Development of a variance prioritized multiresponse robust design framework for quality improvement

Transcription

1 Development of a variance prioritized multiresponse robust desin framework for qualit improvement Jami Kovach Department of Information and Loistics Technolo, Universit of Houston, Houston, Texas 7704, USA Bun Rae Cho Advanced Qualit Enineerin Laborator, Department of Industrial Enineerin, Clemson Universit, Clemson, South Carolina 9634, USA Jiju Anton Strathclde Institute of Operations Manaement, Desin Manufacturin and Enineerin Manaement, Universit of Strathclde, Glasow, G1 1XY, UK Submitted to: International Journal of Qualit and Reliabilit Manaement Februar 008 * Correspondin author

2 Development of a variance prioritized multiresponse robust desin framework for qualit improvement Jami Kovach Department of Information and Loistics Technolo, Universit of Houston, Houston, Texas 7704, USA Bun Rae Cho Advanced Qualit Enineerin Laborator, Department of Industrial Enineerin, Clemson Universit, Clemson, South Carolina 9634, USA Jiju Anton Strathclde Institute of Operations Manaement, Desin Manufacturin and Enineerin Manaement, Universit of Strathclde, Glasow, G1 1XY, UK Abstract Purpose Robust desin is a well-known qualit improvement method that focuses on buildin qualit into the desin of products and services. Yet, most well established robust desin models onl consider a sinle performance measure and their prioritization schemes do not alwas address the inherent oal of robust desin. In this paper, we propose a new robust desin method for multiple qualit characteristics where the oal is to first reduce the variabilit of the sstem under investiation and then attempt to locate the mean at the desired taret value. Desin/methodolo/approach In this paper, we investiate the use of a response surface approach and a sequential optimization strate to create a flexible and structured method for modelin multiresponse problems in the context of robust desin. Nonlinear prorammin is used as an optimization tool. Findins Our proposed methodolo is demonstrated throuh a numerical example and the results are compared to that of the traditional robust desin method. The proposed methodolo provides enhanced optimal robust desin solutions consistentl. Oriinalit/value This paper is perhaps the first stud on the prioritized response robust desin with the consideration of multiple qualit characteristics. The findins and ke observations of this paper will be of sinificant values to the qualit and reliabilit enineerin/manaement communit. Kewords Qualit, robust desin, multiresponse problems Paper tpe Reserch paper

3 Introduction In toda s industrial production environment, it is essential that companies create and retain a competitive advantae over their competitors with respect to qualit in order to be successful. The robust desin (RD) methodolo, first developed b Tauchi (1986, 1987), is important to industrial qualit improvement initiatives. This approach focuses on buildin qualit into the desin of products and services thouh the determination of the optimum operatin conditions in order to minimize performance variabilit and deviation from the taret value of interest (i.e. process bias). Since it was first introduced, this approach has come under serious criticism due to the statistical analsis methods and optimization approaches utilized. In his method of RD, Tauchi advocates minimizin sinal-to-noise ratios to determine the best overall combination of desin parameter settins and identifin adjustment factors, which are used to adjust the mean to the desired taret value. Yet, Nair and Shoemaker (1990) arue that b simpl collapsin experimental data into sinal-to-noise ratios much of the information concernin the sstem s behavior is lost. Additionall, Tauchi ives no real justification for the use of these ratios, and the details surroundin the use of adjustment factors to achieve the taret value of interest are sketch at best. To address these issues, there have been several attempts in the literature to improve the analsis and optimization phases of the RD methodolo. Several RD optimization models that are relevant to the work presented here are based on the dual response approach, which was first considered b Mers and Carter (1973). Here, the process mean and variance of a sinle qualit characteristic are modeled separatel usin response surfaces. These functions are then optimized simultaneousl to determine the sstem s optimum operatin parameters. The first attempt to combine this tpe of optimization model within the RD methodolo was developed b Vinin and Mers (1990). As an extension to this 3

