different planning scenarios, each similar to model (M2). If index p denotes each scenario as a period, then a multiperiod planning problem of the form (MP) results.
max φ ) zp,up,vp
s.t.
∑p ωpψp(zp,up,vp;θp)
hzp (zp,up,vp,d; θp) ) 0 ∀p hpp(zp,up,vp,d; θp) ) 0 ∀p d ∈ D, zp, up ∈ Z, vp ∈ P, θp ) (θzp,θpp)
(MP)
Subscript p on a term denotes its value in the pth scenario. A weighted objective, made of individual periodic objectives is optimized, where ωp is the probability of the pth scenario. Some aspects of the problem (d) such as facility design variables are invariant across scenarios and are treated in the set D. These variables couple the otherwise independent planning periods. 2. Problem Description and Approach The problem addressed in this work can be stated as a simultaneous incorporation of closed loop dynamic processing decisions under uncertainty, in the planning of multiproduct batch plants. We address this problem through a multiperiod formulation with a focus on the implementation of input profiles. An implementation of closed loop processing decisions that is invariant in the face of uncertainty is desired. In this study, uncertainty is considered in process parameters only, as instances, and not in the planning parameters such as market demands. However, this
approach could be extended to include uncertainty in planning parameters at the cost of a larger problem. 2.1. Background and Discussion. Grossmann and co-workers have integrated various design aspects in planning, excluding dynamic processing decisions (e.g., Voudouris and Grossmann, 1993). They, and others, have also addressed uncertainty in planning (e.g., Varvarezos et al., 1992; Pistikopoulos, 1995). Uncertainty in dynamic processing at single batch stages and for continuous processes has also been addressed (e.g., Terwiesch and Agarwal, 1994; Morari and Perkins, 1994). Pistikopoulos et al. (1996) address batch plant design with uncertainty in processing time and size factors in addition to market and production uncertainty, but without detailed process models. Chiotti et al. (1996) address uncertainty in process parameters via posynomial models in a multiperiod formulation with adaptive operating policies. Also, Mohideen et al. (1996) integrate closed loop performance in the design of dynamic systems under uncertainty. Open loop techniques for dealing with uncertainty require an uncertainty representation. This is either probabilistic, where uncertain parameters are considered with known probability distributions, or deterministic, where parameters assume instances from within a set. A deterministic representation of uncertainty space can be treated as a grid with each point representing a probable scenario. This becomes identical to the multiperiod design formulation in (MP) (Varvarezos et al., 1992). Closed loop techniques address uncertainty directly without any prior representation and rely on estimating a situation and taking appropriate corrective action. These techniques are preferred over resolving processing decisions for every possible operation at a processing stage. For instance, initial batch state uncertainty can be treated by on-line correction of a nominal profile. This mitigates the effect of model mismatch and reduces the need for a detailed uncertainty description in terms of probability bounds or functions, that in turn lead to a large problem. One way to correct for uncertainty is on-line reoptimization (Eaton and Rawlings, 1990), where current state estimates are used to optimize the remaining part of the profile successively. This approach could serve our purpose, if it were possible to generally derive feedback laws for the optimal profile and conveniently express them in the planning problem. A more general approach is to correct the nominal control profile through linear state feedback laws (Terwiesch and Agarwal, 1994). In aerospace applications similar techniques are used for correcting the path of space vehicles with limited on-board computing (Pesch, 1986). The linear state feedback law approach permits profile implementation and planning to be addressed together and is discussed later. Operating under uncertainty requires prior knowledge of the input processing decisions to be implemented in each scenario. Identifying the scenario is not easy during operation. One approach is to solve for the processing decisions as design variables that remain invariant across periods. This is the case with robust input policies. However, if it is possible to identify the scenario perfectly during operation, then a better approach is to implement processing decisions specific to the identified period. This corresponds to a perfect information situation. These perfect information (PI) and robust input (RI) cases can be modeled as special cases of the overall multiperiod planning formulation.
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3711
These are used to evaluate our proposed approach based on feedback correction. Perfect Information (PI). Here the input profile is allowed to change for each instance of an uncertain parameter (or period). In this study, we assume that process parameters do not change during a batch run or from one batch to another during any planning horizon. If process parameters change across batches of a product that are processed at different times in the campaign, such as an initial condition dependent on the season or source, then a multiperiod formulation can still be used. For brevity we consider the following perfect information problem (MPI) as a starting point.
max φ ) zp,up,vp
s.t.
