Multiobjective Framework for Modular Design Generation

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Ind. Eng. Chem. Res. 2005, 44, 3594-3606

Multiobjective Framework for Modular Design Generation Incorporating Demand Uncertainty Vishal Goyal and Marianthi G. Ierapetritou* Department of Chemical and Biochemical Engineering, Rutgers University, Piscataway, New Jersey 08854

Recent developments in process design have focused on establishing optimization-based approaches to support decision making under uncertainty by creating customized designs under specified production requirements. Modular-based design has long been recognized as a preferred alternative because it results in more flexible solutions in a dynamically changing environment. This paper introduces an integrated data analysis, stochastic design optimization approach to generate a portfolio of standardized designs that span the random demand space. The main idea is to cluster the demand and generate a Pareto set of designs by modeling cost and robustness in a multiobjective framework. Two case studies are presented to illustrate the applicability of the proposed approach. 1. Introduction Process design has been defined as a creative activity in which ideas are generated and translated into equipment and processes for producing new materials or for significantly upgrading the value of existing materials.1 A successful plant design consists of various iterative loops of which approximately 85% of the time is consumed by the early R&D stages where the design team understands the large amount of information and data flow.2 The current approach for process design is based on developing customized designs that satisfy a specific customer. However, standardization of parts, interfaces, or manufacturing units has long been recognized as an interesting alternative to customized process design.3 Because of substantial market dynamics, the best way to satisfy a customer would be to include the demand requirements within the design stage. This idea of incorporating demand data analysis within the design optimization phase is the main philosophy of the presented work, with the aim of generating a portfolio of standardized designs instead of traditional customized designs. Because of rapid and uncertain changes in the current market, the inclusion of uncertainty represents a key issue in process design. To handle uncertainty in process design, three different approaches have been reported: (a) the scenario-based approach, where the uncertainty is described by a set of scenarios resulting in a multiperiod optimization problem;4-6 (b) the stochastic approach, where the uncertain parameters are described through probability density functions resulting in a stochastic optimization problem;7-9 (c) the parametricbased approach, where the problem in solved parametrically over the space of the uncertain parameters.10,11 A detailed survey of state of the art in stochastic optimization approaches is presented by Sahinidis.12 Most of the existing approaches result in an optimal design in terms of an economic objective, but only a * To whom correspondence should be addressed. Tel.: (732) 445-2971. Fax: (732) 445-2421. E-mail: marianth@ sol.rutgers.edu.

limited number of publications pose the problem as a multicriteria optimization problem where the tradeoffs between conflicting objectives are systematically exploited.13 In the presence of uncertainty, design feasibility and robustness represent objectives of equivalent importance in design economics. A number of algorithms have been proposed to solve multiobjective optimization problems.14,15 The solution approaches can be categorized in 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 Pareto optimal,16 which is defined as the solution for which any improvement in one objective results in worsening of at least one of the other objectives. The basic idea of the proposed approach is to extend the boundaries of process design to incorporate demand analysis. In our previous work, we addressed this problem by assuming discrete demand points following a robust optimization framework.17 It was also illustrated that better designs could be obtained if the problem is addressed in a multiobjective fashion considering economic and feasibility objectives. Thus, the objective of this paper is to extend the previous work in order to develop an optimization framework for determining optimal standardized designs considering stochastic demand distribution so as to better capture realistic case studies where customer orders are clustered around expected mean values of high probability. In the proposed approach, the demand data are clustered using a fuzzy-clustering algorithm, and the Pareto set of designs are obtained over each cluster by solving a nonlinear multiobjective optimization problem. The approach is performed in an iterative fashion so as to obtain a number of Pareto optimal designs that span the demand space balancing the different objectives of cost, flexibility, and robustness. The paper is organized as follows. Following the Introduction, in section 2 the detailed proposed frame-

10.1021/ie049336y CCC: $30.25 © 2005 American Chemical Society Published on Web 04/02/2005

Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3595

multiobjective stochastic design optimization problem is formulated over each cluster to generate a set of Pareto designs:

min (a) EΘ{capital and operating cost} (b) variance in operating cost (c) demand infeasibility penalty h(d,x,c,θ) ) 0 g(d,x,c,θ) e 0 ∀θ∈Θ prodθ + zθ g demθ P ∩ prodθ g demθ g R Θ