4 dual response approach to RD, Lin and Tu (1995) proposed the mean-squared error (MSE) model. This approach relaxed the zero-bias assumption of the previous model to provide solutions that are better (or at least equal), in terms of the variance achieved, throuh a more flexible optimization model. Lin and Tu (1995) further suested assinin different weihts to process bias and variabilit as a wa of prioritizin the optimization procedure. Alon these lines, Park and Cho (003), Shin and Cho (005), and Cho and Park (005) considered more complex RD sstems from the viewpoints of non-normal data, process oriented modelin, and unbalanced desin space, respectivel, Finall, Kovach and Cho (006) and Lee et al. (007) developed RD models for unbalanced data structures and irreular experimental situations, respectivel. The RD optimization models discussed thus far ma be considered multiresponse approaches because the simultaneousl optimize the response models for both the process mean and variance. These approaches, however, onl consider the response models for the process mean and variance for a sinle performance measure. Yet, customer jude products on multiple scales simultaneousl. Further, the prioritization scheme of these optimization models do not alwas address the inherent oal of RD. Tauchi (1986, 1987) believed it was vital to qualit improvement efforts to not simpl focus on adjustin the process mean to the desired taret value. He felt stronl that it was also critical to reduce the variabilit of the process around the mean. Therefore, when optimizin a sstem, RD should consider multiple qualit characteristics simultaneousl and place top priorit on minimizin process variabilit. In this paper, we propose a new approach to the multiresponse RD problem to address these issues, called variance-prioritized multiresponse robust desin (VPMRD). In this approach, the optimization phase of RD is formulated as a nonlinear oal prorammin problem where the prioritization 4

5 scheme is specified usin a preemptive optimization procedure. Throuh a numerical example, we demonstrate the use of this methodolo to determine the optimum operatin conditions for the sstem under investiation based on experimentation and sstem modelin. In the next section, we describe our proposed methodolo in detail. Then, an example is used to demonstrate the practical implementation of our proposed approach. Next, a comparison stud is conducted to validate the proposed model. Finall, we discuss conclusions concernin the practical implications of the proposed approach. The proposed methodolo While there are several optimization models that can be emploed within the RD methodolo, few consider the inherent oal of RD. Recall that Tauchi s qualit philosoph suests that qualit improvement efforts should not simpl focus on the mean; it also involves the variabilit around the mean. Yet, well-established optimization approaches, includin the dual response (Vinin and Mers, 1990) and MSE models (Lin and Tu, 1995), ma have failed to place top priorit on minimizin the variance. Therefore, we propose a new method of RD that creates a structured, et flexible modelin approach for multiresponse RD problems where the oal is to first reduce the variabilit of the sstem under investiation and then attempt to reduce the process bias. In this new approach, we utilize nonlinear oal prorammin techniques [3] to determine the sstem s optimum operatin conditions based on response surface models of the sstem derived thouh experimentation and analsis. The methodolo proposed in this paper is shown in Fiure 1, which consists of first determinin the problem to be considered and choosin the responses of interest, as well as the desin factors, to be included in the investiation. This information is then used to create the 5

6 appropriate experimental desin, and the experiment is carried out accordin to plan. Next, data collected from the experiment are analzed usin the least squares method of reression analsis. Based on these results, response surface models for the mean and variance of each response are created. Finall, the proposed VPMRD model is used to simultaneousl optimize the responses to obtain the settins for the desin parameters that make the sstem under investiation perform optimall relative to minimizin the process variance and bias. [Fiure 1 Approximatel Here] Plannin In an RD stud, the determination of the qualit characteristics of interest is stronl related to the nature of the problem bein investiated. Similarl, the choice of experimental parameters is often derived from the responses under investiation usin prior enineerin knowlede concernin the sstem bein studied. In man situations, however, practitioners ma not have sufficient experience to effectivel choose the appropriate experimental parameters. Therefore, the alternative is to collect data relative to potential experimental factors and then use multivariate studies, correlation analsis, and/or screenin experiments to determine the specific factors to consider in an RD investiation. Experimentation In desinin an experiment, there are man standard approaches from which to choose. These include, but are not limited to, factorial desins, Tauchi desins, and response surface desins. The ke, therefore, is to choose the desin that best addresses the experimental question and supports the desired data analsis iven the available resources (i.e. time and experimental budet). The size of the experiment is determined in part b the number of factors included, the 6

