Robust Optimization Method for the Economic Term in Chemical

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Ind. Eng. Chem. Res. 2001, 40, 5950-5959

Robust Optimization Method for the Economic Term in Chemical Process Design and Planning Min-ho Suh and Tai-yong Lee* Department of Chemical Engineering, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea

This paper presents a framework for robust optimization of the economic term in chemical process design and planning. Robust optimization models are considered as a multiobjective optimization problem whose objectives are the expected performance and the robustness. The Pareto-optimal subset condition for robust economic optimization is developed. Various types of robustness measures are discussed from the point of view of the Pareto-optimal subset condition and numerical implementation. As a result, the cost of the worst-case scenario is proposed as the robustness measure for cost-minimization problems. A decision-making procedure is proposed to select the best robust design among the robust design alternatives obtained by the proposed robust model. Examples illustrate the adequateness of the proposed robustness measure and the efficacy of the proposed decision-making procedure. 1. Introduction It has long been accepted that the effects of various sources of uncertainty should be considered during both process design and operation.1 Uncertainties can arise from variations in either external parameters, such as the quality of the feed streams, product demand, environmental conditions, and economic cost data, or internal process parameters, such as physical properties and reaction constants.2 Both external and internal parameter uncertainties can cause unexpected performance variations, so that robust design and operation is conceived as a task of great significance. Many robust optimization models have been proposed to cope with the uncertain parameter variations. In this context, the major research goals of robust optimization study are to develop a quantitative representation of uncertainties and a robust optimization model that can cope with the uncertain parameter variations.3 In representing the uncertain parameters, the scenariobased approach is one of the most commonly applied methods. The implicit scenario-based approach is applied when the uncertain parameters are described by (joint) probability density functions, so that the stochastic flexibility or the probability of feasible operation can be evaluated and controlled.1,4,5 The explicit scenariobased approach is often used if one knows a priori which combinations of parameter realizations and associated probabilities are important.6-8 When the uncertain parameters cannot be represented as well-known distribution functions (e.g., uniform, Gaussian, etc.) and when large discrepancies between forecasted scenarios are seen, then a reasonable choice is to represent the uncertain parameters as the realizable scenarios and their probabilities. The explicit scenario-based approach is applied to the uncertain product demand in this paper. The multiscenario economic optimization problem driven by applying any scenario-based approach to the * To whom all correspondence should be addressed. Telephone: +82-42-869-3926. Fax: +82-42-869-3910. E-mail: tylee@ mail.kaist.ac.kr.

corresponding deterministic model can be regarded as a multiobjective problem in which the cost vector of the scenarios should be optimized. In stochastic programming approaches, the expected value of the objective performance is taken as the objective variable in multiscenario models.9,10 We can expect that the system would behave optimally in the mean sense if the stochastic programming model solution were implemented. However, the system might show poor performance at a particular realization of scenarios such as the worst-case scenario. This means that the stochastic model cannot reflect the variability of performances for each scenario realization. A robust optimization model based on the stochastic model has the additional objective of controlling the variability of performances of individual scenarios. In the context of the number of objectives, a robust optimization problem is clearly another multiobjective optimization problem in which the expected performance and the robustness measure are the two objectives. We should resort to a multiobjective optimization (MOO) technique11 to solve robust optimization models. The Pareto-optimal solutions obtained by a MOO technique would be the robust solutions that we want to generate in a robust optimization. The -constraint method is one of the most favored methods of generating Pareto-optimal solutions. According to the -constraint method, the expected performance is taken as the objective function value, while a robustness measure is constrained so as not to exceed a tolerance value. A measure for evaluating the robustness of a feasible solution is essential for the robustness to be considered as an objective together with the expected performance in robust optimization models. We denote the economic term as the cost (or profit), which is the objective performance in most optimization problems. Reducing variability itself cannot be an objective in robust optimizations of the economic term because the robustness concept for optimizations of the economic term should be focused on reducing the costs of scenarios as much as possible for any realization of scenarios from the point of view of cost minimization. This can be paraphrased by saying that any optimum solution of the

10.1021/ie0005147 CCC: $20.00 © 2001 American Chemical Society Published on Web 11/16/2001

