Expert Systems with Applications 42 (2015) 7036–7045

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

Determining the optimal allocation of parameters for multivariate measurement system analysis Jeh-Nan Pan a,⇑, Chung-I Li b, Szu-Chen Ou a a b

Department of Statistics, National Cheng Kung University, Tainan 70101, Taiwan, ROC Department of Applied Mathematics, National Chiayi University, Chiayi 60004, Taiwan, ROC

a r t i c l e

i n f o

Article history: Available online 5 May 2015 Keywords: Multivariate measurement system analysis Multivariate gauge repeatability and reproducibility Correlated characteristics Precision-to-tolerance ratio

a b s t r a c t Measurement system analysis (MSA) plays an important role in helping organizations to improve their product quality. Usually, a gauge repeatability and reproducibility (GRR) study needs to be conducted prior to the process capability analysis for assessing the adequacy of gauge variation. In performing multivariate MSA for multivariate measurement systems, the existing multivariate precision-to-tolerance (P/T) ratio does not take the correlation coefficients among tolerances into account. Moreover, the optimal allocation of several parameters such as the number of quality characteristics (v ), sample size of parts (p), number of operators (o) and replicate measurements (r) have not been considered in the multivariate GRR study either. As the total number of measurements (n) increase, the estimated total variation becomes more precise, but the related inspection time and costs will be increased as well. Striking the right balance between the precision of measurement system while still maintaining cost-effectiveness in determining the optimal allocation of opr parameters for correlated quality characteristics becomes an important issue in practical applications. In this paper, a revised precision-to-tolerance (P/T) ratio for multivariate MSA with correlated quality characteristics is proposed. The simulation results show that our revised P/T ratio outperforms the existing ones in terms of MSE and MAPE. We have also found that two parameters (number of parts p and a total number of measurements n) significantly affect the expected length of confidence interval of P=TR . A reference table with the optimal allocation of por parameters is constructed accordingly if the inspection cost is limited. Finally, a numerical example with a step-by-step procedure for conducting a MGRR study is given to illustrate the appropriateness of our proposed P/T ratio. Hopefully, it can be served as a useful guideline for quality practitioners when performing a multivariate measurement system analysis in industries. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Measurement system analysis (MSA) plays an important role in helping organizations to improve their product quality. Generally speaking, the gauge repeatability and reproducibility (GRR) study is performed according to the MSA handbook stated in QS9000 standards. Usually, a GRR study needs to be conducted prior to the process capability analysis for assessing the adequacy of gauge variation. Good quality products can only be achieved through an adequate measurement system. Hence, finding ways to ensure the quality of a measurement system becomes an important task for quality practitioners. Moreover, in performing the GRR study, ⇑ Corresponding author. E-mail addresses: [email protected] (J.-N. Pan), [email protected] (Chung-I Li), [email protected] (S.-C. Ou). http://dx.doi.org/10.1016/j.eswa.2015.04.038 0957-4174/Ó 2015 Elsevier Ltd. All rights reserved.

most industries today are using the approval criteria of precision to tolerance (P/T) ratio as stipulated in QS9000. Traditional MSA only considers a single quality characteristic. With the advent of modern technology, industrial products have become very sophisticated with more than one quality characteristic and there is high correlation among them, which can be found in solder paste stencil printing process (Pan & Lee, 2010). In performing the multivariate MSA for correlated quality characteristics, quality practitioners are also expected to determine the optimal allocation of sample size of parts (p), number of operators (o) and repeated measurements (r) for economic reasons. As the total number of measurements (n) increase, the estimated total variation becomes more precise, but the related inspection time and costs will be increased as well. Striking the right balance between the precision of measurement system while still maintaining cost-effectiveness in determining the optimal allocation of por parameters when conducting