7 number of levels of each factor to be tested, and the number of replications planned. The results obtained form the experiment provides the information necessar to create response surface models that describe the behavior of the sstem under investiation. Sstem modelin To create response surface models for the mean and variance of each qualit characteristic under investiation, we utilize the least squares method of reression analsis. Usin this approach, consider that each of the n treatment combinations in an experiment consists of r replicates. For each response of interest, let uj represent the j th response at the u th treatment where j = 1,,, r and u = 1,,, n. Then, for each qualit characteristic, the mean and variance for the u th treatment can be estimated usin the followin equations: u r ( ) = = r r 1 j= 1 uj j= 1 uj u and su r (1) for u = 1,,, n. Assumin the underlin distribution of the experimental data is normal with constant variance, the estimators iven in Equation (1) are then used to create the response surface functions of the mean and variance based on the method of least squares reression analsis as follows. The model for the mean, therefore, is written as where x = Xβ, () µ T 1 T β = X X X, (3) 7

8 X is the desin matrix, β is the estimate of the vector of unknown model parameters, and is the vector of estimated means for each treatment combination in an experiment. In addition, the response surface function for the variance is iven as where ( x) σ = Xγ, (4) 1 γ is the estimate of the vector of unknown model parameters, and T T γ = X X X s, (5) s is the vector of estimated variances for each treatment in an experiment. However, if the underlin distribution of the experimental data is skewed (i.e. is sinificantl non-normal) or the assumption of constant error-variance is violated sinificantl, the data will have to be transformed in order to proceed with this tpe of analsis. Commonl used transformations include, but are not limited to, a lo transformation or square root transformation; the transformation to be used depends on the specific situation and the deree to which the necessar assumptions are violated. Assumin the necessar assumptions can be verified, the fitted response functions for each qualit characteristic under investiation are then optimized simultaneousl to determinin the sstem s optimum operatin parameters. Optimization The framework for the proposed optimization model is shown in Table 1 and is structured in such a wa that iven the sstem parameters and the models of their behavior, the oal is to find the desin factor settins based on satisfin the sstem requirements and oals b minimizin the objective function. In our proposed model, k responses of tpe t are considered. 8

9 These desinations are necessar to accommodate the different tpes of qualit characteristics that ma be considered in an RD investiation. Accordin to Tauchi, responses are usuall cateorized as one of the followin three tpes. 1. Smaller-the-better (S-tpe): Minimize the qualit characteristic of interest (i.e. the taret value equals zero).. Nominal-the-better (N-tpe): The qualit characteristic of interest has a specific taret value. 3. Larer-the-better (L-tpe): Maximize the qualit characteristic of interest (i.e. the taret value approaches infinit). Further, µ kt ( x ) and kt σ x are the response surface functions for the process mean and variance of response kt, respectivel, which are assumed to be independent of each other, and τ and τ are the desired taret values for the mean and variance of response kt, µ k t σ k t respectivel. [Table 1 Approximatel Here] In the proposed model, the constraints consist of the sstem s technical requirements and its desired oals. The sstem requirements are characterized b the upper and/or lower limits on the sstem variables under consideration, which must be satisfied in order for the solution to be feasible. Further, the constraints representin the sstem oals model the deviation from the desired taret values for both the mean and variance of each response. Overall, the sstem oal is to minimize the deviation from the desired taret values. Additionall, we can specif the priorit of individual oals in the objective function iven this framework. The objective function for the proposed model is formulated as a nonlinear oal prorammin problem (Hillier and Liberman, 001). This tpe of approach makes the eneral multiresponse problem inherentl easier to solve because the objective function can be modeled 9

