Product and Process Development via Sequential Pseudo-Uniform

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Ind. Eng. Chem. Res. 2004, 43, 4278-4292

Product and Process Development via Sequential Pseudo-Uniform Design Jyh-Shyong Chang* and Jia-Ping Lin Department of Chemical Engineering, Tatung University, 40 Chungshan North Road, 3rd Section, Taipei, Taiwan, ROC

The application of the uniform design (UD) method to nonlinear multivariate calibration by an artificial neural network (ANN) can be used to build a model for an unknown process efficiently because it allows many levels for each factor. If the cost of each experiment is high, low partitioned levels are usually proposed first to carry out the experiments. However, if a reliable ANN model cannot be obtained from the designed experiments, the sequential pseudo-uniform design (SPUD) method developed here can be employed to locate additional experiments in the experimental region. An information free energy index is used to validate the identified ANN model. Once the identified model is verified as reliable, the optimal operating conditions can be determined to guide the process to the desired objective. The simulation results demonstrate that the product and process development based on the proposed method require only a reasonable number of experiments. Introduction In the current competitive market, the manufacturing of innovative products, such as specialty chemicals, ceramics, and composite materials; the finding of recipes; and the development of new processes over time are increasingly urgent. It is also imperative that a new product be developed at minimum cost without sacrificing quality. Traditionally, the models adopted in analyzing the experimental data designed by experimental design methods1,2 are either linear or quadratic. To cope with nonlinear characteristics in many processes, the uniform design (UD) method can be applied to nonlinear multivariate systems.3-5 The main idea behind the UD method is to find a set of representative points scattered uniformly and regularly in the domain to be investigated. The so-called good lattice point (glp) set, which can be generated with the help of number-theoretic methods, might be used to fulfill such a task.6 An experimental design table based on UD principles can be constructed by means of the glp set.6 The multi-input/ multi-output (MIMO) experimental data collected through the UD method can be molded by an artificial neural network (ANN), and the results have proven to be successful in establishing nonlinear and complex relationships between process variables without any prior knowledge of system behavior.3 No guidelines can be found in the related works3,4,6 on how to choose suitable levels in advance for each control factor when designing experiments for an unknown process. Because it is usually time-consuming and costly to obtain experimental data in practice, low partitioned levels are usually proposed first to carry out the experiments. If a reliable mapping model (ANN is adopted in this study) cannot be established on the basis of the previously designed experiments, we propose that the sequential pseudo-uniform design (SPUD) method developed in this study be used to locate additional * To whom correspondence should be addressed. Tel.: +8862-5925252-2561. Fax: +886-2-5861939. E-mail: jschang@ ttu.edu.tw.

experiments in the experimental region. Partition of the coarse levels into fine levels resembles multigrid methods7 adopted in solving high-dimensional nonlinear model equations. The success of multigrid methods stems from the fact that the problem never has to be solved on the fine grid but only has to be solved on relatively coarse grids with dramatically reduced computational efforts.8 Furthermore, to determine the possible multiple optima of an unknown process, a new experimental design scheme that uses UD, SPUD, ANN, random search, fuzzy classification, and information theory is proposed in this study. The main objective of the proposed experimental design scheme is to build a reliable model and locate the optima of an unknown process simultaneously using only a reasonable number of experiments. It is expected that the developed experimental design scheme can be applied in the real world. An alternative approach can be found in the work of Chen et al.9 In the experimental design scheme proposed by Chen et al.,9 an initial batch of experimental data is first collected to construct an ANN model, and then a random search is performed to generate a number of candidates for the next batch experiments. A fuzzy classification algorithm is then used to find the cluster centers of these candidates. An information free energy index is defined to balance the need for better classification and the relevance of each class in optimization. New experiments are performed at these cluster centers to validate the model. The procedure is repeated until an optimal solution is reached. The case studies reported in the original work9 reveal that this scheme focuses on locating the experiments in the investigated space toward the optima of an unknown process, resulting in the need for too many experiments. Development of the Sequential Pseudo-Uniform Design Method If one applies the UD method to the organization of an experimental design, the UD tables provided by Fang and Ma10 can be adopted accordingly. These are expressed as Un(qs), where U stands for the uniform

10.1021/ie034294j CCC: $27.50 © 2004 American Chemical Society Published on Web 06/05/2004

Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 4279

Figure 1. Sequential pseudo-uniform design (SPUD) for the case of two factors (O, initial uniform design points; 4, sequential pseudo-uniform design points).

design, n for the number of experimental trials, q for the number of levels, and s for the maximum number of factors. In essence, for a system with n factors and each factor divided into q levels, the UD method requires q experiments. The number of experiments based on the UD method is limited. If the experimental design based on a first selection of the number of levels, q0, from a UD table cannot provide enough information to build an accurate ANN model, then we might need to interpolate sequentially another q1 levels of each factor among the original levels q0 to run another q1 experiments. In this study, a way to arrange these q1 experiments such that these q0 + q1 experiments can scatter uniformly and regularly in the investigated domain is proposed. The design principles of the UD method10,11 are as follows: (a) The occurrence of each level of each factor in the experiments is once only. (b) The number of experiments equals the number of levels of the factors. (c) The selected experimental points are distributed uniformly in the experimental domain. These three principles are also considered in the developed SPUD method. There are q1

k)

[q1s-1 - (l - 1)] ∏ l)1

possible combinations that satisfy constraints a and b above in locating these q1 interpolated experiments. To fit requirement c, the proposed SPUD method suggests that these sequential q1 experiments be arranged by the following max-min problem q1

q0

∑∑

/ maximize ) min ( ||xik′ - xjk′|| + ∀ k combinations of Jk′)1,k i)1 j)1 the interpolated x* q1

/ / ||xik′ - xlk′ ||) ∑ l)1,l*i

(1)

If qt ) q0 + q1 experiments cannot provide enough information to build a reliable model (an ANN used in this study), then the procedure discussed above can be repeated. By way of illustration, an example with two factors is described. Consider a first arrangement of UD experi-

Figure 2. Comparison of the discrepancy measure designed by (a) the UD method (discrepancy )|10/29 - 0.6 × 0.6| ) 0.015 17) and (b) the SPUD method (discrepancy )|10/29 - 0.6 × 0.6| ) 0.015 17) (O, initial uniform design points; 4, sequential pseudouniform design points).

ments by U4(42): C1 ) (1, 3), C2 ) (3, 7), C3 ) (5, 1), C4 ) (7, 5). Four experiments (q0 ) 4) are required (Figure 1). As shown in Figure 1, the coordinates x1 and x2 of these experiments are set to be integers. Transforming the integral coordinates into a real dimension is required in a real application. If the experiments with q0 ) 4 levels cannot provide enough information to build a reliable model, then we suggest new experiments at the points where the x1 and x2 coordinates are located at 2, 4, and 6 as shown in Figure 1. Now, qt () q0 + q1 ) 7) levels are established. In this example, we want to determine these sequential q1 ()3) experiments among the six (k ) 6) possible interpolated experiment combinations, which are located at [(2, 2), (4, 4), (7, 7)], [(2, 2), (4, 6), (6, 4)], [(2, 4), (4, 2), (6, 6)], [(2, 4), (4, 6), (6, 2)], [(2, 6), (4, 2), (6, 4)], and [(2, 6), (4, 4), (6, 2)]. Because k ) 6 is a small number, the maximization-minimization problem (eq 1) can be carried out over these six possible interpolated experiment combinations exhaustively to determine which combination achieves the maximum value of the objective function as defined in eq 1. The resulting SPUD experiments will be the same as those designed using the UD method. However, the number of possible interpolated experiment combina-

4280 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004

Figure 3. Flowchart for carrying out the sequential pseudo-uniform experimental design.

tions will increase rapidly as shown in the equation q1

k)

[q1s-1 - (l - 1)] ∏ l)1

For example, if a problem has two factors (s ) 2) and q0 ) 10, then there will be 362880 possible interpolated experiment combinations (q1 ) 9). Therefore, in this study, if k e 1 × 105, the max-min problem (eq 1) was executed over all the possible interpolated experiments exhaustively; otherwise, a maximum of 1 × 105 experiment combinations were chosen randomly from the interpolated coordinates of the factors that satisfy the design principles of the UD method. A maximum value of the objective function (eq 1) can almost be determined over these 1 × 105 randomly chosen experiment combinations. In this way, a sequential pseudo-uniform design of new experiments can be properly determined. Liang et al.4 proposed the term “discrepancy” for use in measuring the uniformity of the distributed experiment points. Let R ) {xk, k ) 1, ..., n} be a set of points on Cs, where xk () [xk1, xk2, ..., xks]T) is an s-dimensional column vector. For any γ ∈ Cs, let N(γ,R ) be the number of points that satisfy xk e γ. Then

D(n,R ) ) sup|N(γ,R )/n - v([0,γ])|

(2)

is called the discrepancy of R, where v([0,γ]) ) γ1 × ‚‚‚ × γs denotes the volume of the super rectangle [0,γ].