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multivariate GRR study for correlated quality characteristics is the main purpose of this research. However, the adequacy of the current por selection method for conducting multivariate GRR is questionable. Thus, it becomes necessary to provide a guideline of por selection for quality practitioners when conducting multivariate GRR study for correlated quality characteristics. Recently, principle component analysis (PCA) related methods are proposed by He, Wang, and Cook (2011), Osma (2011), Peruchi, Balestrassi, de Paiva, Ferreira, and de Santana Carmelossi (2013) and Wang (2013) for analyzing gauge variation. He et al. (2011) proposed a modified principal component analysis (PCA) method to transform multivariate measurement data with dependent variables into those with independent principal components. Peruchi et al. (2013) and Peruchi, Paiva, Balestrassi, Ferreira, and Sawhney (2014) proposed a weighted principal component (WPA) method for multivariate analysis of a measurement system, which can be considered as a special PCA approach. Wang (2013) applied PCA to perform assessment for multivariate GRR. Although He’s and the other existing PCA methods can be served as an alternative approach to multivariate MSA, these methods cannot provide/remove the causes of production problems as the original ‘‘unit of measure’’ for the quality characteristics will be disappear after performing PCA. On the other hand, multivariate analysis of variance (MANOVA) proposed by Majeske (2008) does not have such problems. But, Majeske (2008) did not take the correlation coefficients among tolerances into account. Hence, in this paper, a revised precision-to-tolerance (P/T) ratio is proposed to fill the research gap in determining the optimal allocation of opr parameters when performing multivariate MSA for correlated quality characteristics.

yijk ¼ l þ P i þ Oj þ ðPOÞij þ eijk

8 > < i ¼ 1; . . . ; p j ¼ 1; . . . ; o > : k ¼ 1; . . . ; r

where yijk is the individual value of measurement, l is the measurement mean (total mean), P i is the effect of product, Oj is the effect of operator, ðPOÞij is the effect of interaction between product and operator, eijk is the effect of replicate measurements, p is the sample size of parts, o is the number of operators, and r is the number of replications. The effects of Pi ; Oj ; ðPOÞij and eijk are assumed to be

r r r re , respectively. This is, Pi  Nð0; r  Nð0; r Nð0; r2po Þ and eijk  Nð0; r2e Þ. By using expected mean square

normally distributed with means of zero and variances and

2

2 p Þ; Oj

r^ 2repeatability ¼ r^ 2e ¼ MSE r^ 2reproducibility ¼ r^ 2o þ r^ 2po

ð4Þ ðp  1ÞMSpo þ MSo  pMSE ¼ pr

r^ 2gauge ¼ r^ 2repeatability þ r^ 2reproducibility ¼

ðp  1ÞMSpo þ MSo þ pðr  1ÞMSE pr ð6Þ

If interaction between product and operator does not exist, then the repeatability, reproducibility, and variability of gauge can be calculated as:

The purpose of MSA is to qualify a measurement system for use by quantifying its accuracy, precision, and stability. Usually, gauge repeatability and reproducibility (GRR) study for assessing the adequacy of gauge precision needs to be conducted prior to the process capability analysis. According to AIAG (2010), the variance of the measured values can be uniquely partitioned into two components:

r^ 2gauge ¼ r^ 2repeatability þ r^ 2reproducibility ¼

where

r2part is the component of variance due to the product and

r2gauge is the component of variance due to measurement.

ð7Þ

MSo  MSE ^ 2o ¼ ¼r pr

r^ 2reproducibility

ð1Þ

ð5Þ

Then the variability of gauge can be calculated through the following:

2.1. The gauge repeatability and reproducibility (GRR)

r2obs ¼ r2part þ r2gauge

2 2 2 p ; o ; po 2 o Þ; ðPOÞij 

(EMS), one can obtain the estimated values of sources of variation, which are listed as follows:

r^ 2repeatability ¼ r^ 2e ¼ MSE

2. Literature review

ð3Þ

ð8Þ MSo þ ðpr  1ÞMSE pr

ð9Þ

2.3. The approval criteria for univariate GRR To determine whether the precision of a measurement system is adequate or not, three commonly used criteria are listed below: (1) Precision-to-tolerance ratio (P/T ratio) According to AIAG (2010), P/T ratio is defined as:

6rgauge  100% TOL

Moreover, the variability of measurement can be further defined as:

P=T ¼

r2gauge ¼ r2repeatability þ r2reproducibility

where TOL is the specification tolerance and rgauge is the standard deviation of the measurement system. The P/T ratio represents the percent of the specification tolerance taken up by measurement error. 6 standard deviation, 6rgauge , accounts for 99.73% of measurement system variation based on normal distribution for underlying gauge error. Moreover, AIAG (2010) suggested the following guidelines for accepting gauge precision of a measurement system as listed in Table 1.

where r

2 repeatability

represents repeatability and r

ð2Þ 2 reproducibility

represents

reproducibility. The information obtained from GRR study can be used to quantify the variations, which provides useful guidance for improving the accuracy and precision of a measurement system. 2.2. ANOVA analysis for univariate GRR To estimate the potential sources of measurement variation, two methods commonly used in GRR study are: (1) X–R chart method and (2) analysis of variance (ANOVA) method. Because the X–R method cannot calculate the variance of operator-by-part interaction, Montgomery and Runger (1993) proposed the ANOVA method which uses a two-way random-effects ANOVA model with the interaction term. The random-effects model is defined as:

ð10Þ

Table 1 GRR approval criterion suggested by AIAG. P/T ratio

Decision

P=T 6 10% 10% < P=T 6 30%

The measurement system is considered to be acceptable The measurement system is considered to be marginally acceptable (may be acceptable for some applications) The measurement system is unacceptable

P=T > 30%

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(2) %R&R The P/T ratio may be made arbitrarily small by increasing the width of the specification tolerance. To overcome this drawback, the %R&R criterion is defined as:



%R&R ¼



rgauge  100% robs

ð11Þ

%R&R represents the percentage of the total process variation taken by measurement error. According to AIAG (2010), an excellent measurement system has a %R&R less than 10%; a value of 30% is barely acceptable. (3) Number of distinct categories (ndc) The number of distinct categories represents the number of non-overlapping confidence intervals that will span the range of product variation. The number of distinct categories also represents the number of groups within your process data that your measurement system can discern. The number of distinct categories is defined by:

ndc ¼





pffiffiffi rpart  2 rgauge

ð12Þ

AIAG (2010) recommends that 5 or more categories indicate an acceptable measurement system; when the number of categories is less than 2, the measurement system is of no value for controlling the process.

2.4. The analysis methods and approval criteria for MGRR Wang and Yang (2007) and Wang (2013) applied principal component analysis (PCA) method to transform multiple characteristics into one or a few independent variables. Then, these independent variables are analyzed using analysis of variance. Their analysis procedures are listed as follows. (1) Test for multivariate normality for the collected data. If the normality test fails, then perform the normal transformation for multiple quality characteristics. (2) Use PCA to obtain new variables and new specifications. 2 3 X 11    X 1n 6 . .. 7 .. Let Xv n ¼ 4 .. . 5 be the sample data matrix, . Xv1    Xvn 2 3 X1 6 7 X v 1 ¼ 4 ... 5 be the sample mean vector and S represents Xv the sample covariance matrix, where v represents the number of quality characteristics, ðn ¼ p  o  rÞ represents the total number of measurements. Then, the ith principal component (new variable) is given by

PCi ¼ eT x;

i ¼ 1; 2; . . . ; v

ð13Þ

where e1 ; e2 ; . . . ; ev are the eigenvectors of S and x are v  1 vectors of the observations on the original variables. The specification tolerance of the ith principal component (new specification tolerance) is

TOLPC i ¼ eTi TOL;

i ¼ 1; . . . ; v

where TOL is the vector of specification tolerance. (3) Identify the significant components. For identifying the significant components, Anderson (1963) proposed the following test statistics

v2r ¼ ðn  1Þ

v X j¼kþ1

lnkj þ ðn  1Þðv  kÞ ln

Pv

j¼kþ1 kj

v k

!

