Robust Design to the Combined Array with Multiresponse

Vol. 10, No. 1 (2017) pp. 51  57 https://doi.org/10.13160/ricns.2017.10.1.51

1. Introduction

1

Products and their manufacturing processes are influenced both by control factors that can be controlled by designers and by noise factors that are difficult or expensive to control such as environmental conditions.

The basic idea of robust design is to identify, through exploiting interactions between control factors and noise factors, appropriate settings of control factors that make the system's performance robust to changes in the noise factors. Robust design(or Parameter design in a narrow sense) is a quality improvement technique proposed by the Japanese quality expert Taguchi, which was described by Taguchi^[1,2], Kackar^[3], and others.

In the Taguchi parameter design, the control factors are assigned to an “inner array”, which is an orthogonal array. For each row in the inner array, the noise factors are assigned to an “outer array”, also an orthogonal array.

Because the outer array is run for every row in the inner array, we call this setup a “product array”. A large number of experimental trials in Taguchi's product array may be required because the noise array is repeated for every row in the control array.

Department of Computer Science and Statistics, Chosun University, Gwangju 501-759, Korea

ˋCorresponding author : ymkwon@chosun.ac.kr (Received: January 21, 2017, Revised: March 17, 2017, Accepted: March 25, 2017)

There have been efforts for integrating Taguchi’s important notion of heterogeneous variability the standard experimental design and modeling technology provided by response surface methodology. They combined control and noise factors in a single design matrix, which we call a combined array. The combined array approach was first proposed by Welch, Yu, Kang, and Sacks^[4]. The initial motivation of the combined array is the run-size saving. Related approaches were discussed by Vining and Myers^[5], Box and Jones^[6], Shoemaker, Tsui and Wu^[7], and Myers, Khuri and Vining^[8], etc.

Treatment of the mean and variance responses via a constrained optimization was discussed in Vining and Myers^[5].

In many experimental situations, a number of responses are measured for a given setting of design variables. Khuri and Conlon^[9] introduced a procedure for the simultaneous optimization of multiple responses using a distance function.

The combined-array approach allows one to provide separate estimates for the mean response and for the variance response. Accordingly, we can apply the primary goal of the Taguchi method which is to minimize the variance while constraining the mean. In this paper, we propose how to simultaneously optimize multiple responses for robust design when data are collected from a combined array. An example is illustrated to show the proposed method.

Robust Design to the Combined Array with Multiresponse

Yong-Man Kwon

Abstract

The Taguchi parameter design in industry is an approach to reducing performance variation of quality characteristic in products and processes. In the Taguchi parameter design, the product array approach using orthogonal arrays is mainly used. It often requires an excessive number of experiments. An alternative approach, which is called the combined array approach, was studied. In these studies, only single response was considered. In this paper we propose how to simultaneously optimize multiresponse for the robust design using the combined array approach.

Keywords: Taguchi Parameter Design, Product Array Approach, Combined Array Approach, Simultaneously Optimize Multiresponse૟

2. Simultaneous Optimization of Multiresponse

2.1. Multivariate Linear Model

Suppose the response Ɨ, depends on control variables (or factors) and noise variables. Let a set of control variables be denoted by

ćƖá ÞƖ_ÎìƖ_ÏìzìƖ_Ɗ and a set of noise variables by

ćƘá ÞƘ_ÎìƘ_ÏìzìƘ_Ƌ . Suppose that all response functions in a multiresponse system depend on the same set of

ćƖ and

ćƘ and that they can be represented by second order models within a certain region of interest. Let § be the number of experimental runs and ^Ɛ be the number of response functions. The Ƈth second order model is

Ɨ_ƇÞ ćƖì

ćƘ ß á ĸ_Ƈ×â ć ćĸ

Ƈâ ć›_Ƈ

ćƖâ ć «_Ƈ

ćƘâ ććĹ

Ƈâ ć_Ƈ

ćƖâ Ļ_Ƈì Ƈ á ÎìÏìzìƐ (1)

where ćĸ

Ƈ is Ɗ Z Î, ćĹ

Ƈ is Ƌ Z Î, ›_Ƈ ›_Ƈ is Ɗ Z Ɗ,

«_Ƈ «_Ƈ is Ƌ Z Ƌ, _Ƈ is Ɗ Z Ƌ, which are vectors or matrices of unknown regression parameters, and ^ĻƇ is a random error associated with the ith response.

