Deterministic Framework for Robust Modular Design with Integrated

This paper introduces an integrated data analysis, robust design optimization approach to generate a set of standardized designs that span the demand ...
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Ind. Eng. Chem. Res. 2004, 43, 6813-6821

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Deterministic Framework for Robust Modular Design with Integrated-Demand Data Analysis Vishal Goyal and Marianthi G. Ierapetritou* Department of Chemical and Biochemical Engineering, Rutgers University, Piscataway, New Jersey 08854

One of the important considerations in plant design is to be able to accommodate changing product demands associated with market variability. Competing markets require a high degree of market adaptation and orientation. Recent developments in process design have focused on establishing optimization-based approaches to support decision making under uncertainty, but all of the current efforts are based on creating customized designs under specified production requirements. This paper introduces an integrated data analysis, robust design optimization approach to generate a set of standardized designs that span the demand space. Two case studies of process design are presented to illustrate the applicability of the proposed approach. 1. Introduction Today’s marketplace reality is characterized by rapid, uncertain, and continuous changes. Keeping the market variability in mind, any industrial sector should realize the importance of managing efficiently the available resources and accomplishing production requirements by developing flexible planning strategies as well as identifying and accessing the potential benefits of exploring new resources in conjunction with the existing ones for process design. Because design specifications of chemical plants, such as product demands, ambient conditions, or model parameters, normally vary during process operation, an optimal design must not only be economically optimal but also be capable of operating in steady state for a range of variable conditions that may be encountered. With respect to the way uncertainty is handled, three different approaches are known: (a) the scenario-based approach (Grossmann and Sargent,11 Halemane and Grossmann,12 Varvarezos et al.24); (b) the stochastic optimization (Pistikopoulos and Ierapetritou,19 Bernardo and Saraiva3); and (c) the parametric-based approach (Acevedo and Pistikopoulou,1 Pertsinidis et al.18). The first two approaches are based on characterization of the uncertain parameter space by considering either discrete scenarios or stochastic distributions, assuming that some information regarding the uncertainty is provided either in the form of the most expected nominal point or specific range of values or in the form of a probability distribution function. In the parametric framework, no assumptions are made on the uncertainty model and the objective function is obtained as a function of the uncertain parameters over the demand space. To handle the tradeoff associated with the mean objective value and its variance, Mulvey et al.14 proposed the concept of robustness. A decision is termed robust if the actual objective value of the realized scenario remains “close” to the optimal expected value of all scenarios. Since then, a number of papers have been published studying robustness (among others, work by * To whom correspondence should be addressed. Tel.: (732) 445-2971. Fax: (732) 445-2421. E-mail: marianth@ sol.rutgers.edu.

Suh and Lee23 and Bernardo et al.4) that differ according to the definition of robustness measure and the process considered. Although adding robustness to the objective function results in a more robust solution, it does not systematically explore the tradeoffs between expected cost and variance. To achieve that, a multiobjective optimization problem should be solved where solution variability is explicitly considered as another objective. A large number of solution strategies have been proposed for multiobjective optimization problems. A literature survey can be found in work by Sawaragi et al.22 The motivation for this paper comes from the realization that significant economic savings can be achieved if standardized designs are developed, taking into account the customer demand space. The relevance of the work stems from the fact that developing modular designs would be substantially cheaper for a manufacturer that develops customized designs and beneficial for the customer because various design alternatives would be available with different capital cost and flexibility. The objectives of this paper are (a) to develop an optimization framework for determining robust optimal designs considering demand probabilities and (b) to expand the design optimization boundaries by incorporating data analysis to improve the decision-making process. The main idea is to cluster the demand data, solve a scenario-based robust design optimization over the clusters, and iteratively improve the robust designs to generate the smallest set of designs that span the demand space with minimum cost. The paper is organized as follows. Following this introduction, in section 2, the detailed proposed framework is presented. In section 3, a line-search methodology is proposed for the solution of the robust design optimization problem as a multiobjective problem that results in the Pareto set of optimal design alternatives. In section 4, two case studies of process design under demand uncertainty are presented to illustrate the applicability of the approach. Finally, section 5 summarizes the work and presents future work directions. 2. Module-Based Design Optimization Framework The main aim of the proposed algorithm is to determine the smallest set of economically optimal designs