Figure 1. Proposed approach.
∑p ωpψp(zp,up,vp;θp)
hzp (zp,up,vp,d;θp) ) 0 ∀p hpp (zp,up,vp,d;θp) ) 0 ∀p d ∈ D, zp, up ∈ Z, vp ∈ P, θp ) (θzp,θpp)
(MPI)
This is the multiperiod problem in (MP), with D as the domain of design variables which are equipment sizing decisions, d, and remain invariant over the periods, Z as the domain of the dynamic variables, and P as the domain of the planning variables. Robust Input (RI). Here the input profiles for all process models of a product stage operation are treated as additional variables invariant of periods. As will be discussed later, processing times for these operations must remain identical across all p periods as well. This leads to (MRI).
max φ ) zp,up,vp
s.t.
∑p ωpψp(zp, u, vp; θp)
hzp (zp,u,vp,d;θp) ) 0 ∀p
that include u(z,K), the correction expressions with K as the correction gains. These gains become optimization variables that are solved within a multiperiod problem formulation and then provide the desired implementation advantage in the face of uncertainty, while allowing closed loop monitoring of the process at the same time. As a result, we see that with perfect information, only equipment parameters Vj are treated as design variables in (MPI). For a robust input policy, the input policies (u) are additionally treated as design variables. From (MRI) these include the input profiles as well as their duration in all periods. Finally in feedback correction, closed loop process models hc(zp,up) ) 0 (e.g., see eq 1 below) are added to (MP). This shifts the implementation of processing decisions from the input policies to the correction parameters, which are invariant across periods. However, processing profiles can still vary across periods and yield improved performance over the robust input case. Feedback Correction. This approach recognizes that feedback is often used in operating policies to compensate for uncertainty. Here we treat parameters (K) in the feedback law and processing times as design variables as shown in (MFC), where u˜ are nominal input profiles and δz are deviations from nominal in the state profiles.
hpp (zp,u,vp,d;θp) ) 0 ∀p
max φ ) zp,up,vp
d ∈ D, uij(t), zp ∈ Z, tij, vp ∈ P, θp ) (θzp,θpp)
(MRI)
The perfect information and robust input problems correspond to the “wait and see” and “here and now” situations for planning under uncertainty. The difference in performance of these problems is known as the expected value of perfect information (EVPI) (Infanger, 1994), EVPI ) φMPI - φMRI. 2.2. Proposed Approach. Clearly, perfect information is an ideal situation that is hard to realize, and the more practical robust input approach is conservative. This motivates development of a practical approach that performs better than the robust input approach. We propose a feedback correction policy, where feedback gains in the closed loop correction of the control decisions are treated as invariant design variables. Feedback correction works with a nominal input profile that is corrected and implemented on-line. Deviations in the input and observed state profiles from respective nominal profiles are related by a linear time varying feedback law through correction gain profiles. Closed loop processing decisions are thus addressed via coefficients in the correction gain profiles, that are resolved in an open loop sense. This is further explained through Figure 1, where h(zp,up) ) 0 are dynamic process models and hc(zp,up) ) 0 are the closed loop feedback models
s.t.
∑p ωpψp(zp,u,vp;θp)
hzp (zp,up,vp,d;θp) ) 0 ∀p hpp (zp,up,vp,d; θp) ) 0 ∀p upij(t) ) u˜ ij(t) + KTij(t)δzpij(t)
d ∈ D, zp, up, Kij(t) ∈ Z, tij, vp ∈ P, θp ) (θzp,θpp) (MFC) For the three cases considered the following property holds. Property 1: For problems MPI, MFC, and MRI, φMPI g φMFC g φMRI is satisfied, where φ are objective function values. Proof: The robust input problem (MRI) is the most constrained of all. It corresponds to feedback correction (MFC) with K ) 0 and u ) u˜ and to (MPI) with period invariant constraints for the input policies. The solution to (MRI) is thus feasible for (MPI). Similarly, a solution to (MFC) is feasible for problem (MPI), as it corresponds to problem (MPI) with additional feedback constraints. On the basis of this argument, property 1 holds. Property 1 suggests that feedback correction will recover a part of the EVPI, without perfect information,
3712 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
while providing the same implementation advantage as the robust input case.
period uses this information to switch between uC and operating bounds appropriately, as follows.