Equation (1) can be expressed in matrix notation as ćƗ_Ƈá±

ćľ_Ƈâ

ćĻ_Ƈì Ƈ á ÎìÏìzìƐ (2)

in which

ćƗ_Ƈ is an §^{Z Î} vector of observations on the ith response, ± is an §Z Ǝ full column rank matrix of known constants,

ćľ

Ƈ is the Ǝ Z Î column vector of unknown regression parameters, and

ćĻ

Ƈ is a vector of random errors associated with the ith response. We also assume that

žÞ ćĻ

Ƈß á ć×ì¯ſƐÞ

ćĻ

Ƈß á ň_ƇƇ¢_§ìœƍƔÞ ćĻ

Ƈì ćĻ

ƈß á ň_Ƈƈ¢_§ Ƈìƈ á ÎìÏìzìƐì Ƈ d ƈ í

The Ɛ Z Ɛ matrix whose ÞƇìƈßth element is ň_Ƈƈ will be denoted by ^İ. An unbiased estimator of ^İ is given by

ýİ á² ¢_§à±Þ±±ß^{à Î}± ²î Þ§à Ǝß,

where ²á Þ ć Ɨ_Îì

ć Ɨ_Ïìzì

ć

Ɨ_Ɛß, and ¢_§ is an identity matrix of order §Z§. The Ɛ equations given in (2) may be written in a compact form

ćƗá Ė

Ę ę

ě ćƗ

Î

ćƗ

Ï

{ ćƗ

Ɛ

á Ė

Ę

ę

ě

± × z ×

× ±z × { { | {

× × z± Ė

Ę ę

ě ćľ

Î

ćľ

Ï

{ ćľ

Ɛ

â Ė

Ę ę

ě ćĻ

Î

ćĻ

Ï

{ ćĻ

Ɛ

á ³ ćľ â

ćĻ ì

(3)

where

ćƗ is ^Ɛ§Z Î, ³ is ^Ɛ§Z ƐƎ ,

ćľ is ^{ƐƎ Z Î}, and ćĻ is Ɛ§Z Î. The variance-covariance matrix of

ćĻ is

¯ſƐÞ

ćĻ ß á İ

"

^{¢á Ķ,}

where

"

is a symbol for the direct (or Kronecker) product of matrices.

The BLUE (best linear unbiased estimator) of ćľ in (3) is

ý

ćľ á Þ³ ^{à Î}³ß^{à Î}Þ³ ^{à Î}

ćƗß á Þ³ ³ß^{à Î}³ ćƗ. Thus, the BLUE of

ćľ is ^ý ćľ á Þý

ć ľ_Î

ć ľ_Ï

ć ľ_Ɛ where ^ý

ćľ_Ƈá Þ± ±ß^{à Î}±

ćƗ_Ƈ is the least squares estimator of the Ǝ Z Î vector of regression coefficients for the ith response model^[10]. The variance-covariance of ^ý

ćľ_Ƈ

¯ſƐÞý

ćľßá Þ³ ^{à Î}³ß^{à Î}á Þ± ±ß^{à Î}İ .

The prediction equation for the ith response is given by

ýƗ_ƇÞ ćƖì

ćƘß á ć ćƖì

ćƘßý ćľ

Ƈì Ƈ á ÎìÏìzìƐ (4) where ^Þ

ć ć is the vector of coded input variables, ć ćƖ ì

ćƘ ß is a vector of the same form as a row of the matrix ± evaluated at the point ^Þ