(

Figure 1. Overall proposed approach for modular design generation.

work is described. In section 3, two case studies are presented to illustrate the applicability of the approach. Finally, section 4 summarizes the work and discusses future work directions. 2. Modular-Based Design Optimization Framework The proposed approach uses the supply chain information regarding all different customers at the design optimization stage and comes up with a portfolio of different modular designs that optimally cover the entire demand space to improve the decision-making process for both the manufacturer and the customer. The framework consists of the following integral parts: Objective 1. Application of a data-clustering technique to cluster the data into closely packed groups. Objective 2. Development of a unified data analysis process optimization framework for determining optimal designs. In this section, the main steps of the algorithm are presented in detail. The overall framework is illustrated in Figure 1. Step 1 (Fuzzy Clustering). The demand space considered in this framework is selected to represent a realistic scenario where different customers are clustered around a specific demand point. Thus, each point represents the mean of a stochastic distribution describing a set of customers with similar demands. Examining the demand data further reveals that demands that are closer to each other are more likely to have the same standardized design than demands that are far apart. Thus, the first step of the proposed approach is to cluster the demand data in order to identify regions of similar demand values. Because of the probabilistic nature of the demand, a fuzzy-clustering approach18 provides a systematic methodology to incorporate variance within a clustering framework. The basic idea of fuzzy clustering is to allow clusters to share data points by assigning membership coefficients, thus fusing the cluster boundaries together. The clustering algorithm FANNY19 is utilized to cluster the demand data. A brief introduction to the algorithm is also presented in appendix B. Because it is not possible to determine a priori the minimum number of clusters required to span the demand space, the current approach starts with a small number of clusters, which is then iteratively increased. Step 2 (Nonlinear Stochastic Multiobjective Optimization). When the cluster points are chosen as a representation of the demand space, the following

)

(I) (1) (2) (3) (4) (5)

where d is the set of design variables; x and c are state and control variables, respectively; θ ∈ Θ ) [θ1, ..., θp] is the set of p uncertain demand variables, which is assumed to follow a normal distribution J(θ); zθ represents the unmet demand; and demθ and prodθ are the required demand and production amount for each demand point. In the optimization problem (I), the overall objective (1) consists of the expected cost of design over the demand space, θ ∈ Θ; the variance of the expected operating cost, which is included to penalize large operating cost variability of the optimal design; and the third objective, which is introduced to penalize partial demand satisfaction in the optimal solution. Constraints (2) and (3) are the sets of equalities and inequalities that represent the process operation, capacity constraints, etc., and are case-dependent. Constraints (4) are the demand constraints, which are modeled using the slack variables, zθ, to allow partial demand satisfaction. These constraints are introduced for the points in the demand space that represents the mean of similar customers. Constraints (5), on the other hand, are introduced for all available customers but only when the mean demand is satisfied, zθ ) 0, at constraint (4). Constraints (5) have the form of joint chance constraints to enforce that if the average demand is satisfied, then all of the customers are also satisfied with a certain probability (R), which is a user-defined parameter. It is largely accepted that normality assumptions capture the essential features of product demand uncertainty.9,20 In addition, normality assumptions allow a conversion of the chance constraints into their equivalent deterministic formulations, thus reducing the computational complexity of the approach. Optimization problem (I) involves the calculation of multidimensional integrals, which is computationally very expensive. A number of approaches have been published to evaluate the integrals such as (a) Gaussian quadratures (cubatures), where a quadrature (cubature) formula is applied to each dimension of the uncertainty space using product Gaussian rules,7,21 and (b) random sampling techniques, which approximate the integral by sampling over the uncertainty space.22 Bernardo et al.8 presented a comprehensive review of the effectiveness of the various numerical techniques with changing dimensionality and illustrated that, with an increase in the dimensions of the uncertain variables, random sampling proves to be a more computationally effective technique. Because of a possible high number of demand variables in a typical industrial design problem, a discretization approach through random sampling has been selected to evaluate the integrals in this paper. Because of the joint normal distribution of random