10.1021/ie049771s CCC: $27.50 © 2004 American Chemical Society Published on Web 09/18/2004

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design optimization problem is mathematically modeled as follows (Mulvey et al.14):

min

psξs + λ∑ ps(ξs - ∑ ps′ξs′)2 + ∑ s∈S s∈S s′∈S ω ∑ p s[ ∑ s∈S θ∈Θ hs(d,xs,ys,θs) ) 0

∀s∈S

gs(d,xs,ys,θs) e 0

∀s∈S

prodsθ + zsθ g demsθ Figure 1. Overall proposed approach for modular design generation.

that jointly cover the entire demand space based on customer requirements. The idea is to incorporate data analysis within the design optimization strategy to improve the flexibility of the decision-making process. The proposed algorithm consists of the following stages (Figure 1). Step 1. The demand data are initially clustered in a minimal number of clusters using the facility location methodology described below. Note that it is not possible to know a priori the minimal number of optimal designs that can cover the entire demand space; thus, an iterative procedure has to be utilized that starts with a small number of designs. The facility location approach provides a systematic methodology to incorporate demand weights, which represent the degree of importance of demand point or the mean of the probability distribution of the demand within a data-clustering framework. The problem is mathematically modeled as follows:

min

fiyi + ∑∑ cijxij ∑ i∈F i∈F i∈D

xij ) 1 ∑ i∈F xij e yi

(1)

∀j∈D ∀ i ∈ F, j ∈ D

xij ∈ {0, 1}, ∀ i ∈ F, j ∈ D yi ∈ {0, 1}, ∀ i ∈ F where the binary variables yi, i ∈ F, indicate if demand scenario i is the mean of the cluster, and binary variables xij, i ∈ F, j ∈ D, indicate if the demand scenario j is assigned to cluster i; cij is the distance between scenarios j and i; F and D are the number of required clusters and demand scenarios being clustered, respectively; fi is the cost of choosing the point I as the mean and is proportional to the inverse of the probability pi of the realization of the demand. Details of the facility location model are available in the book by Nemhauser and Wolsey.15 Step 2. The cluster points are then chosen as the representative scenarios, and a multiperiod design optimization problem is solved to generate a robust design for each cluster. The scenario-based robust

(zsθ)2] (2)

∀ θ ∈ Θ, s ∈ S

where ξs ) Cs(d,x,y,θ) represents the total operating and capital cost for scenario s; d, xs, and ys are the sets of design, state, and control variables, respectively; θs is the set of uncertain variables for each scenario s; hs and gs are the vector of equality and inequality constraints, respectively, for each scenario s; zsθ represents the unmet demand in scenario s; demsθ and prodsθ is the required demand and the production amount for demand θ in scenario s, respectively. The first term in the objective function is the expected cost of the design over all possible scenarios. The second term is the variance of the expected cost, weighted using parameter λ, and the last term is the penalty for unmet demand weighted using parameter ω. Due to a possible high variability of demand within a cluster, the second term is included to penalize high variance in the operating cost of the optimal design, although the presence of outliers may still drive the optimal solution toward an overdesign. Hence, the third term is introduced to allow partial demand satisfaction in the optimal solution representing model robustness. The solution of problem formulation (2) results in a robust design over the set of scenarios (cluster points), for a specific value of penalty parameter λ and ω. In section 3, the formulation is extended to a multiobjective framework where the tradeoffs between the different objectives are properly explored. Step 3. The “value” of a design is quantified on the basis of two criteria. The first criterion is the expected total cost consisting of fixed and operating cost, averaged over all feasible scenarios. If a scenario is feasible for two alternative designs, then the most economical design is selected to satisfy the demand. The second criterion is the number of feasible demand points satisfied by the design. The basic idea is to obtain a linear approximation of the boundary of the feasible region to directly validate the feasibility of all of the demand points. To evaluate the operability limits of a given design, the simplicial approximation approach proposed by Goyal and Ierapetritou9 can be utilized for convex and some special cases of nonconvex problems (Goyal and Ierapetritou10). For the case of general nonconvex problems, the approach proposed by Banerjee and Ierapetritou2 can be utilized. The main idea of this approach is to utilize the shape reconstruction algorithm, R shape (Edelsbrunner and Mucke7), to determine the boundary of the feasible region as a piecewise linear approximation and then utilize the point in polygon method (O’Rourke21) to check demand feasibility. The main notion of the R-shape algorithm is that it tries to fit in spheres of radius R in the sample space, without intersecting any sample points, and the complement of the union of spheres forms the desired shape.