3. Problem Formulation and Modeling
up(t) ) (1 - λp(t))[uC] + λp(t)[uU]
The overall problem formulation and modeling aspects are presented in this section. 3.1. Closed Loop Process Modeling. Dynamic process models are extended with additional linear time varying correction equations to allow closed loop process modeling. These express the implemented input profile for an operation u(t) in terms of correction applied to a reference or nominal profile u˜ (t). The correction is a function of observed state profile deviations δz(t) from a reference state trajectory vector and a dynamic gain matrix KT(t) and is of the form
u(t) ) u˜ (t) + KT(t)δz(t)
δ(t) ) uC(t) - uU
This ensures operating bounds are respected without compromising the feedback correction policy in (MFC). 3.3. Overall Formulation. Formulation (MP-1) shows the overall multiperiod problem with feedback correction and bounded inputs. Here certain stages are modeled dynamically (j ∈ Jd), and, of these, some stages have uncertain operations (j ∈ Ju), where Ju is a subset of Jd. Some products have uncertain operations at these stages (i ∈ Iu(j); j ∈ Ju).
max φ )
(1)
In addition, the reference input and state profiles (u˜ and z˜ ), must be consistent with the process dynamics. For this purpose an additional dynamic model corresponding to the reference operation needs to be included in the overall formulation for each uncertain productstage operation. To solve with the robust input strategy, it is not sufficient to constrain the control profile coefficients to be equal in all uncertain process models for an operation. The duration of these profiles must also be identical, suggesting that the uncertain operations last equally long under all situations. In addition, if a reference model is used for the feedback correction strategy, it must also last a duration equal to the operations it is used to correct. This also permits a convenient formulation of feedback correction constraints at collocation points. 3.2. Constraints on the Input. A real situation may need to address bounds on the input policies for reasons of safety or operability. Correction in the presence of active bounds for an input policy requires additional care. For an active period in this situation, the bound must be followed without need for correction. If feedback correction is still imposed, it might result in a compromise in the performance of other periods or a violation of the input constraints. Correction must thus be dropped for a period with an active input bound. This is similar to an override of the correction strategy in regions of bound activity, by a policy to obey the active bound. Correction override requires discrete switching. To achieve this in the simultaneous solution strategy, a continuous switching parameter λ is defined that takes discrete values in different regions with the following constraints. Consider an upper bound uU on the input profiles and uC as the feedback corrected input policy calculated from (1). We now apply the following formulation for each control profile.
∑p wpψp(CTpj,Vj,Qpi)
s.t. Closed loop process models c (z,u,v) ) 0 ∀p, i ∈ Iu(j), j ∈ Ju hpij
zL e z e zU Input constraint handling C δpij(t) ) upij (t) - uU ∀p, i ∈ Iu(j), j ∈ Ju
λpij(t) δpij(t) g 0 ∀p, i ∈ Iu(j), j ∈ Ju λpij(t) δpij(t) g δpij(t) ∀p, i ∈ Iu(j), j ∈ Ju 0 e λpij(t) e 1 ∀p, i ∈ Iu(j), j ∈ Ju Switching U
C upij(t) ) (1 - λpij(t))[upij (t)] + λpij(t)[u ] ∀p, i ∈ Iu(j), j ∈ Ju
Dynamic process models z hpij (z,u,v) ) 0 ∀p, ∀i, j ∈ Jd, j ∉ Ju
zL e z e zU uL e u e uU Process variables in planning
tpij ) tij, ∀p, i, j
(2)
λ(t) δ(t) g 0
0 f zpji ) f(zpij′ ) ∀p, i, j; j * 1, j ∈ Jd, j′ ) j - 1
λ(t) δ(t) g δ(t)
f Bpi ) f(zpij ) ∀p, i, j ) J
0 e λ(t) e 1 A similar formulation for valve switching is used by Gopal and Biegler (1997) . With these constraints, the switching parameter takes a value of 1 when δ is positive and the value zero when δ is negative. The expression for the implemented profile up(t) in any
(3)
Sizing
Vj g SijBpi ∀p, i, j; j ∉ Jd Vj g SFijBSpi ∀p, i, j; j ∈ Jd
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3713 Table 1. Parameter Values for Example 1 Operations
R1
β1
1 2 3 4 5 6 7 8 9 10
2.2 2.2 2.2 2.2 2.2 1.8 1.8 1.8 1.8 1.8
0.46 0.47 0.50 0.53 0.55 0.46 0.47 0.50 0.53 0.55
Figure 2. Example 1: Three stage example with dynamic reactor.