10 as a sinle function usin deviation variables. In mathematical prorammin terms, deviation variables are also known as auxiliar variables. In the objective function for our proposed model, these variables are used to denote the under- and over-achievement of a constraint (i.e. a desired rane or taret value), which are iven as d and d +, respectivel; hence, the deviation variables are also used in formulatin the constraints for the proposed model. For example, consider the constraint associated with the process mean of a particular response of interest. Usin deviation variables, this constraint can be written in eneral terms as where = µ x τ, for = 1,,, r (6) d kt µ k t d = d d (7) + and d d + d if d 0 = { 0 otherwise (8) d if d 0 = 0 otherwise Given these definitions, the constraint in Equation (6) can then be rewritten as + ( d d ) µ x = τ. (9) kt Further, the method described here for modelin constraints usin deviation variables for the mean of a qualit characteristic can also be used for modelin the variance. Given this tpe of approach, the deviation variables associated with the constraints for the mean and variance of each response are then combined to form a sinle objective function as follows: min Z f ( d, d ), f ( d, d + ),, f r ( d, d + r r ) s.t. = (10) Constraints due to sstem requirements: µ k t 10

11 1. S-tpe qualit characteristic: µ USL. N-tpe qualit characteristic: LSL 3. L-tpe qualit characteristic: µ LSL Constraints due to sstem oals: 1. Process mean: + kt d d µ k t ks kl x for k = 1,,, a ks x USL for k = a + 1, a +,, b kn µ kn kn x for k = b + 1, b +,, c µ x + = τ for t = tpe of qualit characteristic (S-, N-, or L-). Process variance: σ kt + + d d = τ x for t = tpe of qualit characteristic (S-, N-, or L-) σ k t Bounds: 1. Desin factors: xi min xi xi max for i = 1,,, n. Deviation variables: d, + 0 and d + = 0 for = 1,,, r d In the proposed model, the constraints are delineated accordin to the tpe of qualit characteristic considered where USL and LSL are the upper and lower specification limits, respectivel, for the sstem s requirements. To establish a prioritization scheme for the optimization procedure, our proposed model utilizes a preemptive approach involvin sequential optimization. Here, weihts of different manitudes are assined to the deviation variables associated with the process mean and variance. In this model, we propose settin priorities in such a wa that the oal is to first minimize the variance and then attempt to achieve the mean equal to the desired taret value. The proposed approach is illustrated throuh the numerical example in the followin section. d kl Implementation of the proposed model Consider the problem of optimizin a chemical filtration process for measurin dosaes with respect to filtration time, volume, and purit. Here, we consider temperature ( x 1 ) and pressure ( x ) as desin factors and filtration time ( ), filtration volume ( N ), and filtration purit 11

12 ( 3L ) as our responses of interest. Hence, this becomes a multiresponse RD stud, iven that there are multiple qualit characteristics of interest in determinin the optimum operatin conditions for the sstem under investiation. In this example, filtration time is considered an S- tpe qualit characteristic since it is desirable to minimize processin time. Additionall, the sstem confiuration has a maximum processin time of 7 seconds that cannot be exceeded. The desired taret for the volume of the filtered chemical dose is 10 ml, where the allowable tolerance is ±0.5 ml; therefore, filtration volume is considered an N-tpe response. Further, filtration purit is required to be as hih as possible, therefore this response is considered an L- tpe qualit characteristic with a natural bound at 100%. In this particular example, it is critical that we reduce the variabilit of each response simultaneousl in order to stabilize the processin cost in terms of filtration time and improve product qualit b minimizin the occurrence of under- and over-filled vials and stabilizin filtration purit. To estimate quadratic models of the responses, a central composite desin with four center points is chosen for this experiment. Additionall, the experiment is replicated three times and data are collected concernin the responses of interest. The experimental desin schemes and the data for each response are displaed in Tables -4, respectivel. [Table Approximatel Here] [Table 3 Approximatel Here] [Table 4 Approximatel Here] Analsis of the data, usin the method shown in Section.3, results in response surface models for the mean and variance of each response as follows: µ x = x x x x x1 x (11) 1