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Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005

variables, demands are generated through the Kaiser and Dickman procedure,23 which involves simulating standard normal random variables and then mapping them on the required domain by using a Cholesky factorization of the variance-covariance matrix. In formulation (I), because partial satisfaction of demands is allowed, the feasible region of a design is defined according to the relative importance of the different objectives. Note that a small number of samples are required to characterize each demand point because an exact representation of the distribution is not needed. This results in the following nonlinear stochastic optimization problem:

min [f(d) + pksξks,

∑ k s

pk |ξk - ∑pks′ξks′|, ∑pk( ∑ zθ ∑ k k k θ∈Θ s

s

s

)]

k

m

s′

h(d,xks,cks,θks) ) 0

∀ k ∈ N, s ∈ S

g(d,xks,cks,θks) e 0

∀ k ∈ N, s ∈ S

k k k + zθm g demθm prodθm

P ∩ prodθ k g demθ k g R s s Θ

(

)

∀ θ ∈ Θ, k ∈ N ∀ s ∈ S, k ∈ N (II)

where f(d) is the fixed cost and ξks ) Cks(d,xks,cks,θks) represents the total operating cost for scenario s for demand point k; zθkm represents the unmet demand of the expected mean θkm for each demand point k; demθkm and prodθmk are the required demand and production amount for the mean of the demand point k; pks is the probability of the demand realization θks; N is the total number of mean demand points; and S is the number of scenarios representing each demand point. In formulation (II), the variance of the operating cost, which corresponds to the second objective function, is evaluated using the expected value of the absolute difference of the operating cost at scenario s from the average operating cost over all feasible scenarios. The third objective of design infeasibility is also evaluated using the expected value of the slack variables over all scenarios. Note that a number of robustness metrics are available in the open literature24 that can be effectively utilized in the above formulation according to the problem definition without affecting the overall framework. In the current framework, the above metrics were utilized to maintain the convexity of the objective function. The main complexity in formulation (II) is the chance constraint, expressing the design reliability with respect to all of the available customers around a mean demand. The complexity arises from the fact that the demand parameters are characterized by joint probability distributions. To address this problem efficiently an approximation is obtained by relaxing the joint chance constraint into individual chance constraints by using the following inequalities:25 k gR P ∩ prodθsk g demθm Θ k ≈ P(prodθsk g demθm ) g Rθ

(

s.t.

)

(1 - Rθ) e 1 - R ∑ Θ

∀θ∈Θ

(6) (7)

It has been shown through a direct application of Boole’s inequality that any θ that satisfies the joint constraint

also satisfies the above inequalities, thus resulting in a tight lower bound of the original problem. Furthermore, because of the normal distribution of random variables, an equivalent deterministic representation of the individual chance constraint is possible,25 resulting in the following constraints:

prodθ k gdemθ k +σθkΦ-1(Rθ) s

∀ θ ∈Θ,k∈N, s ∈ S (8)

m

where Φ-1 is the inverse cumulative function, σ is the standard deviation of the random demand, and Rθ is chosen so as to satisfy constraint (7). Thus, by using constraints (8) in place of the joint chance constraints, problem (II) can be reformulated as a nonlinear deterministic optimization problem. For the case when the demand variables can be considered as independent random variables, no approximation is required because the equivalent deterministic representation (8) can be directly used. However, note that the chance constraints (8) have to be enforced only for the set of customers for which the mean demand is satisfied. This can be ensured by introducing binary variables, yk, ∀ k, which converts the nonlinear programming (NLP) into a deterministic mixed-integer NLP (MINLP) as follows:

min [f(d) + pksξks,

∑ k s

pk |ξk - ∑pk ξk |, ∑pk ( ∑ zθ ∑ k k k θ∈Θ s

s

s

s′

s′

s

s′

h(d,xks,cks,θks) ) 0

∀ k ∈ N, s ∈ S

g(d,xks,cks,θks) e 0

∀ k ∈ N, s ∈ S

k k k + zθm g demθm prodθm

)] (III)