Scheduling
prod 2 R2 β2
prod 3 R3 β3
2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
Table 2. Number of Reactor Operations (+ Reference) in Example 1: Uncertainty in One Product
npi ) Qpi/Bpi ∀p, i
reactor periods prod 1 prod 2 prod 3
Np
∑ NPRSpik ) npi
prod 1
planning period
∀p, i
10 (+1)
1
1
planning periods
total reactor models
distinct models
10
30
12 (+1)
k)1
product B (labeled Cp) is maximized in a fixed time of 0.6 h.
Np
NPRSpik ) npk ∑ i)1
∀p, k
maxupij(t) Cp s.t.
NPRSpii e npi - 1 ∀p, i
z hpij (zpij,upij) ) 0 ∀p, i, j ∈ Jd
tpij ) 0.6 Np
CTpj )
(4)
Np
(npitpij + ∑ SLpik,jNPRSpik) ∑ i)1 k)1
∀p, j
CTpj e H ∀p, j SLpik,j + tpk,j ) tpi,j+1 + SLpik,j+1 ∀p, i, k, j; j < J Vj g 0, npi g 0, Bpi g 0, NPRSpik g 0, tpij g 0, SLpikj g 0 (MP-1) Here, hc and hz include algebraic formulations for feedback models (e.g., with (1)) and dynamic process models respectively. This formulation is now used to solve the examples of Bhatia and Biegler (1996), with uncertainty in process parameters. 4. Example 1. Single Dynamic Stage The first example involves a dynamic reactor and two recipe-based blenders (Figure 2). The reactor supports a competing reaction, with B as the desired product. Three similar products are processed in this plant, using an identical sequence of stages. Each reactor operation involves two reaction kinetic parameters (Ri and βi). 4.1. Single Product Uncertainty. First, uncertainty is considered in the reactor operation for just one product (i ) 1). Here, Jd ) {1} corresponds to the reactor along with Ju ) {1} and Iu(1) ) {1}. For product one, two reaction parameters R1 and β1 are considered uncertain, with five and three instances each. Only 10 of the 15 possible operations for product one are considered likely and lead to ten periods in the multiperiod planning problem, as shown in Table 1. Each period contains three products and three reactor models. However, only the distinct operations for the process models (Table 2) need to be included in (MP-1). Solution Strategy. To provide a good initialization to solve (MP-1), each of the individual reactor operations are optimized separately by solving problems (4) for each reactor operation, where production of desired
Reactor operations for the first product are then combined in a multiperiod corrected reactor problem, where expected reactor conversion is maximized subject to constraints on the correction gains and the duration of each operation. This problem (5) is initialized with solutions of the individual reactor problems (4).
maxupij(t) s.t.
∑p ωpCp
c hpij (zpij,upij,Kpij) ) 0 ∀p, i ∈ Iu(j), j ∈ Ju
Kpij(t) ) K1ij(t) ∀p, i ∈ Iu(j), j ∈ Ju tpij ) 0.6
(5)
Next, individual planning periods are optimized using initializations from the previous step followed by the multiperiod planning problems. The perfect information case (MPI) is solved first. This is followed by the robust input model (MRI) and the feedback correction model (MFC). Results and Discussion. Three planning problems, consistent with the ones in Bhatia and Biegler (1996) were solved for this system. In problem P1, we minimize an operating cost, proportional to stage cycle times, for meeting specified production requirements of each product within an existing facility. In problem P2, we maximize revenues from operating a given facility for a specified planning horizon. In P3, we maximize an overall profit objective that is a function of fixed design costs, revenues and operating costs. In this example we considered both ZW and UIS transfer policies, as in Bhatia and Biegler (1996). The results for individual periods in problem P1 appear in Table 3, for each of the perfect information (MPI), feedback correction (MFC), and robust input models (MRI). The operating cost for each period with feedback correction is close to the corresponding cost for the perfect information case. Also, this is a significant improvement over the robust input case while offering
3714 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 Table 3. Periodic Operating Costs ($) for Problem P1sMinimize Operating Cost p ψMPI p ψMFC p ψMRI
period 1
period 2
period 3
period 4
period 5
period 6
period 7
period 8
period 9
period 10
24 888.84 24 888.93 24 901.24
24 939.60 24 939.66 24 954.18
24 806.11 24 806.20 24 829.84
25 233.10 25 233.20 25 269.43
25 327.08 25 327.26 25 373.64
24 269.25 24 269.41 24 388.40
24 321.48 24 321.67 24 433.78
24 475.53 24 475.65 24 569.91
24 625.90 24 626.02 24 706.00
24 724.23 24 724.37 24 796.70
Table 4. Weighted Objectives ($) for Problems P1, P2, and P3 policy
problem
MPI
MFC
MRI
EVPI ($)
% recovered
ZW
min op-cost max revenue max profit min op-cost max revenue max profit
24 761.11 48 992.61 28 251.28 21 692.01 49 525.63 34 055.09
24 761.23 48 991.76 28 249.69 21 720.68 49 489.31 33 989.93
24 822.31 48 883.15 28 108.59 21 797.05 49 364.26 33 825.78
61.20 109.46 142.69 105.03 161.37 216.10
99.8 99.2 98.9 72.7 77.5 71.6
UIS