ćƖ ì

ćƘ ß. From (4) it follows that

¯ſƐãýƗ_ƇÞ ćƖ ì

ćƘ ßä á ć ćƖ ì

ćƘ ßÞ± ±ß^{à Î} ćƅ Þ

ćƖ ì ćƘ ßň_ƇƇì Ƈ á ÎìÏìzìƐì

œƍƔãýƗ_ƇÞ ćƖ ì

ćƘ ß ìýƗ_ƈÞ ćƖ ì

ćƘ ßä á ć ćƖ ì

ćƘ ßÞ± ±ß^{à Î} ćƅÞ

ćƖ ì ćƘ ßň_Ƈƈì Ƈ ìƈ á ÎìÏìzìƐ èƇ d ƈ í

Hence,

¯ſƐãý ćƗÞ

ćƖ ì ćƘ ßä á

ć ćƖ ì

ćƘ ßÞ± ±ß^{à Î} ćƅÞ

ćƖ ì ćƘ ßİ ì Ƈ á ÎìÏìzìƐ

where ^ý

ćƗ Þ ćƖ ì

ćƘ ß á ÞýƗ_ÎÞ ćƖ ì

Ƙ ßìýć Ɨ_ÏÞ ćƖ ì

ćƘ ßì z ìýƗ_ƐÞ ćƖ ì

ć is the vector of predicted responses at the point ^Þ

ćƖ ì ćƘ ß. An unbiased estimator of ¯ſƐ ãý

ćƗ Þ ćƖ ì

ćƘ ßä is given by ý¯ſƐ ãý

ćƗ Þ ćƖ ì

ć ćƖ ì

ćƘ ßÞ± ±ß^{à Î}ƅÞ ćƖ ì

ćƘ ßýİ. 2.2 Estimated Mean and Variance Models in a Multiresponse

Box and Jones^[6] modeled the mean and variance separately in a single response. But, we are interested in showing the estimated mean and variance response models in multiple responses.

The fitted ith second-order model in (4) can be rewritten as

ýƗ_ƇÞ ćƖ ì

ćƘ ß á ƀ_Ƈ×â ć ćƀ_Ƈâ

ć ›_Ƈ ćƖâ

ć «_Ƈ ćƘâ

ć ćƐ

Ƈâ ć _Ƈ

ćƖ ì Ƈ á ÎìÏìzìƐí

The noise variables

ćƘ are not controllable and they are random variables. In the absence of other knowledge,

ćƘ would be usually uniformly distributed over «_Ƙ.

Let ^ýƋ_ƇÞ

ćƖß ith estimated mean response at an ćƖ averaged over the noise variables

ýƋ_ƇÞ

ćƖß á

õ

_«

Ƙ

ýƗ

Ƈ

ÞćƖ ì ćƘ ßƎÞ

ćƘ ßƂ

ćƘ ì Ƈ á ÎìÏìzìƐì where ƎÞ

ćƘ ß is a probability density function of ćƘ, and

ćƘ has a uniform distribution over «Ƙ(à Î = Ƙ = Î ). Box and Jones[6] showed that the ith estimated mean becomes

ýƋ_ƇÞ

ćƖ ß á ƀ_Ƈ×â ć ćƀ

Ƈâ ć ›_Ƈ

ćƖâ ćР΃Ɛý«_Ƈì Ƈ á ÎìÏìzìƐì

(5)

where ƒƐý«_Ƈ is the trace of the matrix ^ý«_Ƈ. Let us write ýƔ_ƇÞ

ćƖ ß for the ith mean square variation about the ith mean response

ýƔ_ƇÞ

ćƖ ß á

õ

_«

Ƙ

ÞýƗ_ƇÞ ćƖ ì

ćƘ ß àýƋ_ƇÞ ćƖ ßß^ÏƎÞ

ćƘ ßƂ ćƘ ì Ƈ á ÎìÏìzìƐí

Let us call this measure the ith estimated variance, which becomes

ýƔ_ƇÞ ćƖ ß á ćÐ

ÎÞ ćƐ

Ƈâý_Ƈ ć ćƐ

Ƈâý_Ƈ

ćƖ ß âýš_Ƈì Ƈ á ÎìÏìzìƐì

(6)

where ^ýš_Ƈá ãÑİ_{ƈ á Î}^Ƌ ÞƐ_ƈƈ^Ƈß^Ïâ Òİ_{ƈ á Î}^{Ƌ à Î}İ_{Ɖ á ƈâÎ}^Ƌ ÞƐ_ƈƉ^Ƈ ß^Ïäî ÑÒ and ^ƐƈƉƇ is the ^ƈth row and ^Ɖth column element of the matrix ^ý«_Ƈ.