To show the applicability of the SPUD method, we give an example with two factors, each factor being partitioned into 29 levels. The value of D(n,R ) designed by the UD method is 0.015 17 as shown in Figure 2a. For the same problem, if we first choose 15 levels based on the UD method and then proceed to use the SPUD method to locate the additional 14 levels, the same discrepancy of R [D(n,R ) ) 0.015 17] can also be obtained, as shown in Figure 2b. Artificial Neural Network Artificial neural networks are known to be a powerful tool for approximating complex multivariable functions.12,13 In this study, we adopt the most traditional feed-forward neural network (FNN) to build the relationships between any nonlinear inputs and outputs of an unknown process. An FNN is usually composed of three layers of the network structure: input layer, hidden layer, and output layer. The routines provided in the Neural Network Toolbox (MathWorks) were used to build such a neural network. Random Search Two optimization methods are usually available to find the optimum of an objective function constrained by a nonlinear functional model such as an FNN. The first method consists of gradient-based optimization


Figure 5. (a) Experiments performed initially based on the UD method, (b) following interpolated experiments based on the SPUD method (testing points of b, first run; O, second run; 1, third run; 3, fourth run).

quality evaluation of the identified model. The required techniques for retrieving the information index from the experimental data can be found in the article of Chen et al.,9 except that the term Tmax in eq 16 of Chen et al.9 is corrected here as fmax. Figure 4. (a) Three-dimensional modified Himmelblau function, (b) corresponding contour.

techniques such as the Newton method. The other method consists of non-gradient-based optimization techniques such as the random search method.14 The random search method15 sequentially explores the parameter space of an objective function in a seemingly random fashion to find the optimal condition for minimizing (or maximizing) the objective function. Although the random search is a direct application method without the need for differential information, the optimal point obtained is quite dependent on the initial guess. Therefore, a large number of initial guesses must be initiated to locate the most representative candidate points for the best performance. The calculation strategy of the random search method proposed by Solis and Wets15 was employed in this work. Information Index In this study, the information index developed by Chen et al.9 was used not to locate the additional experiments to be explored but was employed as a

Optimization of Products and Process via the SPUD Method The proposed algorithm for determining the optimum operating conditions for producing a product or a process via the SPUD method is shown in Figure 3. In summary, the procedures shown in Figure 3 are as follows: (1) Choose a suitable level q0 for each control factor using the available UD table. (2) Build an FNN model based on the experimental data from step 1. (3) Check the adequacy of the identified FNN model by the interpolated experiments chosen according to the SPUD method (recycle from step 4) or the UD method (recycle from step 6). (4) Find the optimal conditions by the random search method if the identified model is adequate; go to step 5. Otherwise, perform the chosen experiments provided by the SPUD method and identify a new FNN model from the available experiment data; go to step 3. (5) Check the adequacy of the optimal conditions experimentally; if the optimal conditions determined are reliable, stop the procedure.


Figure 6. Optimal experimental design of the modified Himmelblau function. (a) Experimental data for training (O) and testing (2) the FNN. (b) Output error of the testing experiments (O, precess; ×, model). (c) Locations of the inputs corresponding to the local minima. (d) Local minimal outputs Z corresponding to part c. (e) Information analysis plot.

(6) If the condition given in step 5 cannot be met, carry out more experiments around the assumed optimal conditions from the UD table again and identify a new FNN model; go to step 3. The objective function addressed in steps B1 and B2 of Figure 3 are defined according to the identified FNN model

minimize f[y(x)] lexeu

shown in Figure 3, each element of x is rescaled to be in a span of [-1 1]. There are 2dim(x) combinations of the elements of x with either a positive or a negative sign. In the same figure, each time the chosen rRun test experiments based on the SPUD are carried out is considered “one run”. The model error based on the test experiments is defined as

where f[y(x)] is the objective function based on the identified FNN model y(x). To carry out the algorithm

Run )

Y(xi) - y(xi)

∑ | i)1

rRun

(3)

Y(xi) rRun

| (4)


Figure 7. Optimal experimental design of the modified Himmelblau function (continuation of Figure 6).