Table 5. Weighted Objectives for Single Product Uncertainty and Constrained Input policy
problem
MPI
MFC
MRI
EVPI ($)
% recovered
ZW
min op-cost max revenue max profit min op-cost max revenue max profit
24 891.98 48 758.56 27 946.18 21 949.34 49 171.41 33 548.78
24 892.07 48 755.13 27 945.97 21 971.34 49 139.01 33 490.49
24 917.32 48 713.24 27 887.09 22 022.07 49 076.81 33 402.70
25.34 45.32 59.09 72.73 94.60 146.08
99.6 92.4 99.6 69.8 65.8 60.1
UIS
a similar implementation advantage. The results for the weighted objective function, which reflect expected performance for all planning scenarios considered, are presented in Table 4. The EVPI reported is the difference between the performance of the perfect information case and the robust input case.
EVPI ) |φ(MPI) - φ(MRI)| % recovered )
|φ(MFC) - φ(MRI)| × 100% EVPI
This reflects the potential for improvement that exists with perfect information. A part of this potential can be recovered or realized by the feedback correction approach. The EVPI reflected for the ZW policy is lower than UIS, as it is more constrained by the requirement of zero wait for a batch between stages. For a similar reason, almost all of this potential is realizable by (MFC). For all problems (P1, P2, and P3) with the ZW policy, close to 99% of the EVPI is recovered with feedback correction. The value of EVPI is largest for the profit maximization problem (P3), which involves more planning decisions than any other problem. For ZW problems, all reactor operations have an equal duration of 0.6 h. A further reduction in time provides no incentive as the third stage (a blender) becomes the bottleneck. For all problems P1, P2, and P3, final conversions and hence batch sizes for the uncertain product operation in each period are almost the same for the MPI and MFC cases, differing in only two cases (largest uncertainty) by a mere 0.2%. In all these problems, achieving a higher reactor conversion contributes to improving the objective. For the (MRI) formulation, all periods reflect a lower product conversion (up to 1.5%) than with (MPI). The UIS problems, being more relaxed than the ZW problems, are characterized by a larger EVPI for all problems (Table 4). Here, any improvement in a stage operation directly leads to lower operating costs or higher revenues. In the ZW case, this effect is masked to some extent by the presence of idle time in the schedule. The improvements in each period with feedback correction, though aided by the assumption of unlimited storage at zero cost, are still unable to close
Table 6. Number of Reactor Operations (+ Reference) in Example 1: Uncertainty in All Products reactor periods prod 1 prod 2 prod 3 2 (+1)
2 (+1)
2 (+1)
planning periods
total reactor models
distinct models
8
24
6 (+3)
as high a portion of EVPI present in the UIS problems. This may be due to the MPI model having reactor operations of different durations; all operations in the MFC and MRI cases are forced to have equal duration. Here the MRI operations last 2.6% longer with no clear trend for improvement in the reaction conversions. Constrained Input. An upper bound of 5.0 was imposed on the transformed temperature profile for each reactor operation. The switching formulation successfully handled the upper bound. In no period for any problem was the constraint violated. The bounds on the input lead to a deterioration in the weighted objective functions for each planning problem. The EVPI values for planning problems are now smaller, 41% of the EVPI in the corresponding unconstrained problems for ZW and correspondingly about 65% for UIS. The recovery of EVPI with feedback correction is over 92% for all ZW problems and over 60% for the UIS problems. Except for P3 in ZW, these figures are lower than the corresponding figures for an unconstrained policy. Although the input correction is somewhat specific to each policy in view of the override option, the effect of the input constraint is to partially reduce the recovery of EVPI in the constrained case (Table 5). Although the EVPI figures are not large compared to the weighted objective function values of the corresponding problems, the trends for closure with the proposed approach are encouraging. Also, so far uncertainty is considered only in one product. Next, uncertainty is extended to reactor operations for other products and the approach is further analyzed. 4.2. Multiple Product Uncertainty. Uncertainty is now extended to reactor operations for all products. In this case, Jd ) {1}, Ju ) {1}, and Iu(1) ) {1,2,3}. One reaction parameter is considered uncertain with two instances, for each product, leading to eight planning periods, with three reactor models in each. As before, only distinct reactor models (Table 6) are included in the problem with relevant processing information used
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3715 Table 7. Parameter Values for Multiple Product Uncertainty
Ri βi
1
2
3
4
5
6
ref 1