Ind. Eng. Chem. Res., Vol. 40, No. 25, 2001 5951

robust model must be a Pareto optimum of the original multiscenario optimization problem. The worst-case analysis model takes the performance of the worst-case scenario as the objective variable.12,13 Because it concerns only the worst-case scenario realization, the cost of the resulting design can be very high even if the probability of the worst-case scenario is very low. The worst-case analysis model, which has only the one objective of minimizing the worst-case cost, is not recommended for economic terms. We propose the worst-case cost as the robustness measure that, together with the expected cost, forms a multiobjective costminimization problem. It is guaranteed that the optimum solution obtained from the resulting robust model is a Pareto optimum of the multiscenario problem. The advantage of the proposed method is that the design alternatives that are obtained are all acceptable from the viewpoint of the multiscenario cost minimization. This means that the cost realizations of all scenarios are not simultaneously higher than those obtained from the stochastic optimization problem. In particular, the cost realization of the worst-case scenario is always lower than or equal to that of the stochastic optimization problem. A major drawback will be that we cannot use the probability information of the problem in defining the robustness measure. This drawback can be alleviated, however, by the decision-making procedure, which utilizes the probability information. Samsatli et al.1 proposed a one-sided robustness metric that can evaluate the extent to which inequality constraints are violated. This type of partial mean can be used for the economic term because the one-sided feature is actually adequate for robust optimizations of the economic term. Also, the Pareto optimality of the solutions obtained from the robust model for the original multiscenario problem is guaranteed. However, a significant difficulty is that the desired target value in the definition of the measure cannot be determined easily for economic terms. From the viewpoint of reducing the variability of the performance vector of scenarios, the variance was proposed as the robustness measure.8,14 The mean of the absolute deviation might be the next choice because it does not introduce any nonlinearity, and a variant of the mean of absolute deviation was also proposed.15 The variance-based robustness measures, which are mentioned above, are not suitable for robust optimization of economic term because the Pareto optimality of the solutions obtained from the robust models can not be guaranteed. The application of those robustness measures to the optimization of an economic term, namely, to cost minimization, might generate unacceptable results for which the cost realization of each scenario is higher than that obtained from the stochastic optimization problem because the robust optimization penalizes costs both higher and lower than the expected cost. Robust design alternatives are provided with tradeoff information between the expected performance and the robustness measure using the Pareto curve. In most applications of robust optimization, the selection of the best alternative design from among the provided robust solutions is transferred to a decision maker’s preference for the two objectives. However, deciding the best robust solution in the Pareto curve from the point of view of MOO is also an important issue. We will provide a reliable procedure for determining the best alternative

for the proposed robustness measure for a unified framework of robust optimization. The objective of this paper is to propose a framework that includes a robust model equipped with a proper robustness measure and a corresponding decisionmaking procedure for the robust optimization of the economic term. This paper is organized as follows: The general modeling framework for robust optimizations, which includes the theoretical background and the solution algorithm, is explained in section 2. A discussion of robustness measures from the viewpoint of the Pareto optimality for the multiscenario cost-minimization problem and numerical implementation is presented in section 3. In section 4, we propose a decisionmaking procedure for determining the best solution for among the robust design alternatives generated using the proposed robust model. Two examples are addressed in section 5 to illustrate the efficacy of the proposed model and the corresponding decision-making procedure. Finally, conclusions are drawn in section 6. 2. General Modeling Framework of Robust Optimization At the design stage of a process, it is not unusual that the process has a certain degree of uncertainty, which should be treated properly by the designer. When the process is represented by a mathematical model, the uncertainty is often expressed in terms of model parameters, and the distinction between the design and control variables becomes vital to the designer.16 Design variables determine the structure of the process and the size of the production modules, which must be established before the uncertain parameters reveal themselves. In contrast, the actual values of the control variables will be determined after the realization of the uncertain parameters, because these variables are used to adjust the mode and level of production in response to disruptions in processing, changes in demand or production yield, and so on. Consider following deterministic optimization model

minC x,y

s.t. C ) f(x,y;θ) h(x,y;θ) ) 0 g(x,y;θ) e 0 x ∈ U, y ∈ V where x and y are vectors representing the design and control variables, respectively; C is the cost to be minimized; θ is the vector of uncertain parameters; f(x,y;θ) is an economic objective function; and h(x,y;θ) and g(x,y;θ) are vectors of equality and inequality constraints, respectively, that describe the process. Applying the explicit scenario-based approach to the deterministic model by assuming a finite number of scenarios {1, 2, ..., N} to represent the uncertainty results in a multiscenario economic optimization problem, which is actually a multiobjective optimization problem described as follows:

min{C1, ..., CN} x,ys

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Ind. Eng. Chem. Res., Vol. 40, No. 25, 2001

subject to

Cs ) fs(x,ys;θs) s ) 1, ..., N

(1)

hs(x,ys;θs) ) 0 s ) 1, ..., N

(2)

gs(x,ys;θs) e 0 s ) 1, ..., N

(3)

x∈U

(4)

ys ∈ V s ) 1, ..., N

(5)