Case Studies To demonstrate the ability of the proposed SPUD method, two examples are tested. The first example is the modified Himmelblau function of two independent variables studied by Chen et al.9 Fundamentally, this

example is a static problem. Thus, it is easy to visualize a developing model and to conduct a search in a twodimensional case. The other example is concerned with a dynamic problem, which involves determining an optimal temperature trajectory of a batch reactor where


Figure 8. Optimal experimental design of the batch reactor system for two control factors. (a) Experimental data for training (O) and testing (2) the FNN. (b) Output error of the testing experiments. (c) Locations of the inputs corresponding to the local minima. (d) Local minimal outputs Z corresponding to part c. (e) Information analysis plot.

a series reaction is carried out. For these two cases, the SPUD method is used to build a reliable model and to

locate the optima of an unknown process simultaneously using only a reasonable number of experiments.


Figure 9. Optimal experimental design of the batch reactor system for two control factors (continuation of Figure 8).

Example 1: Modified Himmelblau Function. The modified Himmelblau function studied by Chen et al.9 is used to represent an unknown system. This function with two independent variables has multiple local

optima

Z(x1,x2) ) (x12 + x2 - 11)2 + (x1 + x22 - 7)2 + x1 + 3x2 + 57 (5)

4286 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 Table 1. Parameters and Given Conditions for the Batch Series Reaction System parameter

value

parameter

value

CA0, kmol/m3 A10, m3/(kmol s) E1, kJ/(kmol K) Tmin , °C batch time, h

1 1.1 2.09 × 104 25 1

CB0, kmol/m3 A20, 1/s E2, kJ/(kmol K) Tmax, °C

0 172.2 4.18 × 104 125

Figure 10. Optimal temperature trajectory of the two-controlfactor problem developed in Figures 8 and 9 (the initial temperature is set to 125 °C).

defined for - 5 e x1 e 5 and - 5 e x2 e 5. From the mesh surface and the contour plot shown in parts a and b, respectively, of Figure 4, four local optimal points with a global point at (-3.80, -3.32) with a value of Z ) 43.3 can be found. Using a chosen testing error criterion of t ) 3%, we want to locate these local minima of Z (the objective function J ) min Z) with the fewest experiments. Following the flowchart of Figure 3, nine experiments designed using the UD (Figure 5a) were first carried out, and an FNN model was built from these data. Then, the SPUD method was applied to arrange the next eight testing experiments for improving the qualityof the built FNN model (Figure 5b). These eight experiments (step A2 in Figure 3) were carried out sequentially to provide new data for revising the built FNN model. As shown in Figure 5b, nt ) 1 (step C1 in Figure 3) was chosen for each combination of the coordinates x1 and x2, and rRun ) 2 (step C1 in Figure 3). For the system with factors equal to 2 or 3, it is suggested that the rRun experiments be done sequentially across the four (or eight) quadrants of the independent factors. If the factors are in excess of 3, rRun is set to be the number of factors, and the experiments are carried out randomly across the possible combinations of the coordinates of the factor x. Different choices of nt and rRun can be made according to the characteristics of the problem. When all of the testing experiments had been completed, the SPUD method was repeated to provide the new interpolated experiments (step C2 in Figure 4). A total of 27 experiments were performed in this example; therefore, not all locations of the experiments are depicted in Figure 5. Once the experimental data were available (step A1 or F2 in Figure 3), an FNN model was built, and the model error of the testing points is depicted in Figures 6b and Figure 7b. The error index Run (eq 4) shown in the same figure denotes the model quality based on the