k

m

s

∀ θ ∈ Θ, k ∈ N

k e UB(1 - yk) (1 - yk) e zθm

∀k∈N

k - σθkΦ-1(Rθ)] e UByk LB(1 - yk) e [prodθsk - demθm ∀ θ ∈ Θ, s ∈ S, k ∈ N

prodθsk e UByk

∀ θ ∈ Θ, s ∈ S, k ∈ N

where the binary variable yk is 1 if a demand point is satisfied and 0 otherwise; LB and UB are negative lower and positive upper bounds, respectively, for the chance constraints. The last three constraints in formulation (III) ensure the following: (a) If a demand point is satisfied, then yk ) 1 and thus zθmk ) 0, and the chance constraint is imposed for all of the scenario points. Note that the right-hand side of the chance constraint is relaxed in this case with the use of a large positive constant (UB). (b) If a demand point is not satisfied, then yk ) 0 and thus zθmk g , which is a small positive constant. In this case, because the mean value of the demand cannot be completely fulfilled, the chance constraints are not imposed for all of the scenario points and no production is allowed. This is achieved by relaxing the lower bound of the chance constraints and forcing prodθks ) 0, ∀ s, for that demand point k. Formulation (III) corresponds to a nonlinear multiobjective optimization problem, which is solved using the line-search approach proposed by Goyal and Ierapetritou,17 the basic steps of which are summarized in appendix A. Step 3 (Design Selection and Evaluation). At the end of step 2, a set of Pareto optimal solutions are


obtained for each cluster. The decision maker can then select among the alternative designs the one that best compromises one’s objectives. One of these choices is the Pareto design, which corresponds to equivalent weights among the objectives and can be obtained as the design in the Pareto surface that is closest to the utopia point. The utopia point is a point that represents a fictitious design characterized by the best value of each objective when considered independently (appendix A). Independent of the final design choice for each cluster, a couple of criteria are introduced in order to compare different Pareto designs. The first criterion is the expected total cost consisting of fixed and operating costs, averaged over all feasible scenarios. If a scenario is feasible for two alternative designs, then the most economical design is selected to satisfy this demand. In the presence of uncertainty, the second criterion for design comparison should be the performance of the design under different demand values. Because of the probabilistic nature of the current approach, design reliability is utilized to quantify the performance of a design. Design reliability has often been used in chemical engineering literature, usually in terms of equipment maintenance.26 In view of the proposed approach, design reliability is defined as the probability of satisfaction of all demand scenarios by a design such as ∏k∑sFkms, where Fkms is the mass function of discrete sample s of feasible demand point k. Thus, the reliability measure used is the probability measure of the feasibility extent of a design over the feasible demands. A design with higher reliability is chosen as the optimal design when the costs have competing values. Step 4 (Iterative Process). The number of clusters are then increased by 1, and steps 1-3 are repeated for the new clusters. Because at each iteration the demand space for each design optimization is reduced as a result of the 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 (6) while forcing zθmk to be zero. Step 5 (Termination Criteria). The new set of designs obtained at step 4 is compared with the previous set of designs on the basis of cost and reliability. If better designs are obtained at step 4, then the iteration continues and steps 1-3 are performed again followed by a comparison with the designs obtained at the previous iteration. This procedure stops when no new designs are determined. At the end of the proposed approach, a set of standardized design alternatives that span the entire demand space are 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 reliability. In the next section, two realistic case studies are presented to further clarify the steps of the proposed approach.

Figure 2. Reactor-separator flowsheet with reaction kinetics for case study 1.

3. Case Studies 3.1. Reactor-Separator System. The first case study presented in this section involves the design of a continuous stirred tank reactor in series with an ideal separator. The aim of the process is to convert raw material A into two finished products B and E, as shown in Figure 2. An isothermal liquid-phase reaction is considered following the kinetic mechanism described by Rooney and Biegler.27 The model equations for the process are shown in problem (9), where xa, xb, xc, xd, and xe are the mole fractions of components A-E, respectively; V is the volume of the continuous stirred tank reactor; 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 (γ) 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 consists of the fixed cost of reactor design, raw material cost, and recycle cost. A 25-point demand space for products B and E was randomly generated assuming a bivariate normal distribution with a variance (σ2) ) 36 and a correlation coefficient (F) ) 0.7, as shown in Figure 3. The demand of products B and E is assumed to follow a joint distribution function on their similarity (products of the same raw material, A), which is considered to attract similar customers. Note that the points correspond to the mean demand of a set of customers with similar demand orders. The objective of the problem is to determine the Pareto set of designs (V) and their input flow rates (Fao) that can satisfy the entire demand space with minimum overall cost. The process model for a customized design is as follows:

min c1V + c2Fao + c3[γFxa + βF(xc + xd)]