where x is the vector of design variables and ys is the vector of control variables corresponding to scenario s. Cs is the cost under scenario s calculated from the objective function fs(x,ys;θs). The uncertain parameter realization and its associated probability under scenario s are denoted by θs and ps, respectively. hs(x,ys;θs) and gs(x,ys;θs) are, respectively, the vectors of equality and inequality constraints describing the process under scenario s. The multiscenario optimization problem has the objective of minimizing the cost vector {C1, ..., CN}. In many cases, no solution exists that minimizes the cost vector simultaneously, and the concept of Pareto optimality is introduced to handle multiobjective problems. A feasible solution {C1P, ..., CNP} is called Paretooptimal if there is no other feasible solution {C1F, ..., CNF} such that CsP g CsF for s ) 1, ..., N with CsP > CsF for at least one s. The Pareto-optimal solution is generally not unique even if the corresponding deterministic problem is convex. Hence, the solution procedure consists of two phases: generation of a Paretooptimal set and selection of a particular solution from the set.17 From a practical point of view, generation of complete set is neither required nor recommended. In the present study, the following two criteria are used to generate a meaningful subset and to select a particular solution from the subset: (1) The probability of each scenario must be considered so that the scenario with a higher probability is treated with a higher priority. (2) No particular scenario can be realized with an unacceptably high cost. This introduces the robustness concept into the problem.14 The consideration of probability naturally leads to stochastic programming that minimizes the expected cost. The standard stochastic program is described by

h minC x,ys

(6)

subject to

∑psCs

R(C) e

(7)

s)1

and eqs 1-5. C h is the expected cost. Let the optimal solution of the stochastic model be denoted by {C1S, ..., h S. The CNS} and the minimum expected cost by C solution is a Pareto-optimal solution of the original multiscenario problem. Otherwise, at least a feasible solution must exist. Let the solution be denoted by {C1F, h F such that CsS g ..., CNF} and the expected cost by C F S F Cs for s ) 1, ..., N with Cs > Cs for at least one s. h F, but it contradicts the fact This implies that C hS > C that C h S is the minimum expected cost of the stochastic model. This proves that the stochastic model solution is a Pareto solution of the multiscenario problem. Because the stochastic model uses the probability of each scenario as its weighting factor, the cost realization

(8)

R(C) is a function of costs and defines the robustness measure. is the tolerance for the robustness measure. The robust model can be considered as another representation of a multiobjective problem whose objective is to minimize the expected cost C h as well as the robustness measure R(C). We can generate a set of robust solutions that are Pareto-optimal solutions of the two objectives because the robust model is a representation based on the -constraint method,15 which minimizes the expected cost while keeping the robustness measure below an acceptable level. From eq 8, it is clear that should be greater than or equal to the achievable minimum of the robustness measure, which can be determined from the model given by

minR(C) x,ys

(9)

subject to eqs 1-5. The minimization problem will be referred to as the purely robust optimization model, which is a special case of the robust model defined by eqs 1-8. The solution C of the purely robust problem might not be uniquely determined. In this case, the robust model should be subsequently solved with set to the minimum of the purely robust problem. To obtain the Pareto solutions of the robust optimization model, we first solve the stochastic model to obtain the upper bound of the robustness measure. Next, we solve the purely robust model to obtain the lower bound of the robustness measure. Finally, we solve the robust model to obtain the Pareto solutions of the multiobjective problem. The parameter , the tolerance of the robustness measure, is bounded by the solutions of the stochastic model R(CS) and of the purely robust model R(CR)

R(CR) e e R(CS)

N

C h )

might be unacceptably high for a certain scenario with especially low probability. This necessitates the minimization of an additional objective that measures the unacceptably high costs. To handle the tradeoff associated with the expected cost and the additional objective, Mulvey et al.14 proposed the concept of robustness. A general form of a robust optimization model consists of the stochastic model given in eqs 1-7 and a constraint on the additional measure given by

(10)

If > R(CS), then the constraint given by eq 8 is inactive, and the robust model is reduced to the stochastic model. If < R(CR), the robust model becomes infeasible. After the number of the Pareto solutions, NP, is determined, the robust solutions are sequentially obtained by varying the tolerance . The robust optimization algorithm comprises the following steps. Algorithm: Generating the Robust Solution Set. Step 1. Solve the stochastic model given by eqs 1-7 and denote the optimum costs by CsS, s ) 1, ..., N. Set U ) R(CS). Save the obtained objective function value as C h S. Step 2. Solve the purely robust model given by eqs 1-5 and 9 and denote the optimum value by R(CR). Set L ) R(CR).