available experiments. As expected, the more uniformly the experimental data are scattered and the more regular the domain to be investigated, the better the FNN model obtained. The quality of the identified FNN model can be confirmed further by parts c-e of Figures 6 and 7. In carrying out steps B1 and B2 of Figure 3, Ns was set to be 250 in this study; therefore, 250 sets of optimal solutions were achieved using the random search. As the quality of the identified model was improved, the optimal inputs (x) and output Z were clustered into some local extrema. From Figures 7 and 8, we can see that four clusters are achieved, and these four local optima can be determined accordingly on the basis of the objective function J ) Z defined in this example. The possible extrema can be determined by plotting the value of the objective function J against the experiment points (Figures 6d and 7d); the information index including information free entropy, information free enthalpy, and information free energy can also be applied to confirm these extrema. Following the SPUD method, the identified FNN model was developed from the new experiments; therefore, at the same time, the objective function J and the information free energy F were also developed. Step D2 (Figure 3) was achieved for runs 6 and 7, where the cluster number (4) was the same and the testing errors, Run, for both were less than the chosen testing error criterion t (3%). Furthermore, one can find that, as step G of Figure 3 is achieved (run 7 for this example), plots of the value of the objective function J and the information free energy F are also fully developed in Figure 7d,e. As shown in Figure 7e, the calculated information energy F is almost the same whether the cluster number chosen is 4 or larger than 4. At this time (run 7), the testing error, Run ) 0.29%, is much less than the set value t (3%). The global minimum of x is located at (-3.78, -3.30) by the identified FNN model. If the experiment at (-3.78, -3.30) is performed, the output of the experiment system is Z ) 43.31, but the model output is 43.99. Therefore, x* ) 1.55% at step G (Figure 4) and is less than the set value f (3%). Compared to the true minimum x* (-3.80, -3.32) with a value of Z* ) 43.3, the model output (43.99) is very close. Similar results can be obtained for the other three local minima. Because step G of Figure 3 was passed, the algorithm of the proposed SPUD method was stopped. In the end, only 27 experiments were required to achieve the target of this example. When steps F1-F3 Of Figure 3 were carried out, three experiments were designed according to the UD method around the searched global minimum x (-3.78, -3.30). Thirty experimental data points were used to identify the revised FNN model. The searched global minimum x was then located at (-3.79, -3.32), and its model output Z (43.31) was almost equal to the system true minimum Z* ) 43.3. Because the model error based on the previous 27 experiments was x* ) 1.55% (less than f ) 3%), steps F1-F3 were optional. Example 2: Determination of an Optimal Temperature Trajectory for a Batch Reactor. In a batch reactor, the following series reaction is to be carried out k1

k2

A 98 B 98 C It is assumed that A f B is a second-order reaction whereas B f C is a first-order reaction. The component


Figure 11. Optimal experimental design of the batch reactor system for two control factors with measurement noise σCB ) 0.02 kmol/m3. (a) Experimental data for training (O) and testing (2) the FNN. (b) Output error of the testing experiments (O, process; ×, model). (c) Locations of the inputs corresponding to the local minima. (d) Local minimal outputs Z corresponding to part c. (e) Information analysis plot.

dCB ) k1(T)CA2 - k2(T)CB dt

mass balances are as follows

dCA ) - k1(T)CA2 dt

CA(0) ) CA0

(6)

where

CB(0) ) CB0

(7)


Figure 12. Optimal experimental design of the batch reactor system for two control factors (continuation of Figure 11).

( ) ( )

k1(T) ) A10 exp -

E1 RT

(8)

k2(T) ) A20 exp -

E2 RT

(9)

The parameters and given conditions of the batch

reactor system are listed in Table 1.16 The objective of this problem is to find a temperature trajectory such that the concentration of B at the end of the reaction (tf ) 1 h) is maximal. That is

J ) max CB(tf ) 1 h) T(t)

(10)


Figure 13. Optimal temperature trajectory of the two-controlfactor problem developed in Figures 11 and 12 (the initial temperature is set to 125 °C).

The method of designing the temperature function of this case is to use orthogonal collocation.17 Three

orthogonal collocation points are used, whose locations at time t are 0, 0.667, and 1 h. A Lagrangian interpolation polynomial will pass through the desired temperatures of these three collocation points of time t. At t ) 0, T0 is set to be 125 °C; for the rest of the times, the factors to be designed in this example, T1(t)0.667 h) and T2 (t)1 h), are located in the range of 25 °C e T e 125 °C to meet the objective function given in eq 10. By an approach similar to that described in the previous example, six runs were performed in this case (Figures 8 and 9), and 26 experiments were required to locate the optimal operating conditions T/1(t)0.667 h) ) 40.69 °C and T/2(t)1 h) ) 72.46 °C (Figure 10). The progression of the objective function value J or the information free energy F shown in Figure 9d,e is fully developed as the optimal operating conditions are achieved. The output of the built FNN gives CB(t)1 h) ) 0.59843 kmol/m3, whereas the process gives 0.59844 kmol/m3. This means that the model error based on 26