(9)

Fao - xaF(1 - γ) - Vcao(k1 + k2)xa ) 0 -Fxb + Vcaok1xa ) 0 -Fxc(1 - βi) + Vcao[k2xa - (k3 + k4)xc + k5xe] ) 0 -Fxd(1 - βi) + Vcaok3xc ) 0 -Fxe + Vcao(k4xc - k5xd) ) 0 xa + xb + xc + xd + xe - 1 ) 0 Fprod,b ) Fxb Fprod,e ) Fxe 0 e γ, β e 1; 10 e V e 210 40 e Fao e 800; 100 e Cao e 125

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Figure 3. Case study 1: demand space.

Figure 4. Case study 1: Pareto designs for iteration 1.

The above model is then reformulated in the multiobjective optimization formulation (III), where the overall operating cost, ξ, consists of the annualized raw material cost and the recycle cost. The value of the chanceconstraint probability was set at R ) 0.9, expressing the requirement of 90% probability of demand satisfaction for each demand point around the mean. It is assumed that each demand point represents 10 individual customers. The values of the bounds LB and UB are cluster-dependent and can be chosen as (maxθ,ks [prodθks + demθkm + σθkΦ-1(Rθ)], respectively, to provide tighter bounds to improve convergence and ) 10-5. All of the MINLP problems in this paper are solved

using GAMS/SBB28 as the optimization solver on a 933 MHz Dell PC with a Linux operating system. The application of the proposed approach is initialized by assuming three clusters. Steps 1 and 2 of the proposed approach are performed where the demand space is first clustered using fuzzy clustering, and then a multiobjective optimization problem is solved for each of the three clusters. Note that the application of fuzzy clustering results in demand points being shared between different clusters, such as, for example, points X and Y in Figure 4, which are then considered in the multiobjective optimization problem for two clusters. A Pareto design with V ) 44 m3, Fao ) 234 mol/h, and an expected




cost of $1.6 × 104 is obtained for the cluster illustrated by ] in Figure 4, which has a design reliability of 0.27, whereas a design with V ) 111 m3, flow rate of Fao ) 392 mol/h, and an expected cost of $4.4 × 104 is obtained with a design reliability of 0.53 for the cluster illustrated by +, and for the cluster illustrated by O, a Pareto design with V ) 170 m3, Fao ) 594 mol/h, and an expected cost of 6.2 × 104 is obtained with a design reliability of 0.91. For illustration purposes, the feasible regions of the designs are obtained using grid-search optimizations and are shown with dots in Figure 4. Steps 1-3 of the proposed approach are then applied by increasing the number of clusters to 4. This results in three additional Pareto designs for the new clusters,

as illustrated in Figure 5: (V, Fao, capital cost, reliability) ) (76, 269, $2.2 × 104, 0.59), (143, 471, $5.0 × 104, 0.91), and (187, 695, $6.9 × 104, 0.91) for the clusters shown by +, f, and ], respectively. The feasible region for each of these designs is shown with dots in Figure 5. The fourth obtained design is the same as that in the previous iteration. All of the new design configurations are better than the previously 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 6 (3 from the previous iteration) robust designs have been determined with different costs and degrees of flexibility. Note that, by forcing zθkm ) 0, additional customized designs can be gener-

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ated for any unsatisfied demand point k. The number of clusters is then increased to 5, which results in one new design configuration for the cluster shown by O in Figure 6: (V, Fao, capital cost, reliability) ) (33, 116, $8.2 × 103, 0.45). Thus, at the end of the 3rd iteration, a total of 7 robust designs have been obtained to cover the demand space. The number of clusters is then increased to 6, resulting in two new design configurations: (V, Fao, capital cost, reliability) ) (64, 254, $2.4 × 104, 0.48) and (128, 438, $4.8 × 104, 0.31) for the clusters illustrated by + and 4, respectively, in Figure 7. The ranges of feasible operation are also illustrated in Figure 7. Thus, at the end of the 4th iteration, a total of 9 robust designs have been obtained to cover the demand space. Furthermore, Figure 8 illustrates the Pareto surface obtained for the cluster shown with 4 in Figure 7 using the line-search methodology proposed by Goyal and Ierapetritou.17 Another iteration of the procedure is carried out by increasing the number of clusters to 7, which results in the formation of two new clusters. The Pareto optimiza-

Figure 8. Case study 1: Pareto surface at the 4th iteration.