Step 3. Calculate the increment of , ∆, using

∆ )

U - L , NP ) number of robust solutions NP

Step 4. For iteration count k ) 0, set 0C h )C h S and 0R(C) ) U. Step 5. k ) k + 1. ) U - k∆. Solve the robust model given by eqs 1-8. h )C h and kR(C) ) R(Ck). Set kC Step 6. If k ) NP, then go to step 7. Otherwise, go to step 5. h and kR(C), k ) 0, ..., NP. Stop. Step 7. Print kC Using this algorithm, we can obtain NP + 1 robust h , kR(C)), k ) 0, ..., NP. design alternatives, (kC Pareto optimality between the expected cost and the robustness measure of the robust model solution obtained by the above algorithm does not necessarily meet the Pareto optimality of the original multiscenario problem. We need to verify the Pareto optimality of robust solutions for the multiscenario problem under the assumption that the robust solutions are Paretooptimal solutions between the expected cost and the robustness measure. 3. Robustness Measure As mentioned earlier, robust optimization models are characterized by the presence of a robustness measure R(C) as a constraint, so that the choice of the robustness measure is an important issue in robust optimization. A necessary condition for R(C) is that the Paretooptimal set of the associated robust model be a subset of the Pareto-optimal set of the original multiscenario problem. This condition will be referred to as Paretooptimal subset condition of the robust problem. The Pareto-optimal subset condition is indeed necessary because the final solution will be chosen as one of the Pareto-optimal solutions of the robust problem, and the failure to meet this necessary condition might lead to the choice of an inferior solution in the decision-making phase. We claim that the optimum solution, if it exists, of the robust model defined by eqs 1-8 is a Pareto solution of the original multiscenario problem if

R(C(1)) e R(C(2)) for all C(1) and C(2) such that C(1) e C(2). The inequality C(1) e C(2) means Cs(1) e Cs(2) for all s ) 1, ..., N, i.e., R(C) is a monotonically increasing function of each Cs. To prove this claim, let {C1R, ..., CNR} be the optimal solution of the robust model with C h R as the corresponding minimum expected cost, and let {C1F, ..., CNF} be a feasible solution of the original multiscenario problem with C h F as the corresponding expected cost. Assume that CsR g CsF for s ) 1, ..., N with CsR > CsF for at least one s. Then, CF satisfies eqs 1-5 because it is a feasible solution of the original problem, and it satisfies eq 8 because R(CF) e R(CR). Hence, it is a feasible solution h F, which contradicts of the robust problem and C hR > C the robust optimality of CR. Therefore, the monotonicity condition of R(C) is sufficient for a robust problem to meet the Pareto-optimal subset condition. The worst-case analysis model

R(C) ) Cw ) max{C1, ..., CN}

serves as a proper robustness measure of any costminimization problem because the robustness measure is a monotonically increasing function of each Cs and the Pareto-optimal subset condition is fulfilled by the associated robust problem. The robust worst-case (WC) model is described by eqs 1-7 and

Cs e WC s ) 1, ..., N

(11)

where CwR e WC e CwS, CwR ) R(CR) ) min{C1R, ..., CNR}, and CwS ) R(CS) ) min{C1S, ..., CNS}. We propose to use the worst-case cost as the robustness measure of the economic term in multiscenario optimization problems for several reasons. As we mentioned, this approach guarantees the Pareto-optimal subset condition. Second, the associated robust WC problem is well-defined in the sense that the problem has only one undetermined parameter WC, which varies in the predetermined range to generate design alternatives. Finally, the numerical implementation shows that the design alternatives generated using the robust WC model include all of the acceptable alternatives generated by other models and even more, as shown later. The major drawback of worst-case analysis model is that it does not count the probability of scenarios, and the solution might be too conservative even if the probability of the worst-case scenario is extremely small. In section 4, a decision-making procedure is proposed that enables us to overcome this drawback of using the worst-case analysis model as the robustness measure. The partial mean model can be stated as N

R(C;C*) )

∑ps max{0,Cs - C*}

s)1

where C* is a target value above which costs of scenarios should be penalized. If a fixed target value is used for C*, the associated robust partial mean (PM) model can be described by eqs 1-7 and N

R(C;C*) )

∑ps max{0,Cs - C*} e PM s)1

(12)

where R(CR;C*) e PM e R(CS;C*). The robust PM model also satisfies the Pareto-optimal subset condition because the robustness measure, the partial mean in this case, is a monotonically increasing function of each Cs, regardless of the choice of C*. Samsatli et al.1 proposed the concept of a one-sided robustness metric and presented successful implementations of the partial mean for penalizing the extent of an inequality constraint violation on a desired target value. The desired target value could be set to the required level of the final product concentration for the reactor systems in their applications. The meaningful range of C* is given by

CbS e C* e CwS where CbS ) min{C1S, ..., CNS} and CwS ) max{C1S, ..., CNS}. For C* < CbS, the robust PM problem either reduces to a stochastic problem or becomes infeasible depending on PM. For C* > CwS, the robust PM problem always reduces to the stochastic problem. Although the meaningful range of C* is given on the basis of the stochastic model solution, the choice of C* is still a nontrivial task because no specific target value is known

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a priori to be better than the others for producing design alternatives. If we vary both C* and PM, the robust PM problem requires an excessive amount of computation and still presents difficulties in the choice of C*. Ahmed and Sahinidis15 proposed to use the so-called upper partial mean for which C* ) C h in eq 12. Despite their claim that the upper partial mean N

R(C) )