ð14Þ

where v2r follows a chi-squared distribution with degrees of freedom r ¼ 12 ðv  kÞðv  k þ 1Þ  1. After performing the above testing, the number of significant components are determined. Accordingly, the new variables and new specifications can be obtained. (4) Calculate P/T ratio. Perform the ANOVA method to calculate P/T ratio for all new variables. Then, a composite index combining all new variables is given by w Y

!w1 P=Tj

;

j¼1

where w is the number of significant components and P=Tj is the P/T ratio for the jth new variable. Other multivariate measurement system analysis methods using PCA related methods can be referred to He et al. (2011), Osma (2011) and Peruchi et al. (2013, 2014). Although the PCA method can be used to transform the correlated characteristics into independent variables and these independent variables are analyzed using a random factor experiment, these methods cannot provide/remove the causes of production problems. Majeske (2008), on the other hand, applied multivariate analysis variance method (MANOVA) to estimate the variance–covariance matrices for a multivariate measurement system. He considered the two-way random-effects MANOVA model with the interaction term

X ¼ l þ P i þ Oj þ ðPOÞij þ eijk

8 > < i ¼ 1; . . . ; p j ¼ 1; . . . ; o > : k ¼ 1; . . . ; r

ð15Þ

where l represents the total mean vector, vector P i represents the effect of product, vector Oj represents the effect of operator, vector ðPOÞij represents the effect of interaction between product and operator, and vector eijk represents the effect of replicate measurements. Analogous to univariate GRR, they assumed that P i  Nð0; Rp Þ; Oj  Nð0; Ro Þ; ðPOÞij  Nð0; Rpo Þ, and eijk  Nð0; Re Þ. After performing MANOVA, one can obtain the estimated values of the repeatability, reproducibility, and variability of gauge as shown by:

^ repeatability ¼ R ^E R ^ ^O þ R ^ PO Rreproducibility ¼ R

ð16Þ

^ repeatability þ R ^ reproducibility ^G ¼ R R

ð18Þ

ð17Þ

When the gauge error is not independent, Majeske (2008) indicated that the constant-density contour ellipsoid has principle axes of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi length v299:73;v kGl , resulting in a volume of

Q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v v v299:73;v kGl p2 l¼1   C 1 þ v2 ^ G . Thus, Majeske where kG1 ; . . . ; kGv are the eigenvalues of matrix R (2008) further proposed a multivariate precision-to-tolerance ratio as

31 2 Yv qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v v 2 2 v k p G 7 6 99:73;v l l¼1 7 6   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1v1 0 v 7 6 v 2 v299:73;v kGl 7 6 C 1þ Y 7 6  2 A P=T ¼ 6 ¼@ Yv TOLl  v 7 7 6 TOLl l¼1 2 7 6 p l¼1 2 7 6 5 4  

C 1þ

v

2

ð19Þ

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where TOLl denotes the specification tolerance for the lth quality characteristic and v21a;v denotes the (1  a)th percentile of a chi-square distribution with v degrees of freedom. Although the dependence of gauge errors is considered in Majeske (2008)’s P/T ratio, the correlation coefficients among tolerances are not taken into account. Hence, in this paper, a revised precision-to-tolerance (P/T) ratio is proposed to fill the research gap in determining the optimal allocation of opr parameters when performing multivariate MSA for correlated quality characteristics. 2.5. The optimal allocation of parameters for GRR Based on the confidence interval of measurement variability proposed by Montgomery and Runger (1993) and Pan (2004) discussed the suitability of (p, o, r) selection using a statistical method for conducting a GRR study. The procedures for searching the optimal combination of por parameters are listed below. (1) Consider various combinations of p; o and r. Let sample size of parts (p) be 5, 10, 15, and 20, the number of operators (o) be 2, 3, 4, and repeated measurements (r) be 3 and 4. Thus, there are a total of 24 combinations as shown in Table 2. Then choose various values of sources of variation r2o ; r2po , and r2e . Notice that the adoption of sample size 5, 10, 15 and 20 of parts is based on the conventional rule used by industry. (2) Define the expected mean squares for different sources of GRR variations as shown in Table 3. Substitute the (p, o, r) combination and values of r2o ; r2po , and r2e in Step (1) to the expected mean squares listed in Table 3, then the estimated mean square ho ; hpo , and he can be obtained. Begin computer simulation for chi-square random variables including ðp1ÞMSo ho

;

ðp1Þðo1ÞMSpo , hpo

and

poðr1ÞMSE . he

Then, the estimated mean

square MSo ; MSpo , and MSE can be obtained. (3) Substitute the expected mean square MSo ; MSpo , and MSE from Step (2) to the 100ð1  aÞ% confidence interval of measurement variability as proposed by Montgomery and Runger (1993) in Eq. (20).