13 σ x = x x x x x1 x (1) µ N x = x x x x x1x (13) σ N x = x x x x x1 x (14) µ 3L x = x x x x x1 x (15) σ 3L x = x x x x x1 x (16) Usin Equations (11) throuh (16), the proposed VPMRD optimization model is then applied to obtain the optimum operatin conditions for the filtration process, as shown in Table 5. In this particular example, the constraints are comprised of the bounds on the responses that must be strictl adhered to. These bounds are denoted b the technical requirements of the sstem stated as the USL and/or LSL. Specificall, filtration time must be less than 7 seconds, filtration volume must be within the rane of ml, and filtration purit must be reater than zero. Further, the oals are determined based on the desire to minimize the process bias and variance. For this example, the taret values of interest are theoreticall zero seconds for filtration time, 10 ml for filtration volume, and 100% for filtration purit. Additionall, we specif that the taret for the variance is equal to zero because the oal of an RD stud is alwas to minimize the variance. Given these constraints and their associated deviation variables, the deviation function to be minimized considers the over-achievement for the S- and N-tpe qualit characteristics and the under-achievement for the N- and L-tpe qualit characteristics of interests for the mean and the over-achievement of each qualit characteristic for the variance. Here, the streamlined procedure for preemptive oal prorammin is used to first minimize the variance and then attempt to achieve the mean equal to the desired taret value where all 13

14 responses of interest are weihted equall. The bi M method [3] is utilized to establish a prioritization scheme within the optimization process. Here, the deviation variables associated with the variance are iven substantiall larer weihts, desinated b M, over that of the deviation variables associated with the process bias. [Table 5 Approximatel Here] The results of the proposed method, which are shown in Table 6, indicate the oal of zero variance for filtration purit was achieved; et, the variance of both filtration time and volume slihtl exceed zero. In addition, the mean of each response is achieved with minimal amounts of bias allowed; however, none of the mean values achieved the exact taret value desired. [Table 7 Approximatel Here] Comparison stud In this section, we validate the proposed methodolo b comparin its results to those obtained usin traditional RD optimization models, includin both the dual response approach (Vinin and Mers, 1990) and the MSE model (Lin and Tu, 1995). The dual response approach minimizes the variance with the constraint that the process mean equals the desired taret value, which can be written as min σ ( x ) (17) µ = τ s.t. ( x ) x where is the reion of interest. This optimization model is tpicall used for problems in which it is critical that the qualit characteristic of interest adheres strictl to the desired taret 14

15 value. Given this approach, we obtain optimum operatin conditions that result in the mean located on taret with some amount of variation around the mean. To improve upon the dual response approach, Lin and Tu (1995) relaxed its zero-bias assumption (i.e. the constraint requirin that the process mean must equal the desired taret value) and proposed a model that simultaneousl minimizes the squared difference of the mean from the taret value and the variance as follows: min µ x τ + σ ( x ) (18) s.t. x When minimizin the variabilit of the response of interest is of equal or reater importance than achievin the desired taret value, this optimization strate is often utilized. Based on such an approach, it is observed that b allowin some difference between the mean and the desired taret value, the resultin process variance is less than or at most equal to the variance of the dual response approach. For the purpose of comparison, we utilize an approach similar to that demonstrated b Tan and Xu (1995) in which we approximate the traditional RD models usin the VPMRD framework developed in this paper. Here, we reformulate the oriinal RD models as oal prorammin problems and appl them to the example used previousl to illustrate our proposed methodolo. The formulations of and the results obtained from these equivalent models are presented in the followin sections. The results obtained usin these models are then compared to the results obtained from our proposed methodolo. Expansion of the dual response approach for multiresponse problems 15