∑

N

ps max{0,Cs -

s)1

ptCt} ∑ t)1

is asymmetric, it is not monotonic because R(C) ) R(-C). The upper partial mean differs from the previously mentioned monotonically increasing robustness measures from the viewpoint of the Pareto-optimal subset condition. In fact, it is always one-half of the mean of absolute deviation defined by N

R(C) )

∑ps|Cs - Ch | s)1

The mean of the absolute deviation is not monotonic for the same reason as the upper partial mean. The robust mean of the absolute deviation (MAD) model is described by eqs 1-7 and N

R(C) )

∑ps|Cs - Ch | e MAD

(13)

s)1

The robust solutions obtained by the robust MAD model might not satisfy the Pareto-optimal subset condition. Later in this paper, an example will be presented whose robust solution is not a Pareto solution of the original multiscenario optimization problem. The variance model is stated as N

R(C) )

∑ps(Cs - Ch )2 s)1

and it also has a possibility of violating the Paretooptimal subset condition because R(C) ) R(-C) and R(C) not monotonic. Neither the mean of absolute deviation (and the upper partial mean) nor the variance should be used as the robustness measure of the economic term because the final decision based on the Pareto optima of the associated robust problem does not guarantee the Pareto-optimal subset condition. 4. Decision-Making Procedure Using the proposed robust WC model, we can generate a Pareto set for two objectives, the expected cost and the worst-case cost. The solution of the multiobjective problem consists of two phases. In the first phase, the Pareto set is determined, and in the second phase, the decision maker selects the preferred best solution from among all of the candidates according to the decision maker’s own criteria. Any attempt to select the best from among the Pareto solutions is a subjective decision in nature, but we still need the guideline that reflects the probability information of the problem. An important issue in multiobjective decision making is how to scale the objective variables. A widely used scaling technique is so-called zero-one normalization,18 which scales each objective variable so that it varies from 0 to 1 on the Pareto curve. In many multiobjective

Figure 1. Pareto curves with different scales. The scales of the squares and circles are 1 (not scaled) and pw, respectively. The point k represents the Pareto solution with WC ) WCU - k∆WC.

decision-making situations, especially those that suffer from the difficulty of incommensurability, the normalization technique is the natural choice unless a problemspecific scaling factor exists. Let us now consider the present multiobjective stochastic problem, in which two objective variables are expressed in terms of monetary values indicating that incommensurability is not severe. Usually, the worstcase scenario is characterized by an extremely low probability, which must be taken into account during the decision making. The probability of the worst-case scenario cannot be properly represented, however, if zero-one normalization is used to depict the Pareto curve. Hence, the scaling factor must be a function of the probability of the worst-case scenario pw to satisfy the representational theory of preference measurement,23 so that the selected robust optimal solution is closer to the stochastic model solution for smaller pw and to the worst-case analysis solution for larger pw. This suggests that the scaling factor be Wpw where W is a kind of weighting factor reflecting the decision maker’s preference for the worst-case cost. Given the Pareto solutions (C h ,R(C)) ) (C h ,Cw), the scaled solutions can be stated as (C h ,WpwCw). It seems reasonable to set W ) 1, but nonunity values can be selected to reflect the decision maker’s preference. Theoretically, as the relative weighting factor between the two objectives approaches 0, the decision converges to the stochastic model solution. Conversely, as W f ∞, the decision converges to the worst-case analysis solution. Figure 1 shows two different representations of the h ,pwCw) Pareto curve, (C h ,Cw) denoted by squares and (C denoted by circles, on the same graph. The point E stands for the stochastic model solution, and the point


W stands for the worst-case analysis solution. C h S and R Cw are the expected cost of the stochastic model solution and the worst-case cost of the worst-case analysis solution, respectively. The point I stands for the ideal point (C h S,CwR), which is the reference point for decision making. The best robust solution is chosen as the point on the Pareto curve that is closest to the ideal point.19 Any type of metric can be used to evaluate the closeness of the robust solutions to the ideal point. The figure shows that the best solution is moved from the point 8 to point 4 because of the reflected low probability of the worst-case scenario. The decision-making procedure for the best robust solution can be summarized as follows. Procedure: Finding the Best Robust Solution. h and kCw, k ) 0, ..., NP, Step 1. Given kC Set the weight W. (A typical setting will be equal to 1.) Step 2. Draw the scaled Pareto curve (C h ,WpwCw). Step 3. Find the best robust solution k by evaluating the robust solutions k

k

S

h ,Wpw Cw) - (C h ,WpwCw )|| min||( C k

A parameter case study is carried out to illustrate the performance of the proposed robustness measure and other measures, and the results are addressed in subsection 5.1. The important observation in this case study is not that the results of our approach are similar to those of other approaches, but rather that our approach generates only meaningful solutions from the point of view of the optimization of the economic term. In subsection 5.2, we present the results from the application of the proposed framework to a long-term planning problem. The aim of our experiment is to demonstrate how the proposed decision-making procedure serves to determine the best robust solution. For all example problems, the robust optimization formulation was solved using the MIP solver of OSL.20 The modeling system GAMS21 was used to implement the model, and all computations were carried out on a Sun SPARC 10 station. 5.1. Comparison of Robustness Measures. Various robust optimization models are investigated, including the proposed WC model, the robust PM model with C* ) C h S, and the robust MAD model. The robust variance model is not included in this case study because the performance of the variance as the robustness measure is similar to that of MAD. To treat the maximization of profit, the cost C should be replaced by the profit P in the previous discussion, and eqs 6, 11, and 12 should be changed to

h maxP

(6′)