Figure 14. Optimal experimental design of the batch reactor system for three control factors. (a) Experimental data for training (O) and testing (2) the FNN. (b) Output error of the testing experiments (O, process; ×, model). (c) Locations of the inputs corresponding to the local minima. (d) Local minimal outputs Z corresponding to part c. (e) Information analysis plot.


Figure 15. Optimal experimental design of the batch reactor system for three control factors (continuation of Figure 14).

experiments is 0.0016% and the result is acceptable. Compared with the true optimum C/B(t)1 h) ) 0.603 kmol/m3 calculated by Ray and Szekely,16 the error is only 0.8%. To show how the SPUD method performs

when the output CB is corrupted by random errors (B ) 0.02kmol/m3), eight runs were performed in this case (Figures 11 and 12), and 33 experiments were required to locate the optimal operating conditions T/1(t)0.667


greatly reduced. Application of the proposed SPUD method in the real world is possible. Acknowledgment We thank the National Science Council (Grant NSC 92-2214-E-036-002) and Dr. T. S. Lin, President of Tatung University, Taipei, Taiwan, ROC, for all the support conducive to the completion of this work. Nomenclature

Figure 16. Optimal temperature trajectory of the three-controlfactor problem developed in Figures 14 and 15 (the initial temperature is set to 125 °C).

h) ) 43.14 °C and T2*(t)1 h) )80.1 °C (Figure 13). The progression of the objective function value J or the information free energy F shown in Figure 12d,e is fully developed as the optimal operating conditions are achieved. The output of the built FNN gives CB(t)1 h) ) 0.603 kmol/m3, whereas the process gives 0.593 kmol/ m3. For the same example, the initial temperature T0 was instead allowed to be free, and seven runs were performed (Figures 14 and 15). Fifty-one experiments were required. The temperature collocation points of the minimum objective function value found by the model built at the time of the last run are located at 89.88, 54.99, and 59.38 °C (Figure 16), and CB(t)1 h) is 0.6087, which is quite close to the true optimum CB*(t)1 h) ) 0.603 kmol/m3. Conclusions In industrial production, a desired product recipe or an optimal operation approach is usually sought. Because the system under study is usually unknown, it is hoped that the best product recipe or the optimal operating path of a process can be obtained with minimal experiments. In this research, the experiment design is first based on the UD method. Then, the developed SPUD method is applied to locate additional experiments in the experimental region. An information free energy index is used to validate the identified ANN model. Once the identified model is verified as reliable, the optimal operating conditions can be determined to guide the process to achieve the desired objective. The modified Himmelblau function of example 1 could be regarded as a problem of recipes of unknown products. We applied the SPUD method in this case study. It involved only 27 experiments before the best recipe could be determined. The batch series reaction process of example 2 could be regarded as the operational optimization problem of an unknown process. Application of the SPUD method required 26 and 51 batches for two and three control factors, respectively; we could obtain the optimal temperature trajectory for this case study. In the meantime, a reliable FNN model can be obtained for the unknown process. Compared with the method cited in the work of Chen et al.,9 the number of experiments required to reach the same target can be