^ 2gauge ur

2 gauge

6r

v2a2;u

6

^ 2gauge ur

ð20Þ

v21a2;u

where

u¼



^ 2gauge

r

" 2  1 2 MS2

r12 2 #1 MS2po MSE þ þ r p1 ðp  1Þðo  1Þ poðr  1Þ o

or

o12

Note that the procedure for determining the optimal combination of por parameters for multivariate measurement system is still lacking as the above procedures are designed for univariate GRR. 3. Revised P/T ratio 3.1. Developing revised P/T ratio To take the correlation among multiple quality characteristics into account, Pan and Lee (2010) proposed a novel multiple process capability index NMC p . They revised Taam’s et al. (1993) modified engineering tolerance region based on the assumption that the correlation of multiple quality characteristics is consistent with the correlation among specifications. The revised engineering tolerance region is defined as

n o 1 Ed;A ;T ¼ X 2 Rv jðX  TÞ0 ðA Þ ðX  TÞ ¼ v299:73;v

ð21Þ

where X 2 Rv denotes X is a v-dimensional random vector and the elements of matrix A are given by

0

10

USL  LSL

1 USL  LSL

i i CB j jC qij B @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A; i; j ¼ 1; . . . ; v 2 v299:73;v 2 v299:73;v

ð22Þ

where T is the target vector, ðUSLi  LSLi Þ denotes the ith specification tolerance, and qij represents the correlation coefficient between the ith and jth quality characteristics. The volume of the revised engineering tolerance region can be calculated as: 1

jA j2



pv299:73;v

v2  v

C

2

1 þ1

ð23Þ

In this paper, we consider the two-way random-effects MANOVA model with the interaction term as shown in Eq. (15). According to Eq. (19), the numerator in P/T ratio can be rewritten as 1

jRG j2



pv299:73;v



C 1 þ v2

v2



where RG is the variance–covariance matrix of gauge error. To take the correlation among multiple quality characteristics into account for MGRR, we adopt the idea proposed by Pan and Lee (2010) and revise the denominator of the P/T ratio proposed by Majeske (2008). Accordingly, the revised P/T ratio is defined as

2

or

(4) Repeat the computer simulation from Step (1) to (3) for 10,000 times. By calculating the mean as width of confidence interval for variability of measurement, one can obtain an optimal combination of (p, o, r) with the shortest width of confidence interval of measurement variability.

Table 2 Various combinations of sample size of parts, number of inspectors and repeated measurements for conducting a GRR study. p o

r

5

10

15

20

2 2 3 3 4 4

3 4 3 4 3 4

30 40 45 60 60 80

60 80 90 120 120 160

90 120 135 180 180 240

120 160 180 240 240 320

1  v2    3v 1

1 2 v þ 1 1 2 jR C G j pv99:73;v 2 jRG j 2v 6 7 P=TR ¼ 4 ¼ 5 v     1 1 2 jA j jA j2 pv299:73;v C v2 þ 1

ð24Þ

where the elements of matrix A are given in Eq. (22). It is worthy to note that both of the matrices RG and A are positive definite. When the matrices are not positive definite, the methods as proposed by Amemiya (1985) and Calvin and Dykstra (1991) are able to deal with this issue. 3.2. Comparing the simulation results for P/TR and other P/T ratios In this section, various simulation studies are conducted to compare the performance of the revised P/T ratio (P=TR ) with that of the existing ones. Two scenarios are considered here: random-effects MANOVA models with and without interaction term. Each will be further explained below. 3.2.1. Random-effects MANOVA model without interaction term If the interaction between the part and operator does not exist, then we consider the random-effects MANOVA model without