From (2.5) the ith estimated mean can be rewritten as ýƋ_ƇÞ

ćƖ ß á ć ćƖ ßý

ćľ_×Ƈ Ƈ á ÎìÏìzìƐ (7) where

ć ćƖ ß áÞ^ÎìƖÎì z ìƖ_Ɗì Ɩ_Î^Ïìz ìƖ_Ɗ^Ïì Ɩ_ÎƖ_Ïì z ì Ɩ_{Ɗ à Î}Ɩ_Ɗì ÎîÐì z ìÎîÐß and ^ý

ćľ_×Ƈá Þƀ_×^Ƈìƀ_Î^Ƈìzìƀ_ÎÎ^Ƈ ìzìƀ_ƊƊ^Ƈìƀ_ÎÏ^Ƈ ì zìƀ_{Ɗà Î}^Ƈ ìƐ_ÎÎ^Ƈ ìzìƐ_ƋƋ^Ƈ is a part of ^ý

ćľ

Ƈ. From the fact that ý

ćľ

×Ƈ is a part of ^ý ćľ

Ƈ, the variance-covariance of ^ý ćľ

× is given by

¯ſƐ

Þ

^ý_ć^ľ×

ß

^{á Þ±±ß×à Îİ ,}

where ^ý ćľ

×á Þý ćľ

×Î ćľ

×Ï ćľ

×Ɛ , ^Þ± ±^{ßà Î} is Ǝ Z Ǝ, and Þ± ±ß_×^{à Î} is Ə Z Ə, where Ǝ á ÞƊ â Ƌ â ƌßÞƊ â Ƌ â Ïßî Ï, and Ə á ÞƊ â ÎßÞƊ â Ïßî Ï â Ƌ. Here ^Þ± ±ß_×^{à Î} is the Ə Z Ə submatrix of ^Þ± ±ß^{à Î}. From (7) and above, we then have

¯^{ſƐ ãý} ćƋ Þ

ćƖ ßä á

ć ćƖ ßÞ± ±ß_×^{à Î} ćƆ Þ

ćƖ ßİ ì where ^ý

ćƋ Þ

ćƖ ß á

Þ

^ýƋÎÞ ćƖ ßìýƋ_ÏÞ

ćƖ ßìzìýƋ_ƐÞ

ćƖ ß

ß

^{is the}

vector of estimated mean responces at the point ćƖ. An unbiased estimator of ¯ſƐãý

ćƋ Þ

ćƖ ßä is given by ý¯ſƐ ãý

ćƋ Þ ćƖ ßä á

ć ćƖ ßÞ± ±ß_×^{à Î} ćƆ Þ

ćƖ ßýİ. (8) 2.3. The Proposed _¦ measure about the estimated mean responses

Let us find conditions on a set of control variables ćƖ which optimize a set of estimated mean responses ý

ćƋ Þ

ćƖ ß subject to maintaining estimated variance responses ^ý

ćƔ Þ

ćƖ ß within some specified upper bounds. If

all the estimated mean ^ý ćƋ Þ

ćƖ ß attain their individual optimum values

ćʼn at the same set

ćƖ of operating conditions, then the problem of simultaneous optimization is obviously solved. This ideal optimum rarely occurs. In more general situations we might consider finding compromising conditions on the control variables that are somewhat favorable to all mean responses. Such deviation of the compromising conditions from the ideal optimum condition can be evalulated by means of a distance function which measures the distance of ^ý

ćƋ Þ

ćƖ ß, from ćʼn. Let

ćʼn be the optimum (or target) value of ^ýƋ_ƇÞ ćƖ ß over «_Ɩ and let

ćʼn á Þʼn_Îìʼn_Ïìzìʼn_Ɛ . We shall consider a constrained-optimization procedure for each response according to the Taguchi's three basic situations as follows.

1. “nominal-is-best characteristics”: target value of ýƋ_ƇÞ

ćƖ ß á ʼn_Ƈ ,

2. “larger-the-better characteristics”:

¦ſƖ

ćƖ«Ɩ

ýƋ_ƇÞ

ćƖ ß á ʼn_Ƈ ,

3. “smaller-the-better characteristics”:

¦Ƈƌ

ćƖ«Ɩ

ýƋ_ƇÞ

ćƖ ß á ʼn_Ƈ . A distance function of ^ý ćƋ Þ

ćƖ ß for the target value ćʼn may be expressed as



ã

^ý

ćƋ Þ ćƖ ßì

ćʼn

ä

^á

ã

^Þý

ćƋÞ ćƖ ß à

ć ¯ſƐãý

ćƋ Þ

ćƖ ßä뫮 ^{à Î}Þý ćƋÞ

ćƖ ß à ćʼn ß

ä

^ÎîÏ

Using the estimate given in (8) for the variance-covariance matrix of ^ý

ćƋ Þ

ćƖ ß, we get a distance function

ƙ

Ɯ

ƚ

_ć

ć ćƖ ßÞ± ±ß_×^{à Î} ćƆÞ

ćƖ ß ÞýćƋ Þ

ćƖ ß à

ć İ^{à Î}Þý ćƋ Þ

ćƖ ß à ćʼn ß ƛ

Ɲ

ƞ

^ÎîÏ (9) If the mean response ^ý

ćƋ Þ

ćƖ ß takes on different degrees of importance, we can imply weights ƕ_Îìƕ_Ïìzìƕ_Ɛ where × ï ƕ_Ƈï Î for each i and İ_{Ƈ á Î}^Ɛ ƕ_Ƈá Î . Then the distance function can be written as