16 Usin the dual response approach, the first priorit of the optimization procedure is to achieve the desired taret value and then the model attempts to minimize the variance. To approximate the optimization procedure of the dual response approach for a multiresponse problem, we use a preemptive oal prorammin approach, which is similar to that used in the proposed model. As discussed previousl, all qualit characteristics are equall weihted; et in this case, substantiall larer weihted priorities are placed on the deviation variables associated with minimizin the process bias over that of the deviation variables associated with minimizin the variance. Given the same response surface models and constraints used to demonstrate the proposed model, the bi M method can be used in the optimization procedure to establish a prioritization scheme that reflects the dual response approach for multiple qualit characteristics as follows: 1 ( + 1 ) ( + 1 ) ( ) ( ) ( ) ( 5 ) Z = d + d + d + M d + M d + d + M d (19) Based on Equation (19), we can interpret the objective of this model as an attempt to locate the mean at the desired taret first and then an attempt is made to minimize the variance. Therefore, this equivalent model replicates the oals of the oriinal dual response approach and allows us to optimize multiple qualit characteristics simultaneousl. The results obtained usin this model are shown in Table 7. [Table 7 Approximatel Here] Expansion of the MSE model for multiresponse problems In terms of the MSE model, minimizin the squared difference of the mean from the desired taret value and minimizin the variance are of equal priorit in the optimization procedure and are considered simultaneousl. The optimization procedure of the MSE model can be 16

17 approximated for multiresponse problems usin a nonpreemptive oal prorammin approach (Hillier and Liberman, 001). To create this equivalent model, equal weihts are iven to the variables in the deviation function for the mean and variance. Aain, based on the response surface models and constraints used to illustrate the proposed model, we establish a prioritization scheme that reflects the MSE model for multiresponse problems, which can be written as min Z = ( d + ) + ( d + 4 ) + ( d 6 ) + ( d + 1 ) + ( d 3 + d + 3 ) + ( d 5 ) (0) From Equation (0), we can observe that minimizin the bias and minimizin the variance are of equal importance. This prioritization scheme effectivel replicates the oriinal intention of the MSE model, and this formulation also allows us to optimize multiple qualit characteristics simultaneousl. The results obtained usin this model are shown in Table 8. [Table 8 Approximatel Here] Discussion of comparison stud results The results of this stud comparin the proposed approach to that of the prioritization schemes associated with traditional RD optimization models are best considered based on the evaluation of the mean and variance at their optimal settins. In terms of the variance, Table 9 shows the optimum operatin conditions for each approach, which also represents the deviation from the taret, iven that the desired taret value is zero. The onl result that indicates the oal of zero variance was achieved is for the response of filtration purit usin the proposed VPMRD approach; all other results slihtl exceed zero. Yet, it is important to note that the proposed method also resulted in the lowest variance for the response of filtration time in comparison to the other approaches. In terms of the mean, Table 10 shows the evaluation of the mean at the optimum operatin conditions and the deviations of the mean from the taret value for each 17

18 optimization method considered here. For filtration time, all optimization methods over-achieve the taret, with the multiresponse expansion of the dual response concept producin the minimum deviation. In terms of filtration volume and purit, all approaches under-achieve the taret. Here, the multiresponse expansion of the MSE and the dual response concepts produce the minimum deviation for the filtration volume and purit, respectivel. [Table 9 Approximatel Here] [Table 10 Approximatel Here] As has been shown with other RD models in the past, we see here that a trade-off exists between minimizin the variance and minimizin the process bias. Based on the results of this comparison stud, it can be concluded that the proposed model achieves its primar oal of first minimizin the variance and then attemptin to achieve the mean equal to the desired taret value. This loic explains wh this method tended to produce the minimum variance, but not the minimum process bias. Yet, this approach provides a flexible and structured method for modelin multiresponse RD problems in the presence of responses with differin objectives. Therefore, this method is useful in situations where minimizin the variance is more critical than achievin a specific taret value for problems involvin multiple objectives. Conclusion In this work, a new RD optimization approach and framework was proposed, called VPMRD, to handle situations in which we wish to determine the optimum operatin conditions for a sstem when the problem entails multiple responses and in cases where it is criticall important that the variance of the responses bein considered is minimized. Here, the proposed method utilized oal prorammin techniques to model a multiresponse problem in a sinle objective function 18