ref 2
ref 3
1.8 0.5
2.2 0.5
1.8 0.4
2.2 0.4
2.8 0.5
3.2 0.5
2 0.5
2 0.4
3 0.5
in each planning scenario. The uncertain parameter values for all periods are shown in Table 7. Solution Strategy. The initialization scheme for this case also used (4) and (5). All individual operations were solved first to initialize solution of the multiperiod reactor problems, this time one for each product. Planning problems were then resolved from the previous solutions leading up to the multiperiod planning problems. Results and Discussion. Results for weighted objectives of problems P1, P2, and P3 are shown in Table 8. The EVPI values in all problems for both ZW and UIS policy are about twice as large when compared to the single product uncertainty case with ten periods. Trends for closure are similar. For ZW problems, over 99% of EVPI is recovered. For UIS the closure is over 68% for problems P1 and P3 and 87% for problem P2. For ZW policy, as before, periods for each product in the MFC case reflect a conversion extremely close to those in the MPI case. For MRI cases, conversions are slightly different (up to 1.2% for product 1, 1.4% for product 2, and 0.2% for product 3). This is true for all problems P1 and P2. For problem P3, the third product reflects a larger deterioration in one of its periods of 0.9%. For the UIS policy, no clear trend is observed. Reactor operations in the MFC and MRI cases respond with less efficient net production of the desired product. The operation times for each product are larger for the MFC than for the MRI case. It is observed that values for EVPI are now larger with uncertainty present in all products. For improvement of the overall profit objective in problem P3, EVPI as a percentage of the MPI objective function is 1.47% for UIS and 1.1% for ZW policies for this problem. Constrained Input. An upper bound was imposed on the reactor input profiles. Weighted objectives for each planning problem with uncertainty in all products and an input constraint are shown in Table 9. Presence of the input constraints lead to a smaller value of EVPI for all problems. Trends for EVPI recovery are similar to those observed in the unconstrained case. For ZW, the EVPI recovery is almost complete. An issue that has not been addressed here so far is one of implementation of scheduling decisions or the plan for different periods. In a scenario where any period is equally likely and it is not known beforehand
which period will turn up next, it would be hard to decide on a production plan. On the other hand, Iyer and Grossmann (1997) consider operational decisions including startup and shutdown of units in different planning periods for utility systems with piecewise constant demand variations. They address the multiperiod problem through a decomposition approach but for known variation of demands in planning periods. Finally consider the situation where uncertainty comes across different production cycles but does not vary within each cycle. In this situation, implementing any robust policy, either robust input or feedback correction, and comparing the observed state profiles with those expected in each period, would help identify θp for each product, implying perfect information. Subsequently, the solution of the perfect information period corresponding to the identified θp could be implemented and performance close to optimal could be achieved. The only EVPI loss would be dictated by the part of the production cycle dedicated to identifying the period. As a result, the proposed approach is more relevant in the case where the periods for each uncertain operation change within a production cycle. In such a scenario, the production plan would require implementation with some checks required. For instance, in problem P2, the production plan would use feedback correction to process all products until the minimum requirements for all are met. The remaining part could be dedicated to making the more expensive product. The UIS policy is sequence independent and the products could be processed in any order. For the ZW policy it turns out that for this problem, all products required processing for an equal duration in each stage. This renders the ZW case to be sequence independent too. For the more general ZW case, where all products are not identical in their time requirements at all stages and the problem is sequence dependent, the robust duration constraints still ensure each period of any operation has the same processing time. In this case, the number of product batches would differ across planning scenarios. For this single dynamic stage example the recovery of EVPI, through feedback correction, is significant, even though the EVPI is small compared to the benefits of integrating dynamic processing and planning decisions. To extend the analysis to more than one uncertain stage we consider the two dynamic stage example given by Bhatia and Biegler (1996). 5. Example 2. Two Dynamic Stages The four-stage example from Bhatia and Biegler (1996) is considered next. It involves two dynamic