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The value of the parameter R controls the level of detail of the reconstructed surface. Step 4. The number of clusters are then increased by one, and steps 1-3 are repeated for the new set of scenarios. Because at each iteration the demand space for each design optimization is reduced due to an increased number of clusters, a better or at least the same design in terms of cost is guaranteed to be obtained for some of the demand points. Note that for the demand points that are infeasible for all of the designs in the k + 1 iteration, an alternative design can be selected from the previous iterations or a feasible design can be determined by solving problem 2 while enforcing zsθ to be zero. Step 5. The new set of designs obtained at step 4 is compared to the previous set of designs on the basis of cost and feasibility. Step 6. If any “better” designs are obtained at step 4, then steps 4 and 5 are performed again and the designs are compared to the previous set of designs, until no new (superior) designs are obtained. At the end of the proposed approach, a set of design alternatives that span the entire demand space is obtained. The increased number of alternatives translates to an increased decision-making flexibility due to better utilization of the available demand data and its integration within the design stage. Specifically, great savings can be achieved because modular-based designs are obtained to cover the entire range of demand and better information is provided regarding different design alternatives involving cost, flexibility, and robustness. It should be noted that the existence of outlier points in the demand space may change the clustering, favoring designs with higher cost, and hence will result in an increased number of iterations for the convergence of the algorithm to the optimal set of designs. However, the outliers can be easily eliminated in an initial distance-based clustering step (Pavel17) or they can also be identified if they persistently appear as unmet demands (zsθ > 0) in the robust design optimization problem. Two case studies of process design optimization are presented in section 4 to illustrate the steps of the proposed approach. 3. Parametric Analysis Adding the robustness terms in the objective function moves the solution toward a design with lower variance and high model robustness, but the solution depends on the parameters λ and ω which control the effect of these terms in the objective value. Sensitivity analysis with respect to these parameters can thus reveal the importance of robustness and feasibility in the solution of the design optimization problem. A more comprehensive analysis requires the consideration of the three objectives within a multiobjective optimization formulation. In the past four decades, multiobjective optimization has emerged as a powerful tool for resolving conflicting objectives in engineering decision-making problems. Numerous algorithms have been proposed to determine the optimal set of solutions (Ehrgott and Gandibleux8) and can be categorized into two broad categories. In the first category, the problem is modeled using an aggregate objective function where each objective is assigned suitable weights resulting in a single objective that is intended to represent all of the

desired tradeoffs, whereas in the second category, a set of candidate designs are determined and the most desirable design among them is then selected as the optimal solution. Using either of the approaches, the aim is that the final solution is a Pareto optimal (Pareto16). Pareto solutions are those for which any improvement in one objective results in worsening of at least one of the other objectives. The complete set of such solutions is referred to as the Pareto surface. In the next section, a new line-search methodology is proposed to locate a Pareto surface. 3.1. Pareto Analysis: Line-Search Optimization. The line-search methodology proposed here leads to the generation of a set of points at the Pareto surface. The approach is based on the idea of determining the convex hull of the individual minima of the objective functions and then of evenly discretizing the convex hull and moving in the outward normal direction toward the Pareto frontier to determine the Pareto surface. In particular, the steps of the proposed approach are as follows. Consider a general multiobjective optimization problem as follows:

min {µ1(x), µ2(x), ..., µn(x)} x

gj(x) e 0

j∈J

hk(x) ) 0

k∈K

(3)

where µi are the n different objective functions to be minimized and gj and hk are the set of constraints. Step 1. The first step of the proposed approach is to determine the convex hull of the individual minima. This is obtained by solving problem 3 with individual objective functions to obtain the global minima, which are referred to as utopia points, as shown by µ/1 and µ/2 for a two-objective problem in Figure 2. Step 2. The second step is to determine the convex hull (utopia plane) of the utopia points, which in a 2D problem would be the line joining the utopia points, in a 3D problem would form a hyperplane, etc. The utopia plane is then evenly discretized into Np points where Np is the number of Pareto points required. Step 3. The last step of the approach is to determine the Pareto frontier. This is done by performing a line search in the outward normal direction toward the Pareto surface from the Np points found in step 2. This problem can be formulated as follows:

max δ x

(4)

gj(x) e 0 hk(x) ) 0 µPareto ) µdis + δN B ∆µi where µPareto is the Pareto point; N B is the outward normal direction of the utopia plane from the point µdis; L and ∆µi ) µU i - µi (see Figure 2). Thus, the algorithm determines a set of points on the Pareto frontier, by initializing the search from a uniform set of points on the utopia plane. However, the algorithm is restricted to continuous Pareto frontiers as the line search might identify non-Pareto solutions as Pareto solutions in the

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Figure 2. Proposed approach for Pareto analysis.

Figure 4. Case study 1: demand space.

design optimization formulation (problem 2) and results in the following model:

∑i [RiFixa,i + βiFi(xc,i +

min ξ ) 20V2 + 100Fao + 10

xd,i)] (5) Figure 3. Reactor-separator flowsheet with reaction kinetics for case study 1.

discontinuous cases. The proposed approach is based on the same basic ideas as the normalized normal constraint method (Messac et al.13), but both of these methods differ in the formulation of the optimization subproblem (problem 4), which is utilized to obtain the Pareto frontier. The steps of the proposed approach are illustrated using the case studies considered in section 4.

Fao - xa,iFi(1 - Ri) - Vcao(k1 + k2)xa,i ) 0 -Fixb,i + Vcaok1xa,i ) 0 -Fixc,i(1 - βi) + Vcao(k2xa,i - (k3 + k4)xc,i + k5xe,i) ) 0 -Fixd,i(1 - βi) + Vcaok3xc,i ) 0 -Fixe,i + Vcao(k4xc,i - k5xd,i) ) 0 xa,i + xb,i + xc,i + xd,i + xe,i - 1 ) 0

4. Case Studies 4.1. Reactor-Separator System. The first case study presented in this section is the design of a CSTR in series with an ideal separator. The aim is to convert raw material A into two finished products B and E as shown in Figure 3. An isothermal liquid-phase reaction is considered following the kinetic mechanism as described in Rooney and Biegler.20 The model equations for the process are shown in problem 5, where xa, xb, xc, xd, and xe are the mole fractions of components A, B, C, D, and E, respectively; V is the volume of the CSTR; cao and Fao are the inlet concentration and flow rate of component A, respectively; and Fprod is the amount of desired product recovered from the top of the separator. The control variables are the recycle fractions of component A (R) and components C and D (β). The nominal values of the kinetic constants are k1 ) 0.0374, k2 ) 0.0195, k3 ) 0.0165, k4 ) 0.2701, and k5 ) 0.0261. The overall annualized cost of the process (ξ) consists of the fixed cost of reactor design, raw material cost, and the recycle cost. The demand plot for products B and E and their corresponding probabilities were randomly generated and are shown in Figure 4. The objective of the problem is to determine the optimal set of designs (V) and their input flow rates (Fao) that cover the entire demand space with the minimum overall cost. The problem is modeled following the scenario-based robust