Ps g WC s ) 1, ..., N

(11′)

and N

∑ps max{0,P* - Ps} e PM

(12′)

s)1

respectively. Also, eqs 8 and 9 should be understood as

R(P) g

product A stage 1 stage 2 stage 3

(8′)

processing time (h)

product B

product A

product B

16 24 12

16 40 16

32 8 8

8 12 16

horizon (h)

8000

Table 2. Economic Data for Example 1a

stage 1 stage 2 stage 3

cost coefficient aj

cost exponent bj

200 200 200

0.6 0.6 0.6

price of product ($/kg) A B

inventory cost ($ ton-1 h-1)

5.5 5.5

A B

7 7

a Plant expected life ) 10 years; taxation rate ) 45%; interest rate ) 10%; setup cost per interchange ) $10.

Table 3. Demand Scenarios and Probabilities for Example 1 demand (kg) for case 1 demand (kg) for case 2

1 2 3

5. Results and Discussion

R(P;P*) )

size factors (L/kg)

scenario product A

R

x,ys

Table 1. Production Data for Example 1

1500 2000 1800

product B

product A

1500 1000 1200

1500 2000 1400

product B probability 1500 1000 1600

0.5 0.3 0.2

and

maxR(P) x,ys

(9′)

respectively, only for the worst-case analysis model. The algorithm described in section 2 is used to test the performances of the robust models. Example 1. Voudouris and Grossmann22 introduced a case study of optimal synthesis of multiproduct batch plants, which involves aspects of synthesis, design, production planning, and scheduling. A simple model, in which one unit in each stage and the fixed allocation of tasks to equipment is considered, is adopted in this example. The model is sufficient to compare the performance of the proposed robustness measure with that of other measures in the robust optimization of the economic term. Design variables such as the selection of equipment of standard size are independent of the scenarios. In contrast, control variables such as the length of the production cycle and the sequencing of the products can be dependent on the scenarios because their optimal values are determined on the basis of the values of the uncertain parameters realized in the scenario. The objective variable is the net present value (NPV) that we want to maximize. Detailed descriptions of the mathematical formulation of this problem can be found in Voudouris and Grossmann.22 The process comprises three serial batch stages with one unit per stage. The equipment on the stages is available in the following eight discrete sizes: 190, 205, 220, 235, 250, 265, 280, and 300 L. Two products are produced at this facility. The production data for this example are shown in Table 1, and the cost data are shown in Table 2. Demand for the products is uncertain, and three demand scenarios are considered. Data for the demand scenarios are given in Table 3. This simple example was taken so that the performances of the robustness measures for generating robust solutions could be compared. The computational time required to obtain a robust solution using the tested models varies from 30 to 90 s depending on the given value of

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Figure 2. Robust solutions for case 1 of example 1. The robustness measure is given by (a) the cost of worst-case scenario, (b) the partial mean, and (c) the mean of absolute deviation.

robustness measure tolerances. Using the three robust models, robust solutions are obtained for two cases with different parameter values. Case 1. Figure 2 shows variations of the expected NPV and the NPV of each scenario as functions of the robustness measure tolerances. For each measure, a spectrum of solutions was generated by progressively enforcing robustness, i.e., by increasing WC for a, and decreasing PM and MAD for b and c, respectively. As shown in Figure 2a, three alternative solutions are obtained from the robust WC model, including the