ƙ Ɯ

ƚ

ć ćƖ ßÞ± ±ß_×^{à Î} ćƆÞ

ćƖ ß 뫭 °Þý

ćƋ Þ ćƖ ß à

ć İ^{à Î}뫭 °Þý ćƋ Þ

ćƖ ß à ćʼn ß뫮 ƛ

Ɲ

ƞ

^ÎîÏ

(2.9) where °^á

Ė

Ę

ę

ě ƕ_Î × z ×

ƕ_Ïz ×

| {

ƑƗƋ ƕ_Ɛ

. From the constrained-

optimization procedure and the distance From the distance measure of ^ý

ćƋ Þ

ćƖ ß for the target value

ćʼn, we propose a simultaneous optimization of ý

ćƋ Þ

ćƖ ß over the region of interest «_Ɩ. From (9), the proposed simultaneous-optimization measure about the estimated mean responces can be written as

_¦ = ćƖ«_Ɩ

¦Ƈƌ

_¦Þ ćƖ ß á

ćƖ«_Ɩ

¦Ƈƌ

ć ćƖ ßÞ± ±ß_×^{à Î} ćƆÞ

ćƖ ß 뫭 °Þý

ćƋ Þ ćƖ ß à

ć İ^{à Î}뫭 °Þý ćƋ Þ

ćƖ ß à ćʼn ß뫮 (10)

The _¦ measure can be used without a prior knowledge about the estimated mean responses. It takes into consideration the variances and correlations of the estimated mean responses.

2.4. The Proposed _¯ Measure About the Estimated Variance Responses

In this section, we propose the simultaneous- optimization measure of multiple responses(quality characteristics) for robust design in a combined array.

If we have a prior knowledge about the estimated mean response ^ý

ćƋ Þ

ćƖ ß, it is possible to minimize the estimated variance response. Let

ýƔ_Ƈ^ÝÞ

ćƖß á ć

ćƖ«_Ɩ

¦_{ſƖ ýƔ}

ƇÞ ćƖß à

ćƖ«_Ɩ

¦_{Ƈƌ ýƔ}

ƇÞ ćƖß ýƔ_ƇÞ

ćƖß à ćƖ«_Ɩ

¦_{Ƈƌ ýƔ}

ƇÞ ćƖß

ì

Ƈ á ÎìÏìzìƐì

where ^ýƔ_ƇÞ

ćƖ ß is the Ƈth mean square variation about the Ƈth mean response which is in (6). Note that ^ýƔ_Ƈ^ÝÞ

ćƖß is a

“standardized” measure of ^ýƔ_ƇÞ

ćƖß. The proposed simultaneous optimization measure about the estimated variance responces can be written as

_¯ = ćƖ«_Ɩ

¦Ƈƌ

_¯Þ ćƖß á

ćƖ«_Ɩ

¦Ƈƌ ć ćƔ^ÝÞ

ćƖß

á ćƖ«_Ɩ

¦Ƈƌ

ā

Ƈ á Î Ɛ

ƕ_ƇýƔ_Ƈ^ÝÞ

ćƖßì Ƈ á ÎìÏìzìƐì (11) where

ćƕá Þƕ_Îìƕ_Ïìzìƕ_Ɛ , ^ý

ćƔ^ÝÞ

ćƖß á ÞýƔ_Î^ÝÞ ćƖßì ýƔ_Ï^ÝÞ

ćƖßìzì ýƔ_Ɛ^ÝÞ

ć , İ_{Ƈ á Î}^Ɛ ƕ_Ƈá Î . If the variance response ^ý ćƔ Þ

ćƖ ß takes on different degrees of importance, we can imply weights ƕ_Îìƕ_Ïìzìƕ_Ɛ where × ï ƕ_Ƈï Î for each i and İ_{Ƈ á Î}^Ɛ ƕ_Ƈá Î .