Fprod,b,i ) Fixb,i Fprod,e,i ) Fixe,i Fprod,b,i + zb,i g Fdem,b,i Fprod,e,i + ze,i g Fdem,e,i 0 e Ri, βi e 1; 20 e V e 50 40 e Fao e 200; 100 e Cao e 165 where the index i represents scenario (i) and corresponds to different demand points. The solution robustness term involves the variability of the operating cost consisting of the recycle cost. The robustness parameters are fixed to λ ) 0.1 and ω ) 100, although the results of Pareto analysis are also presented in section 4.1.2. All of the optimization problems in this paper are solved using GAMS/CONOPT3 (Brooke et al.5) as the NLP solver on a Dell 933MHz PC using the Linux operating system. The results from the application of the proposed approach are summarized in the next section. 4.1.1. Computational Results. The application of the proposed approach is initialized by assuming two clusters. Steps 1 and 2 of the proposed approach are performed where the multiperiod robust design optimi-

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Figure 7. Case study 1: robust designs at the third iteration. Figure 5. Case study 1: robust designs at the first step of the proposed approach.

Figure 6. Case study 1: robust designs at the end of the second iteration.

zation problem is solved for each of the two clusters. A solution with V ) 26.03 m3, Fao ) 111.9 mol/h, and an expected cost of (Σspsξs) ) $2.5 × 104 is obtained for the cluster illustrated by “+” in Figure 5, whereas a design with V ) 40.12 m3, flow rate of Fao ) 152 mol/h, and an expected cost of $4.8 × 104 is obtained for the other cluster shown by “b” in Figure 5. Due to the nonconvex nature of the constraints and the low dimensionality of demand space, the approach introduced in section 2 is utilized to determine a linear approximation of the boundary of the feasible region as illustrated using the polytopes in Figure 5. The first design is feasible for 23 out of 30 demand points, and the second design is feasible for 11 out of 20 demand points in the cluster. Steps 4 and 5 of the proposed approach are then applied by increasing the number of clusters by one. This results in two additional robust designs for the two new clusters, as illustrated in Figure 6: (V, Fao, capital cost, feasibility) ) (23.33, 46.43, $1.6 × 104, 12 over 15 demands) and (31.64, 112.1, $3.1 × 104, 14 over 15 demands), for the clusters shown by “4” and “+”, respectively, where the illustrated polytopes represent the R-shape approximation of the feasible regions. The third cluster is the same as in the previous iteration, and hence no new design is generated for that cluster. Both new design configurations are “better” than the

Table 1. Reactor-Separator Case Study: Final Set of Designs optimal design (m3)

expected cost

feasibility

23.3 26.0 31.6 40.1 44.4

1.6 × 104 2.5 × 104 3.1 × 104 4.8 × 104 5.7 × 104

12/15 23/30 14/15 11/20 9/12

previous obtained designs with respect to cost and hence are stored, and another iteration is carried out. Thus, at the end of the second iteration, a total of four (two from previous iteration) robust designs have been determined with different degrees of flexibility. Note that by enforcing zsi ) 0, two additional designs are generated to cover the infeasible demands. The number of clusters is then increased to four, resulting in two new clusters (Figure 7). The robust design optimization problem is solved for each of the two new clusters resulting in the following one new design configuration for the cluster shown by “4” in Figure 7: (V, Fao, capital cost, flexibility) ) (44.43, 167.65, $5.7 × 104, 9 over 12 demands). Thus, at the end of the third iteration, a total of five robust designs have been obtained to cover the demand space. Another iteration of the procedure is carried out by increasing the number of clusters to five, which results in the formation of three new clusters. The robust design optimization problem is solved over each of the three new clusters, but no new design configurations were obtained. Hence, the iterative procedure is terminated at the fourth iteration. To summarize the results, at the end of the optimization process, a set of five modular robust designs have been obtained that span the demand space (Table 1). All of the robust-optimization problems for this case study require an average of 0.3 CPU s for convergence. For the points that are covered by more than one design, there are some interesting tradeoffs that have to be considered in the decisionmaking process because the more expensive designs have higher flexibility and thus higher profit can be anticipated if higher demand is realized in the future. 4.1.2. Parametric Analysis. To elucidate the dependence of the solution on the parameters λ and ω, sensitivity analysis is performed that results in a solution map in the (λ, ω, capital cost) space. The results for a specific cluster (illustrated by ∆ in Figure 7) are shown in Figure 8. As shown in Figure 8, increasing ω for fixed λ drives the design toward an increased model robustness, at the expense of additional capital cost, and