stochastic solution for small WC and the purely robust solution for large WC. In the intermediate range of WC, namely, 37 926 e WC e 38 220, a new robust solution is found. All of the robust solutions are acceptable because they are the Pareto-optimal solutions of the multiscenario problem. Figure 2b shows the result of the robust PM model. Because the purely robust solution of PM model corresponds to the robust WC solution with intermediate WC, only two alternative solutions are obtained from the robust PM model. Another solution can be obtained for a certain target value, but it can be found by trial-and-error or by the aid of other robustness measure, indicating the relative inefficiency of PM as a robustness measure. Using the robust MAD model, we found three alternative solutions, which are the same as those found using the robust WC model. In this specific case, the robust MAD model is successfully used to generate the robust design alternatives. However, this is not always true, and a robust MAD model might generate unacceptable solutions because the Pareto optimality for the solution of the multiscenario model cannot be guaranteed. This problem will be discussed in case 2. Case 2. The system parameters for case 2 are summarized in Table 3, and its robust optimization results are depicted in Figure 3. The robust WC model finds two design alternatives, as shown in Figure 3a. As shown in Figure 3b, the robust PM model fails to find any design alternatives other than the stochastic solution when the target value is set to the expected NPV of the stochastic problem. The robust MAD model generates three design alternatives as its Pareto solutions. The solutions shown in Figure 3c indicate that scenario 3 is the worst for MAD ) 130, with a corresponding NPV of $38,174. It also shows that scenario 3 is the best for MAD e 26, with a corresponding NPV of $38,048, which is less than the worst-case NPV of the stochastic problem. This means that the robust solution for MAD e 26 is unacceptable in the context of maximizing profit. The robust MAD (or variance) models have a chance of generating unacceptable design alternatives because MAD (or variance) is reduced by sacrificing the absolute value of each profit. We presented parameter case studies for a simple example to support our argument of proposing the worst-case value as the robustness measure for robust optimizations of the economic term. In summary, the proposed robust optimization algorithm based on the robust WC model is devised so that its Pareto solutions are guaranteed to satisfy the Pareto-optimal subset condition and are acceptable from an economic point of view. The robust PM model is not suitable for robust optimizations of the economic term from the viewpoint of numerical implementation. A robust model with MAD or variance should not be applied to robust optimizations of the economic term because there is a possibility of generating unacceptable solutions. 5.2. Decision Making of the Best Robust Solution. A long-term planning problem is addressed to illustrate the efficacy of the proposed framework, including the decision-making procedure for the best robust solution. We can generate a number of robust design alternatives for this problem, so that we need to determine the best robust solution among the alternatives. We show that the best solution can be identified in a reliable way using the proposed decision-making procedure.


Figure 3. Robust solutions for case 2 of example 1. The robustness measure is given by (a) the cost of worst-case scenario, (b) the partial mean, and (c) the mean of absolute deviation.

Example 2. In this example, the investment planning of the Korean petrochemical industry under uncertain demand forecast scenarios is addressed. Bok et al.8 proposed a robust model for this problem, but the robust model used the variance as the robustness measure. This problem is revisited here to illustrate the proposed robust optimization framework, including the decisionmaking procedure. The complete deterministic model and related parametric data for this problem can be obtained from Bok et al.8 The objective of the multiperiod planning problem is to minimize the discounted sum of the total costs in all periods. The total cost in each period is given by the sum of the raw material, operating, transportation, and capacity expansion or investment costs minus the

revenues from selling the products abroad. Because this investment model deals with a multilocation problem that includes eight large cities in Korea, China, and southeastern Asia as the demand areas, the transportation costs for interplant shipment and the trade (import/ export) costs are considered. The feasible solution set is defined by the equality constraints of material balance and the inequality constraints of plant capacity, market demand, and import/export upper limits. The scenarioindependent design variables include the binary variable of capacity expansion timing and the continuous variable of expansion quantity for each time period. The rest of the continuous variables, which include working levels of the plants for each time period, are scenariodependent control variables. Demands for products are given as three forecast scenarios with probabilities under the assumption that product demand will increase in comparison with the present conditions and the rate of increase is uncertain. Scenario 1 expects a 20% annual growth rate of the synthetic resin (HDPE, LDPE, LLDPE, PP, PS, etc.) market and a 10% annual growth rate for the rest of the products in the market, with a probability of 0.3 to represent the prospect of the Chinese market. Scenario 2 expects a 15% annual growth of aromatic derivatives (PS, ABS, caprolactam, TPA, and phthalic anhydride) and an 8% annual growth rate for the rest of the products, with a probability of 0.3 to represent the prospect of the southeastern Asian market. Scenario 3 expects a 10% annual growth of all products with a probability of 0.4. The problem addressed here is different from that of Bok et al.8 in the context of the applied robust optimization framework. The objective function of the robust model proposed by Bok et al. was the weighted sum of the expected cost, the variance, the expected total excess capacity, and the expected total unmet demand. The proposed robust optimization framework has been applied to the problem under the assumption that the expected total excess capacity and the expected total unmet demand should not exceed 909 billion tons and 70 trillion tons, respectively. The computational time required to obtain a robust solution ranges from 200 to 500 s according to the value of robustness measure tolerances. The general MIP solver OSL20 was used to obtain the robust solutions in this paper because our major concern is not developing an efficient solution algorithm but rather proposing the robust optimization framework. A specialized solution algorithm for this multiperiod planning problem, such as the Benders decomposition algorithm, can be used to reduce the computational effort required to generate the robust solutions. Using the proposed robust WC model, 11 robust design alternatives are obtained. Figure 4 shows the Pareto curve for determining the best robust solution. By setting the relative weighting factor W to 1, the resultant solutions are scaled by the probability of the worst-case scenario, pw ) 0.3. Each point represents the robust design alternatives. Any type of metric can be used to evaluate the robust design alternatives in the Pareto curve, but we proposed to use the L2 metric to measure the distance between the ideal point and the robust design alternatives. Application of the L2 metric indicates that the point k ) 8 is the best robust solution for this problem. Bok et al.8 presented many robust design alternatives for this problem. Among the alter-