19 usin deviation variables. The objective of the proposed model addressed the inherent oal of RD where the first priorit was to minimize the variance. The proposed optimization approach was described in detail and illustrated thouh the use of a numerical example. The results obtained from this approach were then compared to that of expanded approaches of the traditional RD optimization models in order to address multiple qualit characteristics. The result of this work provides a method for usin RD in real-world situations where optimal solutions are desired in the face of multiple responses. The approach discussed here illustrates that there are trade-offs in desin between minimizin the variance and achievin the desired taret value. Yet, the specific model proposed here addresses these trade-offs b providin a flexible and structured method for modelin multiresponse RD problems. 19

20 References Cho BR, Park C. Robust desin modelin and optimization with unbalanced data. Computers and Industrial Enineerin 005;48(): Hillier FS, Lieberman GJ. Introduction to operations research. New York: McGraw-Hill, 001. Kim YJ, Cho BR. Development of priorit-based robust desin. Qualit Enineerin 00;14(3): Kovach J, Cho BR. A D-optimal desin approach to robust desin under constraints: a new desin for six sima tool. International Journal of Six Sima and Competitive Advantae 006;(4): Lee SB, Park C, Cho BR. Development of a hihl efficient and resistant robust desin. International Journal of Production Research 007;45(1): Lin DKJ, Tu W. Dual response surface optimization. Journal of Qualit Technolo 1995;7(1): Mers RH, Carter WH. Response surface techniques for dual response sstems. Technometrics 1973;15(): Nair VN, Shoemaker AC, The role of experimentation in qualit enineerin: a review of Tauchi s contributions. In: Ghosh, S (Ed.). Statistical desin and analsis of industrial experiments. New York: Marcel Dekker, p Park C, Cho BR. Development of robust desin under contaminated and non-normal data. Qualit Enineerin 003;15(3): Shin S, Cho BR. Bias-specified robust desin optimization and its analtical solutions. Computers and Industrial Enineerin 005;48(1): Tauchi G. Introduction to qualit enineerin: desinin qualit into products and processes. New York: Krauss International Publications, Tauchi G. Sstem of experimental desin: enineerin methods to optimize qualit and minimize costs. Dearborn, MI: American Supplier Institute, Tan LC, Xu K. A unified approach for dual response surface optimization. Journal of Qualit Technolo 00;34(4): Vinin GC, Mers RH. Combinin Tauchi and response surface philosophies - a dual response approach. Journal of Qualit Technolo 1990;(1):

21 Fiures and Tables $ & # $& #!! " #$% Fiure 1 A Process Map of the Proposed Methodolo 1

22 Table 1 General Model of the Proposed VPRMD Framework Given Desin factors: x, i = 1,,, n Responses: kt i, where k = 1,,, a,, b,, c t = tpe of qualit characteristic (S-, N-, or L-) Fitted response models: Mean: µ kt ( x ) and variance: kt ( x ) Desired taret values: Mean: τ and variance: µ k t τ σ k t Find * Robust desin factor specifications: x, i = 1,,, n Satisf + Deviation variables associated with constraints: d, d, = 1,,, r Constraints: 1. Sstem requirements. Sstem oals Minimize Z f1 ( d 1, d 1 ), f ( d, d ),, fr ( d r, dr ) = i σ Table Experimental Desin and Observations of Filtration Time for the Multiresponse Chemical Filtration Stud Temperature Pressure Filtration Time (seconds) Averae Variance Coded Units 3 Replications Treatment No. x 1 x 1 3 s