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components are then separated to remove products P, E, and C as overheads. Due to the presence of an azeotrope, some of the product (equal to 10 wt % of component E) is retained in the bottoms. The bottoms are then split into a purge and recycle stream, which is mixed with the feed and sent back to the reactor. The objective of the optimization problem is to minimize the overall cost, which consists of annualized plant capital cost, raw-material cost, waste-treatment cost, and the recycle cost. The mathematical model of the process optimization problem is as follows:

∑j Fjpurge + 0.05∑ FjR + 6VF j

min 0.2FA + 0.3FB + 0.01Fwaste + 0.001

log(k1) - log(a1) + 12 000/T ) 0 log(k2) - log(a2) + 15 000/T ) 0

Figure 8. Sensitivity analysis for case study 1: capital cost versus (λ, ω).

increasing λ for fixed ω drives the design toward a more solution-robust design, again at the expense of additional capital cost. The robust design obtained for (λ, ω) ) (0.1, 100) is also shown with the “O” in Figure 8. These results, although intuitive, illustrate the need for a better understanding of the effect of robustness parameters before utilizing them in a design-optimization framework. As highlighted in section 3, a systematic investigation of the tradeoffs between the three objectives of problem 2 requires the solution of a multiobjective optimization problem, which is solved using the line-search methodology introduced in section 3.1. For the same cluster considered above (illustrated by 4 in Figure 7), the approach is utilized considering the capital cost, robustness cost, and infeasibility cost as the three competing objectives. The utopia points are first determined (shown by O in Figure 9), leading to the Pareto frontier as shown by the plane in Figure 9. This is discretized into 50 points (Np ) 50), to obtain points, µdis, on the utopia plane. A line search is performed from each of these points to obtain the required points forming the Pareto surface, requiring a total of 11 CPU s. Note that the number of required Pareto points, Np, is a user-defined parameter that depends on the nature of the problem because a stiff problem would require a larger number of points for characterization. However, in this work, it is chosen on the basis of the computational complexity of the underlying optimization problem. Also shown with a “f” in Figure 9 is the robust design obtained with (λ, ω) ) (0.1, 100) which is one of the Pareto points but could be different from the most desirable design. Also due to the nonconvex nature of the Pareto surface, sensitivity analysis over the robustness parameters (λ,ω) would not be able to capture a complete representation of the Pareto surface,6 thus restricting the choice of the final design to the limited available Pareto points. 4.2. Williams-Otto Process. The second case study considers the design under uncertainty for a modified Williams-Otto flow sheet (Williams and Otto25), shown in Figure 10. Two feed streams containing pure components A and B are fed to a CSTR where the three reactions are taking place resulting in P as the main product, E and C as byproducts, and G as a waste product. The effluent is cooled in a heat exchanger and is sent to a centrifuge to separate G. The remaining

log(k3) - log(a3) + 20 000/T ) 0 A A A A Feff,i - (FA,i + FR,i - k1xr,i xr,iVF) ) 0 B B A C B - [FB,i + FR,i - (k1xr,i + k2xr,i )xr,i VF] ) 0 Feff,i C C A B C B - [FR,i + (2k1xr,i xr,i - 2k2xr,i xr,i Feff,i P C xr,i )VF] ) 0 k3xr,i E E C B - [FR,i + (2k2xr,i xr,i)VF] ) 0 Feff,i P P C B P C - [FR,i + (k2xr,i xr,i - 0.5k3xr,i xr,i )VF] ) 0 Feff,i G G P C - [FR,i + (1.5k3xr,i xr,i )VF] ) 0 Feff,i j )xji ) 0 ∑j Feff,i