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that can be used to graphically represent the Pareto solution. The decision-making procedure was proposed on the basis of the scaled Pareto curve, which provides the probability information to the decision maker and prevents the decision maker from inclining toward the worst-case analysis solution, so that the main drawbacks of proposed worst-case cost measure can be alleviated. It was illustrated through a long-term planning problem under demand uncertainty that a reliable decision can be made using the proposed decisionmaking procedure. Acknowledgment This work was partially supported by the Brain Korea 21 Project and by the Korea Science and Engineering Foundation (KOSEF) through RRC/NMR Project #01C04. Nomenclature

Figure 4. Scaled Pareto curve with W ) 1 for example 2. Table 4. Robust Solutions for Example 2 cost (×109, $) point index

scenario 1

scenario 2 (worst-case)

scenario 3

expected cost (×109, $)

67.88 78.80

135.5 114.3

68.03 79.0

87.63 89.53

83.12 83.35

109.5 113.8

83.28 83.51

91.1 92.55

E k)8 (best proposal) W Bok et al.8

natives that have an expected total excess capacity below 909 billion tons and an expected total unmet demand below 70 trillion tons, we present the best solution in Table 4, along with solutions of the stochastic model, the point k ) 8, and the worst-case analysis model. The solution of the worst-case analysis model is better than the solution by Bok et al.8 for the worstcase cost. The worst-case cost of the point k ) 8 is slightly worse than that of the solution by Bok et al.,8 but the costs of other scenarios for the point k ) 8 are better than that for the solution by Bok et al.8 The results for the point E, k ) 8, and the point W show the tradeoff between the cost of the worst-case scenario and that of other scenarios. The best solution, k ) 8, can be regarded as the compromised solution under the quantified relationship between the worst-case cost and the expected cost. 6. Conclusion A unified framework for robust optimizations of the economic term has been presented in this work. A condition called the Pareto-optimal subset condition was developed as a necessary condition for the robust economic optimization. Based on this condition and the standpoint of numerical implementation, the worst-case cost was proposed as the robustness measure for costminimization problems, and the adequateness was discussed. The solutions obtained by the robust WC model are Pareto-optimal solutions for the original multiscenario cost-minimization problem. The robust optimization problem was considered as a multiobjective problem in which the objectives are the expected cost and the worst-case cost. A scaling rule was proposed

C ) cost Cs ) cost of scenario s C ) cost vector {C1, ..., CN} C h ) expected cost kC h ) expected cost of the kth robust solution C h R ) expected cost of the purely robust model solution C h S ) expected cost of the stochastic model solution C* ) target cost of the partial mean kC ) worst-case cost of kth robust solution w CwR ) worst-case cost of the purely robust model solution CwS ) worst-case cost of the stochastic model solution Cw ) worst-case cost E ) point representing the stochastic model solution f ) objective function fs ) objective function of scenario s g ) vector of inequality constraints gs ) vector of inequality constraints of scenario s h ) vector of equality constraints hs ) vector of equality constraints of scenario s I ) point representing the ideal solution k ) index of robust solutions NP ) number of Pareto solutions N ) number of scenarios P h ) expected profit ps ) probability of scenario s pw ) probability of the worst-case scenario P ) profit Ps ) profit of scenario s P ) profit vector {P1, ..., PN} P* ) target profit of the partial mean R(C) ) robustness measure for cost minimization R(P) ) robustness measure for profit maximization kR(C) ) value of the robustness measure of the kth robust solution s ) subscript, scenario index S ) superscript, stochastic model U ) feasible set of variable x V ) feasible set of variable y W ) relative weighting factor W ) point representing the worst-case analysis solution x ) vector of scenario-independent design variables ys ) vector of control variables under scenario s y ) vector of control variables Greek Letters ∆ ) increment of the worst-case cost measure tolerance ) tolerance of the robustness measure L ) lower bound of the robustness measure tolerance MAD ) tolerance of the mean of absolute deviations PM ) tolerance of the partial mean

Ind. Eng. Chem. Res., Vol. 40, No. 25, 2001 5959 U ) upper bound of the robustness measure tolerance WC ) tolerance of the worst-case cost θ ) uncertain parameters

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Received for review May 24, 2000 Revised manuscript received September 12, 2001 Accepted September 19, 2001 IE0005147

Robust Optimization Method for the Economic Term in Chemical

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