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

expected cost

reliability

33 44 64 76 111 128 143 170 187

8.2 × 103 1.6 × 104 2.4 × 104 2.2 × 104 4.4 × 104 4.8 × 104 5.0 × 104 6.2 × 104 6.9 × 104

0.45 0.27 0.48 0.57 0.53 0.31 0.91 0.91 0.91

tion problem is solved over each of the two new clusters, but no new design configurations were obtained. Hence, the iterative procedure is terminated at the 5th iteration. It is interesting to note that, at the end of the iterative procedure, all but 3 of the 25 demand points have been optimally satisfied by a Pareto optimal design. The three remaining demands form an isolated group at the top of the demand space (illustrated by 4 in Figure 7), which have been consistently rejected as outliers by the optimization procedure, and hence a customized design can be built with the following configuration: (V, Fao, capital cost, reliability) ) (210, 790, $6.5 × 104, 0.91). To summarize the results, at the end of the optimization process, a set of 9 modular robust designs have been obtained that span the demand space (Table 1). For the points that are covered by more than one design, there are some interesting tradeoffs that have to be considered in the decision-making process because the more expensive designs have higher flexibility and thus higher profit can be anticipated if higher demand is realized in the future. Also, the Pareto surfaces offer a variety of possible designs, with different values of cost, flexibility, and robustness that allow a decision maker to systematically choose a robust design, according to specific priorities. For example, the design illustrated by b on the Pareto surface in Figure 8 is a more expensive design than the selected Pareto design but


Figure 9. Case study 2: demand space.

has a higher degree of flexibility and can be selected when a higher demand realization is expected. 3.2. Multiproduct Batch Design. The next case study presented in this section addresses the design of a multiproduct batch plant.29 It is assumed that the plant consists of M processing stages where N products with demand Qi must be produced. The design problem consists of determining the sizes Vj for each stage j and for each product i, the batch sizes Bi, and the cycle times TLi. The data for the problem are given in Table 2 and involve the horizon time H, cost coefficients ηi and βi for the units, size factors Sij, and their processing times tij for each product i at stage j. Although the original model is nonconvex, the model can be convexified through logarithmic transformations. The convexified model equations for a customized design are as follows: N

M

min ηj exp[βjvj] j)1

vj g ln[Sij] + bi tLi g ln[tij]

P i Qi ∑ i)1

(cost)

(10)

i ) 1, N

i ) 1, N

j ) 1, M (volume for stage j) j ) 1, M (cycle time for product i)

N

exp[qi + tLi - bi] e H ∑ i)1

(horizon constraint)

ln[VLj ] e vj e ln[VU j ] L ] ln[TLi

ln[BLi ]

e tLi e

U ln[TLi ]

e bi e

ln[BU i ]

ln[] e qi e ln[QU i ]

j ) 1, M i ) 1, N i ) 1, N i ) 1, N

where vj ) ln[Vj], bi ) ln[Bi], qi ) ln[Qi], and tLi ) ln[TLi]. The design problem is solved for a plant consisting of three stages (reactor, centrifuge, and mixer) and two products with no parallel units. To prevent model

Table 2. Multiproduct Batch Plant Data size factor, Sij (L/kg)

processing time, tij (h)

product

stage 1

stage 2

stage 3

stage 1

stage 2

stage 3

A B

2 4

3 6

4 3

8 16

20 4

8 4

stage

ηj ($/L)

βj

product

profit margin, Pi ($/kg)