j Feff,i -(

j j - Feff,i )0 Fex,i

j ) A, B, C, E, P, G

P Fbot,i

E - 0.1Fd,i )0

j j - ηiFbot,i )0 Fpurge,i j FR,i

- (1 -

j ηi)Fbot,i

j ) A, B, C, E, P, G

)0

j ) A, B, C, E, P, G j ) A, B, C, E, P, G

Fprod,p,i + zp,i g Fdem,p,i Fprod,c,i + zc,i g Fdem,c,i Fprod,e,i + ze,i g Fdem,e,i 0 e ηi,

xji

e 1; 580 e T e 650 °R; 100 e V e 700l

100 e FA e 500 kg/s; 100 e FB e 900 kg/s The design variable is the CSTR volume (V), and the two control variables are the recycle fraction (η) and the reactor temperature (T). Index i corresponds to scenario (i). The problem is modeled following the multiperiod robust design optimization framework (problem 2). The variance of the cost of design is determined using the variability of the operating cost, which consists of the waste-treatment cost (waste product G and total purge) and the recycle cost. The robustness parameters are used as scaling factors and are initially fixed to λ ) 0.1 and ω ) 1000. To illustrate the applicability of the proposed approach in higher dimensions, the designoptimization problem is slightly modified to consider the demand uncertainty of products P, E, and C. The demand plot for the three products and their corresponding probabilities were randomly generated assum-

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Figure 9. Pareto analysis for the reactor-separator case study.

Figure 12. Case study 2: robust designs at the first iteration of the proposed approach.

Figure 10. Williams-Otto process.

Figure 11. Demand space for the Williams-Otto case study.

ing a normal distribution and are shown in Figure 11. Similar to the previous case study, the objective of this problem is to determine the optimal set of designs defined as the smallest number of standardized designs with minimum total cost that cover the entire demand space. The proposed approach is thus applied, and the results are summarized below. 4.2.1. Computational Results. The application of the proposed approach is initialized by considering three clusters. Steps 1 and 2 of the proposed approach are applied where the multiperiod robust design optimization problem is solved considering the corresponding scenarios for each of the three clusters. A design with

V ) 277l and an expected capital cost of $5.4 × 103 is obtained for the cluster shown by “f” in Figure 12, whereas a design with V ) 322.3l and an expected cost ) $6.9 × 103 is obtained for the cluster shown by “+” in Figure 12, and a design with V ) 643.2l and an expected cost ) $1.5 × 104 is obtained for the third cluster shown by “O” in Figure 12. Due to the nonconvex nature of the constraints, the feasible regions of the designs are obtained using the approach introduced in section 2. Thus, it is found that the first design is feasible for 13 over 17 demands, the second for 9 over 12 demands, and the largest design is feasible for 14 over 21 demand points. Steps 4 and 5 of the proposed approach are then applied by increasing the number of clusters by one to four. This results in two additional robust designs for two of the new clusters with (V, expected cost, feasibility) ) (528.3l, 9.4 × 103, 12 over 13 demands), for the cluster shown by “f” in Figure 13, and (V, expected cost, feasibility) ) (700l, 11.3 × 104, 8 over 8 demands), for the cluster shown by “+” in Figure 13, respectively. The robust designs for the remaining clusters are the same as in the previous iteration. Because new designs are obtained, another iteration is carried out. Thus, at the end of the second iteration, a total of five (three from previous iteration) robust designs have been determined with different degrees of flexibility. Note, that, by enforcing zsi ) 0, additional designs can be generated to cover the infeasible demands. The number of clusters is then increased to five, resulting in three new clusters (Figure 14). The robust design optimization problem is solved for each of these new clusters, resulting in the following two new standardized designs with: (V, expected cost, feasibility) ) (238.4l, 3.9 × 103, 11 over 13 demands), for the cluster shown by “f” in Figure 14, and (V, expected cost, feasibility) ) (299l, 4.4 × 103, 6 over 8 demands), for the cluster shown by “+” in Figure 14, respectively. Thus, at the end of the third iteration, a total of seven robust designs have been developed that cover the demand space.

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Figure 15. Case study 2: robust designs at the fourth iteration of the proposed approach.