Table 8. Weighted Objectives for P1, P2, and P3 with Multiproduct Uncertainty policy
problem
MPI
MFC
MRI
EVPI ($)
% recovered
ZW
min op-cost max revenue max profit min op-cost max revenue max profit
24 750.12 49 136.97 28 269.46 21 657.04 49 589.75 34 136.46
24 750.90 49 135.49 28 269.46 21 724.18 49 551.58 33 976.38
24 869.48 48 878.63 27 959.01 21 875.53 49 282.72 33 635.12
119.36 258.34 310.45 218.49 307.03 501.34
99.3 99.4 100 69.3 87.6 68.1
UIS
Table 9. Weighted Objectives for Uncertainty in All Products with Constrained Input policy
problem
MPI
MFC
MRI
EVPI ($)
% recovered
ZW
min op-cost max revenue max profit min op-cost max revenue max profit
24 881.47 48 842.92 27 963.24 21 913.02 49 213.75 33 636.68
24 881.83 48 842.01 27 963.24 21 963.71 49 196.47 33 518.00
24 951.76 48 698.57 27 767.20 22 084.03 49 017.23 33 251.62
70.29 144.35 196.04 171.01 196.52 385.06
99.5 99.4 100 70.4 91.2 69.2
UIS
3716 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 Table 10. Number of Process Models (+ Reference) in Example 2 total operations processing planning process distinct stage prod 1 prod 2 prod 3 periods models models reaction 2 (+1) 2 (+1) 2 (+1) distillation 4 (+1) 4 (+1) 4 (+1)
4 4
12 12
2 (+1) 12 (+3)
stages, a reactor that benefits from a dynamic temperature profile and a batch distillation column that allows a dynamic reflux ratio profile. Column dynamics are accounted using the shortcut batch distillation model (Logsdon et al., 1990) and involves equilibrium relations, Gilliland’s correlation, and Underwood’s equations apart from the residual collocation equations. Here we will consider the overall profit maximization problem (P3) for both the constrained and unconstrained input cases. Processing for all products is identical in the reactor, where a binary feed is processed to give a six-component mixture via a mechanism involving three reversible reactions. The next two recipe-based stages cool the batches and remove one waste component. The feed to the final column stage is treated as a binary mixture of the desired component and other lumped components. In the column, products of different purity are separated. These different purities distinguish the three products. Detailed information on this example is given in the work of Bhatia and Biegler (1996). Uncertainty is considered now in both dynamic stages. The situation considers two feeds. These lead to a difference in one reaction parameter for the reactor operations. Final batch condition after the reactor from the two feeds will thus be different. We assume recipes, which involve cooling and waste separation in the second and third stage, do not require any modification. The initial binary composition for the column operations could be identical for the two feeds, as both of the fivecomponent mixtures at the end of the third stage could result in identical binary compositions. The column feed is one uncertain parameter. Additionally, the different batches are considered with two different values for relative volatility. The uncertainty representation for this situation involves two dynamic stages, both uncertain and hence Jd ) Ju ) {1,4}. All product operations in both stages are uncertain, hence Iu(1) ) Iu(4) ) {1,2,3}. There is one uncertain parameter for the reactor operations with two instances arising from the difference in feed. This gives two different feeds for the column operations of all products. In addition, relative volatility in the column parameter is considered uncertain with two instances. Each product operation differs only in the purity specification imposed at the column. Consequently, there are a total of four planning periods, and the distinct process models are accounted in Table 10. With a reaction and distillation model each for the three products, the problem has a total of 12 reactor
and 12 column models, in addition to the reference models. However only two reactor models are distinct for this situation and only one reference reactor model is required. Solution Strategy. The solution strategy also relies on subproblems similar to (4) and (5) to provide good initializations that lead up to the final overall formulation (MP-1). This example was solved with an upper bound constraint imposed on both input policies, the temperature in the reactor and the reflux ratio profile in the distillation column. Results and Discussion. Weighted planning objectives for the profit maximization problem for ZW policy are presented in Table 11. It is observed that EVPI values in this example are now considerably larger. For the unconstrained case, EVPI is 8.2% of the weighted profit objective for the perfect information case. This is larger than the 7% incentive in simultaneously incorporating dynamic processing and planning decisions for this example. EVPI recovery for this