2.5. The Proposed Simultaneous-Optimization ©_Ƌ Measure

Let us find conditions on a set of control variables ćƖ which optimize simultaneously for a set of estimated mean responses and estimated variance responses. If all the estimated mean and all the estimated variance attain their individual optimum values at the same set

ćƖ of operating conditions, then the problem of simultaneous optimization is obviously solved. This ideal optimum rarely occurs. In more general situations we might consider finding compromising conditions on the control variables that are somewhat favorable to all mean responses and all the estimated variance. Such deviation of the compromising conditions from the ideal optimum condition can be formulated by means of the desirability function.

We propose a simultaneous optimization for a set of estimated mean responses and estimated variance responses over the region of interest «_Ɩ using proposed

_¦^Þ

ćƖß and _¯^Þ

ćƖß. From Equations (10) and (11), the proposed simultaneous-optimization measure can be written as

©_Ƌ á ćƖ«_Ɩ

¦Ƈƌ

©_ƋÞ ćƖß á

ćƖ«_Ɩ

¦Ƈƌ ãŁ_¦Þ

ćƖß â ÞÎ à Łß_¯Þ ćƖß ä

(12)

where «_Ɩ is the region of interest on a set of control variables

ćƖ and × = Ł = Î . This is a criterion in which

_¦Þ

ćƖß and _¯^Þ

ćƖß take on different degrees of importance.

As a way of finding the optimal solution of the control factors according to the proposed formula, genetic algorithm was used in the MATLAB Optimization Toolbox.

3. Numerical Example

In this section we give a numerical example, consisting of a multiresponse system of two response variables, Ɨ_Î and Ɨ_Ï, and two control variables, Ɩ_Î and Ɩ_Ï, and a noise variable ^Ƙ. The design is somewhat similar to the standard central composite design. The cube portion of the experimental arrangement is chosen to be a Ï^Ð design and star points are added only for the two control variables. The following Table 1 gives the factor levels and a set of data.

Each of the two responses was fitted to a second order regression model. The estimated response models by the method of least squares are given by

ýƗ_ÎÞ

ćƖìƘ ß á ÔÓí×× à ÎÏíÐÔƖ_Îà ÕíÖÓƖ_Ïà ÔíÏÏƖ_Î^Ïà ÕíÑÒƖ_Ï^Ï à ÕíÎÎƖ_ÎƖ_Ïâ ÒíÐÕƘ^Ïà ÎíÑÑƘ â ÏíÖÓƖ_ÎƘ à ÎíÕÓƖ_ÏƘ ì

(13)

ýƗ_ÏÞ

ćƖìƘß á Î×Ðí×× à ÎÏíÏÎƖ_Îâ ÓíÓÕƖ_Ïà ÎÐíÖÓƖ_Î^Ïà ÕíÒ×Ɩ_Ï^Ï à ÏíÖÐƖ_ÎƖ_Ïâ ÓíÏÐƘ^Ïà ÎíÐÕƘ â ÎíÔÒƖ_ÎƘ à ÏíÖÒƖ_ÏƘ í

(14) From (13) and (14), using the mean and variance response equation (5) and (6), the estimated mean and variance response models are given by

ýƋ_ÎÞ

ćƖ ß á ÔÔíÔÖ à ÎÏíÐÔƖ_Îà ÕíÖÓƖ_Ïà ÔíÏÏƖ_Î^Ï à ÕíÑÒƖ_Ï^Ïà ÕíÎÎƖ_ÎƖ_Ïì

ýƋ_ÏÞ

ćƖ ß á Î×Òí×Õ à ÎÏíÏÎƖ_Îâ ÓíÓÕƖ_Ïà ÎÐíÖÓƖ_Î^Ï à ÕíÒ×Ɩ_Ï^Ïà ÏíÖÐƖ_ÎƖ_Ïì

ýƔ_ÎÞ

ćƖ ß á Þà ÎíÑÑ â ÏíÖÓƖ_Îà ÎíÕÓƖ_Ïß^ÏîÐ â ÏíÒÔ ì ýƔ_ÏÞ

ćƖ ß á Þà ÎíÐÕ à ÎíÔÒƖ_Îà ÏíÖÒƖ_Ïß^ÏîÐ â ÐíÑÒ í The region of interest «_Ɩ is given by the inequality