Figure 13. Case study 2: robust designs at the second iteration of the proposed approach.

Figure 16. Sensitivity analysis for case study 2: capital cost versus (λ, ω). Table 2. Williams-Otto Process: Final Set of Designs

Figure 14. Case study 2: robust designs at the third iteration of the proposed approach.

Another iteration of the procedure is carried out by increasing the number of clusters to six, which results in one new robust design with: (V, expected cost, feasibility) ) (128.9l, 2.1 × 103, 8 over 8 demands), for the cluster shown by “+” in Figure 15, thus resulting in a total of eight designs at the end of the fourth iteration. Another iteration of the procedure is carried out by increasing the number of clusters to seven, but after solving the robust design optimization problem over the new clusters, no new design configurations were obtained. Hence, the iterative procedure is terminated at the fifth iteration. All of the robust-optimization problems for this case study require an average of 3.3 CPU s for convergence. Thus, to summarize the results, at the end of the optimization process, a set of eight modular robust designs have been developed that span the 3D demand space (Table 2).

optimal design

expected cost

feasibility

128.9l 238.4l 277.0l 299l 322.3l 528.3l 643.2l 700.0l

2.1 × 103 3.9 × 103 5.4 × 103 4.4 × 103 6.9 × 103 9.4 × 103 1.5 × 104 11.3 × 104

8/8 11/13 13/17 6/8 9/12 12/13 14/21 8/8

4.2.2. Parametric Analysis. The sensitivity analysis with respect to capital cost and robustness parameters of a cluster obtained at the first iteration (illustrated by “f” in Figure 12) reveals trends similar to those of the previous example. In particular, as shown in Figure 16, by increasing λ and ω, the design is moved toward designs with higher costs but with an increased flexibility. The robust design obtained for (λ, ω) ) (0.1, 1000) is also shown with the “f” in Figure 16. As for the previous case study, the multiobjective optimization framework was also utilized to study the tradeoffs between the different objectives. The three competing objectives were the expected capital cost, the robustness cost, and the infeasibility cost. The utopia points were

Ind. Eng. Chem. Res., Vol. 43, No. 21, 2004 6821

Literature Cited

Figure 17. Pareto analysis for the Williams-Otto case study.

first determined leading to the Pareto frontier as shown by the plane in Figure 17. This is discretized into 50 points (Np ) 50), to obtain points, µdis, on the utopia plane. A line search is performed from each of these points to obtain the required points forming the Pareto surface, requiring a total of 370.5 CPU s. Also shown with a (/) in Figure 17 is the robust design obtained with (λ, ω) ) (0.1, 1000), which is one of the local Pareto solutions. Similar to the previous case study, sensitivity analysis on the robustness parameters would not result in a complete representation of the Pareto surface due to the nonconvex nature of the surface. 5. Summary and Future Directions A novel framework is presented in this paper for the integration of data analysis and robust design optimization stages that have been traditionally performed separately. The main idea is to cluster the demand data, solve a scenario-based robust design optimization over the clusters, and iteratively improve the robust designs to generate the smallest set of designs that span the demand space with minimum cost. The importance of the research is based on the fact that developing a portfolio of modular designs results in substantial savings for the manufacturer and in a larger number of choices for the customer due to the availability of designs having different degrees of flexibility and costs. Two-process-design case studies have been solved to illustrate the applicability and efficiency of the proposed approach, resulting in a number of modular design alternatives that span the demand space. Sensitivity analysis has also been performed with respect to robustness parameters, illustrating the need for a better understanding of the effect of the parameters in the design optimization. To address this problem, a simple framework for obtaining the Pareto surface is proposed and utilized for both case studies. Research is currently underway to extend the proposed approach to a stochastic framework by considering the probabilities of demand points that will be part of a later publication. Acknowledgment Financial support from the National Science Foundation under the NSF CAREER program CTS-9983406 is gratefully acknowledged.

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Received for review March 23, 2004 Revised manuscript received July 17, 2004 Accepted August 13, 2004 IE049771S