case with the proposed approach is 93%. For two dynamic stages with uncertainty, the proposed approach is able to recover a considerable portion of a more significant EVPI. For the constrained case, EVPI is only slightly smaller but the recovery with the feedback approach is still 92.9%. This example reflects that the effect of uncertainty, in terms of the performance gap between the perfect information and the robust input case, becomes more pronounced with the number of uncertain stage operations as compared to the number of uncertain products. For example 2, this is of the same order as the savings that were realized with integrating processing decisions in planning. The proposed approach is promising, in that it is able to realize most of the predicted gap while providing the same implementation advantage as with the robust input case. This practical approach can help realize the true potential of processing decisions in planning. Computational Results. All problems in this work were solved using the NLP solver CONOPT through the GAMS modeling system (Brook et al., 1988). Typical problem size and solution time data are reported in Table 12. Reported CPU times are for an HP9000 work station but it must be noted that the problems were solved using good initializations, through sequential solution of the MPI, MFC, and MRI models. 6. Conclusions and Future Work The proposed approach in this work can be effective in realizing the potential of incorporating processing and planning decisions under process model uncertainty. Here dynamic models were included and discretized to algebraic form using collocation on finite elements. It
Table 11. Example 2, Weighted and Individual Objectives in Each Period for Uncertainty in All Products (unconst)
MPI
MFC
MRI
EVPI ($)
% recovered
φw ) ∑iwφi φ1 φ2 φ3 φ4
18 517.59 17 551.35 20 054.70 16 996.85 19 467.46
18 411.74 17 516.15 19 876.26 16 893.12 19 361.45
16 996.85 16 996.85 16 996.85 16 996.85 16 996.85
1520.74 554.50 3057.85 0 2470.61
93.04 93.7 94.2
(const)
MPI
MFC
MRI
EVPI ($)
% recovered
18 448.35 17 479.23 19 978.33 16 934.47 19 401.39
18 341.26 17 440.71 19 801.03 16 833.67 19 289.64
16 934.47 16 934.47 16 934.47 16 934.47 16 934.47
1513.88 544.76 3043.86 0 2466.92
92.93 92.9 94.2
φw φ1 φ2 φ3 φ4
) ∑iwφi
95.7
95.5
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3717 Table 12. Example 2, Problem Size and CPU Times model
equations
variables
CPU time (min)
MPI MFC MRI
3294 3774 3594
3091 3319 3091
14.7 343 5
should be noted that this is only an approximation to the differential equations. However, in Bhatia and Biegler (1996) we have carefully verified the accuracy of the optimal profiles. Moreover, the problem formulations can be extended to include an automatic determination of a finite element grid that leads to accurate profiles. A description of this approach is provided in the work of Tanartkit and Biegler (1997). Using the MPI or MFC formulations, performance close to perfect information can be realized during operations when all planning scenarios can be accounted for beforehand. However, one drawback of this approach is the explosive growth in problem size as the number of periods increases, especially since additional scenarios might need to be considered. Moreover, if uncertainty in planning parameters is also allowed to contribute scenarios, then planning periods as well as dynamic process models would lead to many more periods. Applications of the approach will thus rely on efficient computation of the overall multiperiod problem, using decomposition strategies tailored to the proposed approach. In addition, to keep the number of periods from growing too large for process uncertainty ranges, one needs to focus on carefully selecting critical uncertain scenarios or uncertain parameter instances. While this work addresses instances of uncertain parameters, continuous parameter variations could also be addressed directly by adapting the approach of Mohideen et al. (1995). In this approach, multiperiod problems are solved in an inner loop with given instances of the uncertain parameters. In the outer loop, critical values of the uncertain parameters are determined and these are used to form new operating periods for the inner loop. Consequently, solving large multiperiod problems efficiently is crucial to the treatment of uncertainty. With the success of Interior Point methods for solving QP problems Albuquerque et al. (1997), we feel this could improve the performance of multiperiod design algorithms. This will be the subject of our future research. Acknowledgment Financial support from the Department of Energy under Grant No. DE-FG02-85ER13396 is gratefully acknowledged for this work.
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Received for review December 4, 1996 Revised manuscript received April 23, 1997 Accepted April 24, 1997X IE960752V
Abstract published in Advance ACS Abstracts, July 1, 1997. X