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Table 3 The expected mean squares (EMS) for sources of variations. EðMSo Þ ¼ r2R þ r r2po þ pr r2o EðMSpo Þ ¼ r2R þ r r2po EðMSE Þ ¼ r2e

interaction term. Let Xijk denote the measurement collected by operator j on part i at k replication, then the random-effects MANOVA model without interaction can be written as

Xijk ¼ l þ Pi þ Oj þ Eijk

ð25Þ

where l is the total mean vector, Pi is the effect of the ith part, Oj is the effect of the jth operator and Eijk is the random error representing the repeatability. All of the above three effects are assumed to be the random effects that are multivariate normally distributed with mean vector of zero and variance–covariance matrix of RP ; RO , and RE , respectively. Without loss of generality, in this simulation study, we assumed that a MGRR study with three quality characteristics is conducted and a 3  3 variance–covariance matrix for the measured values is set as

2

1 6 RP ¼ 4 qp

qp

Fig. 1. Comparison of the simulation results of bias for various P/T ratios under different product correlation coefficients qp (without considering PO interaction).

3

qp qp 1 qp 7 5 qp 1

Table 4 Various combinations of qp12 ; qp13 and qp23 under high, medium and low correlation.

where the correlation coefficients qp are generated from a uniform distribution on the interval (0, 1). Assuming that the operators are independent and the variance–covariance matrix for the operator factor is set as

2

qp12 qp13 qp23

1

2

3

4

5

6

7

8

9

10

0.9

0.9

0.9

0.9

0.9

0.9

0.5

0.5

0.5

0.1

0.9

0.9

0.9

0.5

0.1

0.5

0.5

0.5

0.1

0.1

0.9

0.5

0.1

0.5

0.1

0.1

0.5

0.1

0.1

0.1

3

1 0 0 6 7 RO ¼ 4 0 1 0 5 0

0 1

and the variance–covariance matrix for the error terms is set as

2

1 6q RE ¼ 4 E12

qE13

3

qE12 qE13 1 qE23 7 5 qE23 1

where qE12 ; qE13 , and qE23 are generated from a uniform distribution on the interval (0, 1); then the true value of P/T can be obtained based on Eq. (24). The simulated data is generated for given p ¼ 25 parts, o ¼ 5 operators, and r ¼ 3 repeated measurements. d are computed based For each setting, the average values of P=T on 1000 simulation runs. The comparison of the simulation results d  P=T) under different correlation coefficients of proof bias ( P=T duct is shown in Fig. 1. As one can see, the biases of P/T ratios proposed by Majeske (2008) and Wang and Yang (2007) are deviated from zero. In contrast, the biases of our revised P/T ratio are close to zero. This indicates that our revised P/T ratio outperforms other P/T ratios in terms of bias. To further investigate the effects of degree of product correlation on the bias, we set the variance–covariance matrix for the measured values as

2

1 6 RP ¼ 4 qp12

qp13

3

qp12 qp13 1 qp23 7 5 qp23 1

ð26Þ

where qp12 ; qp13 , and qp23 are generated from a uniform distribution with the interval (0, 1). Without loss of generality, in Table 4, we consider ten (10) different combinations of qp12 ; qp13 and qp23 under high, medium and low correlation for conducting various simulation runs. The simulated data is generated given that p ¼ 25 parts,

Fig. 2. Comparison of the simulation results of bias for various P/T ratios when the product correlation coefficients qp12 ; qp13 and qp23 are under different combinations (without considering PO interaction).

o ¼ 5 operators, r ¼ 3 repeated measurements and the average vald are computed based on 1000 simulation runs. Based on ues of P=T the simulation results as shown in Fig. 2, one can find that the biases of Majeske’s (2008) and Wang and Yang’s (2007) P/T ratios are deviated from zero. In contrast, the biases of our revised P/T ratio are closer to zero.