1 2 3

250 500 340

0.6 0.6 0.6

1 2

5.5 7.5

horizon time

6000 (h)

discontinuity, a small bound on Qi g is enforced, in case the demand point is feasible. A 25-point demand space for products A and B was randomly generated assuming a bivariate normal distribution with a variance (σ2) ) 36 and a correlation coefficient (F) ) 0.7 and is shown in Figure 9. Note that the points correspond to a set of customers with similar demand orders. The objective of the problem is to determine the Pareto set of designs, Vj, that can cover the entire demand space while balancing cost and flexibility. The above model is reformulated in the multiobjective optimization formulation (III), where the overall operating cost, ξ, consists of the annualized production costs. The value of the chance-constraint probability was set at R ) 0.9, expressing the requirement of 90% probability of demand satisfaction for each demand point around the mean. It is assumed that each demand point represents 10 individual customers. The application of the proposed approach is initialized by assuming three clusters. Steps 1 and 2 of the proposed approach are performed where the demand space is first clustered using fuzzy clustering and then a multiobjective optimization problem is solved for each of the three clusters. A Pareto design with an expected profit of $3.5 × 104 is obtained for the cluster illustrated by ] in Figure 10, whereas a design with an expected

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Figure 10. Case study 2: Pareto designs for iteration 1. Table 3. Multiproduct Batch Plant: Final Set of Designs design Figure 10 (]) Figure 10 (+) Figure 10 (f) Figure 11 (+) Figure 11 (4) Figure 12 (4) Figure 12 (f) Figure 13 (f) Figure 14 (+) Figure 14 (4) Figure 15 (])

mixer reactor centrifuge expected (L) (L) (L) profit reliability 1797 3265 4368 3062 3351 3648 4446 1986 2985 3314 4020

2696 4898 6552 4594 5027 5473 6670 2979 4342 4971 6029

3595 6531 8736 6125 6703 7297 8893 3972 5789 6628 8039

3.5 × 104 7.4 × 104 1.0 × 105 6.7 × 104 4.3 × 104 5.1 × 104 9.9 × 104 2.5 × 104 3.9 × 104 4.5 × 104 9.0 × 104

0.53 0.87 0.86 0.71 1.0 0.91 0.71 0.95 1.0 0.75 0.75

profit of $7.4 × 104 is obtained for the cluster illustrated by + and, for the cluster illustrated by f, a Pareto design with an expected profit of 1.0 × 105. The volumes of the stages along with the design reliabilities are listed in Table 3. The feasible regions of the designs are obtained with grid-search optimizations and are illustrated with dots in Figure 10. Steps 1-5 of the proposed approach are then repeated, increasing the number of clusters at each iteration until the iterative procedure converges at the 7th iteration, which resulted in 9 clusters over the demand space. The Pareto designs obtained are illustrated in Figures 11-15, and the results are summarized in Table 3. As illustrated in Figure 16 for the cluster shown by ] in Figure 15, the decision maker has a number of available optimal designs on the Pareto surface from which he can select according to his priorities. For example, although the selected Pareto design at this case was the design (X), closest to the utopia point, with an expected profit of 9.0 × 104 and a reliability of 0.75, design (Y) can be selected with an expected profit of 5.1 × 104 if reliability is of higher priority. To summarize the results, at the end of the optimization process, a set of 12 robust standardized designs have been obtained that span the demand space. Furthermore, the set of designs on the Pareto surfaces

allows a robust selection of customized designs based on specific customer preferences. 4. Summary and Future Directions A novel framework is presented in this paper for the integration of data analysis and design optimization under demand uncertainty to create a portfolio of standardized designs to span the demand space. The main idea is to cluster the demand data using fuzzy clustering, solve a multiobjective optimization problem involving cost, model robustness and solution robustness as three independent objectives, and iteratively improve the set of designs to generate a Pareto optimal portfolio of designs that span the demand space. The importance of the research is based on the fact that developing a portfolio of modular designs would result in substantial savings for the manufacturer and in a larger number of choices for the customer because of 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. There are a number of interesting extensions such as following: (a) Clustering, which is an important part of the proposed approach, is an active area of research, and a good understanding of the nature of the data aligned with a shape-dependent clustering algorithm with outlier detection schemes could provide tighter designs. (b) The authors have used cost and reliability as design validation factors. Additional user or casedependent objectives could be integrated in the overall approach for a customized set of Pareto designs. (c)

Tighter bounds on the relaxed chance constraints could produce different alternatives. An important extension to this problem is that of modular process synthesis, which would involve the selection of the types and sizes of process units from various alternatives, along with the optimal operating conditions, in order to minimize the overall cost for satisfying the overall demand space. The presence of nonconvexities and uncertainties, modeled in a multiobjective framework, would result in complex mathematical problems, and extensive research needs to be carried out to generate the optimal designs in a computationally feasible time.