A10 ) frequency factor of reaction A f B, m3/(kmol s) A20 ) frequency factor of reaction B f C, 1/s C ) number of clusters CA ) concentration of species A, kmol/m3 CB ) concentration of species B, kmol/m3 Ci|i)1,4 ) experiment locations Cs ) s-dimensional unit cube D ) discrepancy of R E ) activation energy, kJ/(kmol K); error f ) objective function or performance index f[yˆ (ci)] ) performance index at the ith cluster center F ) information free energy J ) objective function k ) possible combinations of the interpolated experiments k1 ) rate constant of reaction A f B, m3/(kmol s) k2 ) rate constant of reaction B f C, 1/s l ) lower bound of input vector m ) weight exponent N ) total number of data points in the random search Ni ) fuzzy number of data points of the ith cluster Ns ) number of the initial guesses in the random search p ) probability q ) number of levels q0 ) number of the first selection levels q1 ) number of the new inserting levels qt ) total number of levels ) q0 + q1 r ) number of experiments R ) ideal gas constant, kJ/(kmol K) s ) number of experimental factors S ) information entropy t ) time, h; test T ) temperature, °C u ) upper bound of input vector U ) uniform design; information enthalpy v ) volume xi|i)1,2 ) element of input vector x x ) input vector y ) model output vector Y ) process output vector Z ) output of eq 5 Greek Symbols γ ) s-dimensional column vector ) model error based on the test experiments; measurement noise, kmol/m3 R ) set of points on Cs Superscript * ) optimal Subscripts 0, 1, 2 ) collocation point max ) maximum min ) minimum n ) number of experimental trials Run ) each time the chosen rRun test experiments were performed t ) testing f ) final

4292 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 Acronyms ANN ) artificial neural network FNN ) feed-forward artificial neural network SPUD ) sequential pseudo-uniform design UD ) uniform design

Literature Cited (1) Logothetis, N.; Wynn, H. P. Quality Through Design: Experimental Design, Off-line Quality Control, and Taguchi’s Contributions; Oxford University Press Inc.: New York, 1994. (2) Myers, R. H.; Montgomery, D. C. Response Surface Methodology: Process and Product Optimization Using Designed Experiments; John Wiley & Sons. Inc.: New York, 1995. (3) Zhang, L.; Liang, Y. Z.; Jiang, J. H.; Yu, R. Q.; Fang, K. T. Uniform Design applied to Nonlinear Multivariate Calibration by ANN. Anal. Chim. Acta 1998, 370, 65. (4) Liang, Y. Z.; Fang, K. T.; Xu, Q. S. Uniform Design and its Applications in Chemistry and Chemical Engineering. Chemom. Intell. Lab. Syst. 2001, 58, 43. (5) Cheng, B.; Zhu, N.; Fan, R.; Zhou, C.; Zhang, G.; Li, W.; Ji, K. Computer Aided Optimum Design of Rubber Recipe Using Uniform Design. Polym. Test. 2002, 21, 83. (6) Fang, K. T.; Wang, Y. Number-Theoretic Methods in Statistics; Chapman & Hall: London, U.K., 1994. (7) Wesseling, P. An Introduction to Multigrid Methods; Wiley & Sons: Chichester, U.K., 1991. (8) Briesen, H.; Marquardt, W. An Adaptive Multigrid Method for Steady-State Simualtion of Petroleum Mixture Separation Processes. Ind. Eng. Chem. Res. 2003, 42, 2334.

(9) Chen, J.; Wong, D. S. H.; Jang, S. S. Product and Process Development using Artificial Neural-Network Model and Information Analysis. AIChE J. 1998, 44, 876. (10) Fang, K. T. and Ma, Z. X. Orthogonal and Uniform Experimental Design; Hong Kong Baptist University: Hong Kong, 2000. (11) Fang, K. T. Uniform Design: Application of NumberTheoretic Methods in Experimental Design. Acta Math. Appl. Sin. 1980, 3, 363. (12) Hornik, K.; Stinchcombe, M.; White, H. Multilayer Feedforward Neural Networks are Universal Approximators. Neural Networks 1989, 2, 359. (13) Hornik, K.; Stinchcombe, M.; White, H. Universal Approximation of an Unknown Mapping and Its Derivatives Using Multilayer Feedforward Networks. Neural Networks 1990, 3, 551. (14) Jang, J. S. R.; Sun, C. T.; Mizutani, E. Neural-Fuzzy and Soft Computing; Prentice Hall International: Englewood Cliffs, NJ, 1997. (15) Solis, F. J.; Wets, J. B. Minimization by Random Search Technique. Math. Oper. Res. 1981, 6, 19. (16) Ray, W. H.; Szekely, J. Process Optimization; John Wiley & Sons, Inc.: New York, 1973. (17) Villadsen, J.; Michelsen, M. L. Solution of Differential Equation Models by Polynomial Approximation; Prentice Hall: Englewood Cliffs, NJ, 1984.

Received for review December 8, 2003 Revised manuscript received March 20, 2004 Accepted April 26, 2004 IE034294J

Product and Process Development via Sequential Pseudo-Uniform

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