à Î = Ɩ_ÎìƖ_Ï= Î . The ranges for ^ýƋ_ÎÞ ćƖß, ^ýƋ_ÏÞ

ćƖß, ýƔ_ÎÞ

ćƖß and ^ýƔ_ÏÞ

ćƖ ß over «_Ɩ are, respectively, ÐÏíÓÕ = ýƋ_ÎÞ

ćƖ ß = ÕÐíÏÒ, ÓÓíÓÓ = ýƋ_ÏÞ

ćƖ ß = Î×ÖíÓÒ, ÏíÒÔ = ýƔ_ÎÞ

ćƖ ß = ÎÒíÓÐ and ÐíÑÒ = ýƔ_ÏÞ

ćƖ ß = ÎÒíÔÔ. Suppose that the quality characteristics for Ɨ_Î and Ɨ_Ï are the nominal-is-best characteristics and the larger-the-better characteristics. Let us assume that the target value of^ýƋ_ÎÞ

ćƖ ß is taken to be 75.00 and the target value of ^ýƋÏÞ

ćƖ ß is taken to be ¦ſƖ

ćƖ «_ƖýƋ_ÏÞ

ćƖ ß á Î×ÖíÓÒ. From (12), we obtained the results of simultaneous optimization based on the minimization of the ©_ƋÞ

ćƖß measure over «_Ɩ. Table 2 indicates that the optimal setting for the Ʌ=0.1, ƕ_Î=ƕ_Ï=1.00 is Ɩ_Îáà ×íÎ× and Ɩ_Ïá ×íÎÕ , which produces a predicted value of 77.21, 107.14, 4.00, and 4.45 for ^ýƋÎÞ

ćƖ ß, ^ýƋÏÞ ćƖ ß, ^ýƔÎÞ

ćƖ ß, and ýƔ_ÏÞ

ćƖ ß, respectively.

If a simultaneous optimum value is much different

from its corresponding individual optimum value, we may reoptimize ©_Ƌ. Also we may analyze ©_Ƌ sequentially as the acceptable values for Ł and weights ƕ_Îìƕ_Ïìzìƕ_Ɛ are varied.

4. Conclusion

The combined-array approach allows one to provide separate estimates for the mean response and for the variance response. Accordingly, we can apply the primary goal of the Taguchi methodology which is to obtain a target condition on the mean while achieving the variance, or to minimize the variance. In this study we proposed the simultaneous-optimization measure ©_Ƌ of multiple responses. The proposed concept of ©Ƌ

measure is minimizing the deviation of the mean responses from the target values and also ©_Ƌ measure is minimizing the deviation of the variance responses from

Run Ɩ_Î Ɩ_Ï Ƙ Ɨ_Î Ɨ_Ï

1 2 3 4 5 6 7 8 9 10 11 12 13 14

-1 -1 -1 -1 1 1 1 1 -1.41

1.41 0 0 0 0

-1 -1 1 1 -1 -1 1 1 0 0 -1.41

1.41 0 0

-1 1 -1 1 -1 1 -1 1 0 0 0 0 0 0

80.6 74.9 83.1 71.2 66.8 74.2 38.1 36.8 80.9 42.4 73.4 45.0 77.4 74.6

81.4 95.9 105.0 103.0 74.0 76.8 81.2 76.9 100.0 50.5 71.2 101.0 102.0 104.0 Table 1. Experimental design and response values

Weight Location of optima Simultaneous optima

Ʌ Ɩ_Î Ɩ_Ï ýƋ_ÎÞ

ćƖ ß ýƋ_ÏÞ

ćƖ ß ýƔ_ÎÞ

ćƖ ß ýƔ_ÏÞ ćƖ ß 0.10

0.30 0.50 0.70 0.90

-0.10 -0.07 -0.05 -0.03 -0.01

0.18 0.21 0.27 0.29 0.26

77.21 76.49 75.46 74.92 75.03

107.14 106.94 106.88 106.68 106.37

4.00 3.95 4.03 4.00 3.84

4.45 4.62 4.90 5.04 4.96 Table 2. Simultaneous optimization for ©_Ƌ

the target values of the variance responses that is minimum values of the variance responses. The ©Ƌ

measure is easy to apply, and permits the user to make subjective judgements on the importance of each response.

In this study we assume that the noise variables would be uniformly distributed over the region of interest of noise variables. It will be of interest to consider the case when the noise variables are not uniformly distributed over the region of interest of noise variables.

Acknowledgments

This study was supported(in part) by research funds from Chosun University, 2015.

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