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Acknowledgment Financial support from the National Science Foundation under the NSF CAREER program CTS-9983406 is gratefully acknowledged. Appendix A. Pareto Analysis: Line-Search Optimization The line-search methodology summarized here17 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 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: (i) 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

(11)

where µi are the n different objective functions to be minimized and gj and hk are the sets of constraints.


the utopia plane. However, the algorithm is restricted to continuous Pareto frontiers because the line search might identify non-Pareto solutions as Pareto solutions in the discontinuous cases. Appendix B. Fuzzy Clustering Partitional clustering algorithms divide a data set into clusters or classes. In real applications, there are often no sharp boundaries between clusters so fuzzy clustering is often better suited for the data. Fuzzy clustering is a generalization of partitioning where each object is spread over various clusters and the degree of belonging of an object to different clusters is quantified by means of membership coefficients, which range from 0 to 1, with the constraint that the sum of their values is 1. It has the advantage that it does not force every object into a specific cluster. The fuzzy clustering algorithm, FANNY,19 aims at the minimization of the following problem: n

Figure 16. Case study 2: Pareto surface at the 6th iteration. k

min

∑

µiv2µjv2d(i,j) ∑ i,j)1 n

v)1

2

µjv2 ∑ j)1

s.t. µiv g 0

∀ i, v

n

∑ µi,v ) 1

v)1

Figure 17. Proposed approach for Pareto analysis.

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 (11) 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 17. Step 2. The second step is to determine the convex hull (utopia plane) of the utopia points, which in a twodimensional problem would be the line joining the utopia points, in a three-dimensional 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

(12)

gj(x) e 0 hk(x) ) 0 B ∆µi µPareto ) µdis + δN 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 17). Thus, the algorithm determines a set of points on the Pareto frontier, by initializing the search from a uniform set of points on

where d(i,j) represents the given distances (or dissimilarities) between objects i and j, whereas µiv is the unknown membership of object i to cluster v. The constraints imply that membership cannot be negative and that each object has a certain total membership distributed over different clusters. When each object has equal membership in all clusters, the clustering is entirely fuzzy. On the other hand, when each object has a membership of 1 in some cluster and zero membership in all other clusters, the clustering is entirely hard. Additional details are available in the book by Kaufman and Rousseeuw.19 Literature Cited (1) Douglas, J. M. Conceptual Design of Chemical Processes; McGraw-Hill Inc.: New York, 1988. (2) Herder, P. M.; Weijnen, M. P. C. Quality Criteria for Process Design in the Design Process-Industrial Case Studies and an Expert Panel. Ind. Eng. Chem. Res. 1998, 22, S513. (3) Seepersad, C. C.; Cowan, F. W.; Chamberlain, M. K.; Mistree, F. Strategic Design: Leveraging and Innovation for a Changing Marketplace. Comput.-Based Des.: Eng. Des. Conf. 2002. (4) Grossmann, I. E.; Sargent, R. W. H. Optimal Design of Chemical Plants Design with Uncertain Parameters. AIChE J. 1978, 24, 1021. (5) Halemane, K. P.; Grossmann, I. E. Optimal Process Design under Uncertainty. AIChE J. 1983, 29, 425. (6) Varvarezos, D. K.; Grossmann, I. E.; Biegler, L. T. An OuterApproximation Method for Multiperiod Design Optimization. Ind. Eng. Chem. Res. 1992, 31, 1466. (7) Pistikopoulos, E. N.; Ierapetritou, M. G. A Novel Approach for Optimal Process Design under Uncertainty. Comput. Chem. Eng. 1995, 19, 1089. (8) Bernardo, F. P.; Pistikopoulos, E. N.; Saraiva, P. M. Integration and Computational Issues in Stochastic Design and Planning Optimization Problems. Ind. Eng. Chem. Res. 1999, 38, 3056.

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Received for review July 26, 2004 Revised manuscript received February 21, 2005 Accepted March 2, 2005 IE049336Y

Multiobjective Framework for Modular Design Generation

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