3.2.2. Random-effects MANOVA model with interaction term If the interaction between the part and operator does exist, then we consider the random-effects MANOVA model with interaction

J.-N. Pan et al. / Expert Systems with Applications 42 (2015) 7036–7045

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term. Let Xijk denote the measurement by operator j on part i at replication k, then the random-effects MANOVA model with interaction can be written as

Xijk ¼ l þ Pi þ Oj þ ðPOÞij þ Eijk

ð27Þ

where l is the total mean vector, Pi is the effect of the ith part, Oj is the effect of the jth operator, ðPOÞij is the effect of part-operator interaction and Eijk is the random error representing the repeatability. All of the above four effects are assumed to be random effects that are multivariate normally distributed with mean vector of zero and variance–covariance matrix of RP ; RO ; RPO and RE , respectively. Without loss of generality, in this simulation study, we assume that a MGRR study with three quality characteristics is conducted. The settings of the variance–covariance matrix for RP ; RO and RE are the same as the previous section, while the variance–covariance matrix for interaction is set as

2

1

6 RPO ¼ 4 qpo12

qpo13

3

qpo12 qpo13 1 qpo23 7 5 qpo23 1

where qpo12 ; qpo13 , and qpo23 are generated from a uniform distribution with the interval (0, 1). The simulated data is generated given d are computed that p ¼ 25; o ¼ 5; r ¼ 3 and the average values of P=T based on 1000 simulation runs. Based on the simulation results shown in Fig. 3, one can find that the biases of Majeske’s (2008) and Wang and Yang’s (2007) P/T ratios are deviated from zero. In contrast, the biases of our revised P/T ratio are closer to zero. To investigate the effects of degree of product correlation on the bias, we set the variance–covariance matrix for the measured values as shown in Eq. (26) and the values of qE12 ; qE13 , and qE23 generated from a uniform distribution with the interval (0, 1). For the variance–covariance matrix RP , ten (10) different combinations of qp12 ; qp13 and qp23 as shown in Table 4 are considered for conducting various simulation studies. The simulated data is generated given that p ¼ 25; o ¼ 5, and r ¼ 3, then the average vald are computed based on 1000 simulation runs. Based on ues of P=T the simulation results shown in Fig. 4, one can find that the biases of our revised P/T ratio are closer to zero and the biases of Majeske’s (2008) and Wang and Yang’s (2007) P/T ratios are deviated from zero.

Fig. 4. Comparison of the simulation results of bias for various P/T ratios when the product correlation coefficients qp12 ; qp13 and qp23 are under different combinations (considering the PO interaction).

3.3. The evaluation criteria for the accuracy of P/TR In this section, the mean square error (MSE) and mean absolute percent error (MAPE) are used to evaluate the performance of P/T ratios under different MGRR methods. The MSE and MAPE are defined as

MSE ¼

2 Pn ^ i¼1 hi  hi

MAPE ¼

n !  n ^ 1X hi  hi    100%  n i¼1  hi 

ð28Þ ð29Þ

hi is the estimated value for P/T where hi is the actual P/T ratio and ^ ratio. Simulation studies are conducted given o ¼ 5 and r ¼ 3 under various qp and p. A comparison of simulation results for MSE and MAPE under three P/T ratios using different MGRR methods is summarized in Tables 5 and 6 respectively. Notice that P=TW denotes P/T ratio using Wang and Yang’s (2007) PCA method, P=TM denotes P/T ratio using Majeske’s (2008) MANOVA method, and P=TR denotes P/T ratio using our proposed method. In order to reach a steady state for simulation results of MSE and MAPE for P=TW ; P=TM , and P=TR ratios, a minimum of 1000 simulation runs is required. As one can see from Tables 5 and 6, both MSE and MAPE for our P=TR are smaller than P=TW and P=TM . This indicates that our P=TR outperforms the existing P=TW and P=TM . 3.4. Deriving the confidence interval for P=TR and its expected width In order to determine the optimal allocation of various combinations among sample size of parts (p), number of operators (o) and replications (r), different widths of confidence intervals for our proposed P=TR based on the MANOVA random effect model are calculated. First, we derive the confidence interval for P=TR . According to Eq. (24), the estimator of P=TR is given by