23 Table 3 Experimental Desin and Observations of Filtration Volume for the Multiresponse Chemical Filtration Stud Temperature Pressure Filtration Volume (ml) Coded Units 3 Replications Averae Variance Treatment No. x 1 x N1 N N 3 N s N Table 4 Experimental Desin and Observations of Filtration Purit for the Multiresponse Chemical Filtration Stud Temperature Pressure Filtration Purit (%) Coded Units 3 Replications Averae Variance Treatment No. x 1 x 3L1 3L 3L3 3L s 3L

24 Table 5 Proposed VPMRD Optimization Model for the Multiresponse Chemical Filtration Stud Given Desin factors, x : x = Temperature (ºF) 1 x = Pressure (psi) Responses, : = Filtration time (seconds) = Filtration volume (ml) N 3L = Filtration purit (%) Fitted response models: Mean: µ k t ( x ) from Equations (11), (13), and (15) Variance: Desired taret values: Mean: Mean: Variance: σ x from Equations (1), (14), and (16) k t τ = 0 µ τ = 0 σ τ and variance: µ k t τ = 10 µ N τ = 0 Find * Robust desin factor specifications: x, i = 1, Satisf i σ N τ σ k t τ = 100 µ 3L τ = 0 + Deviation variables associated with oals: d, d, = 1,,, 6 Constraints: Goals: 1. µ ( x ) ( x ) ( x ) 7 µ N 1. µ + ( x ) + = 4. ( x ) S d d σ ( x ) + = 5. ( x ) S d d µ + ( x ) + = 6. ( x ) σ 3L µ N + d d = 3L 0 + σ L + d d = µ N d3 d3 10 L + d d = Bounds: 1. Desin factors: x for i = 1, i + σ Deviation variables: d, d and d d = 0 for = 1,,, 6 Minimize Z = M ( d ) + M ( d + 4 ) + M ( d + 6 ) + ( 3)( d + 1 ) + ( 3)( d 3 + d + 3 ) + ( 3)( d 5 ) 4

25 Table 6 Results of the Proposed VPMRD Approach for the Multiresponse Chemical Filtration Stud Optimal Settins x* = (1.4117, ) Mean Variance µ x = N µ x = µ 3L ( x ) = σ x = σ x = N σ x = 0 3L Table 7 Results of the Multiresponse Expansion of the Dual Response Approach for the Multiresponse Chemical Filtration Stud Optimal Settins x* = (0.5444, 1.85) Mean Variance µ x = N µ x = µ 3L ( x ) = σ x = σ x = N σ x = L 5

26 Table 8 Results of the Multiresponse Expansion of the MSE Model for the Multiresponse Chemical Filtration Stud Optimal Settins x* = (0.4759, 1.135) Mean Variance µ x = N µ x = µ 3L ( x ) = σ x = N σ x = L σ x = Table 9 Comparison of Optimization Methods in Determinin the Optimal Desin Factor Settins and the Evaluation of the Variance for the Multiresponse Chemical Filtration Stud Method Multiresponse Expansion of the Dual Response Approach Multiresponse Expansion of the MSE Model Proposed VPMRD Approach Optimal Settins (x 1 *, x *) σ x σ N x 3L ( x ) σ (0.5444, 1.85) (0.4759, 1.135) (1.4117, )

27 Table 10 Comparison of Optimization Methods in Determinin the Optimal Desin Factor Settins and the Evaluation of the Mean for the Multiresponse Chemical Filtration Stud Method Multiresponse Expansion of the Dual Response Approach Multiresponse Expansion of the MSE Model Proposed VPMRD Approach Optimal Settins µ x 1 (x 1 *, x *) d + µ x 3 N d µ x 5 (0.5444, 1.85) (0.4759, 1.135) (1.4117, ) L d 7