21v dR ¼ jSG j P=T  jA j where SG is the sample variance–covariance matrix of gauge error. dR , we have: Based on the definitions of P=TR and P=T Fig. 3. Comparison of the simulation results of bias for various P/T ratios under different product correlation coefficient qp (considering the PO interaction).

dR jSG j 21v P=T ¼ jRG j P=TR

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Table 5 Comparison of MSE for three P/T ratios using different MGRR methods.

qp = 0.1

qp = 0.5

qp = 0.9

p

P/TW

P/TM

P/TR

P/TW

P/TM

P/TR

P/TW

P/TM

P/TR

25 50 100

20.864 20.536 36.188

0.057 0.058 0.059

0.018 0.017 0.018

6.180 38.818 96.883

0.163 0.159 0.160

0.024 0.023 0.022

53.117 84.555 156.276

1.894 1.898 1.894

0.060 0.056 0.058

Table 6 Comparison of MAPE for three P/T ratios using different MGRR methods.

qp = 0.1

qp = 0.5

3.5. The simulation results for the optimal allocation of por parameters

qp = 0.9

P

P/TW (%)

P/TM (%)

P/TR (%)

P/TW (%)

P/TM (%)

P/TR (%)

P/TW (%)

P/TM (%)

P/TR (%)

25 50 100

30.5 66.2 177

15.9 15.7 15.8

8.1 7.7 7.8

61.5 160 275

24.3 23.8 24

7.9 7.7 7.7

129 219 266

53.1 53.4 53.5

7.9 8.2 7.8

According to Anderson (2003), jSG j=jRG j is the distribution of Qv 2 vnpi W ¼ ði¼1 , where n is the total number of measurements and n1Þv we have



jSG j Pr x1a2 6 6 xa2 ¼ 1  a jRG j ( )  21v P=T dR  21v ¼1a Pr x1a2 6 6 xa2 P=TR 8 9 > > < P=T dR d P=TR = Pr   1 6 P=TR 6  21v > ¼ 1  a > : xa 2v x a ;

ð30Þ

12

2

where wa is a constant such that PrfW < wa g ¼ 1  a. According to Eq. (30), a 100ð1  aÞ% confidence interval for P=TR is given by:

2

3

dR dR 7 6 P=T P=T 6 7 21v 5 4 21v ; 

xa2

ð31Þ

x1a2

Then, the expected width of confidence interval for P=TR can be obtained:

3

2

3

2

dR dR 7 6 P=T 6 P=T d 7 6 E6 21v   21v 5 ¼ Eð P=TR Þ  4 4

x1a2

xa2

¼ E P=TR 



jS G j jRG j

¼ P=TR  E

jS G j jRG j

21v !

21v !

x1a2

2 6 6 4 2

1

1

x1a2

6 6 4

1

x1a2

21v  

21v  

1. If the precision of measurement system is satisfactory, then we assume P=TR ¼ 0:1. 2. If the precision of measurement system is marginally acceptable, then we assume P=TR ¼ 0:3. 3. If the precision of measurement system is unacceptable, then we assume P=TR ¼ 0:5. Using the smallest expected width of confidence interval as a criterion for selecting the optimal allocation of por parameters, the simulation results of various expected lengths of confidence intervals for P=TR under different (n, p) combinations are summarized and illustrated in Fig. 5. As expected, we found that the expected width of confidence interval decreases as the total number of measurements increases. In addition, for a given total number of measurements, the expected width of confidence interval decreases as the sample size of parts decreases. In other words, two parameters (number of parts p and a total number of measurements n) significantly affect the expected length of confidence interval of P=TR . Note that, in Fig. 5, the decreasing trend of various expected widths of confidence intervals for P=TR becomes stabilized and the three lines of p = 30, p = 20, p = 10 are almost coincided when the total number of measurements (n) reaches between 180 and 300 for the first scenario (P=TR ¼ 0:1). Striking the right balance

1 7 7 21v 5

xa2

1 7 7 21v 5

xa2

21v  

3

The expected width of confidence interval in Eq. (34) can be used as a criterion for searching the optimal allocation of por parameters. The number of quality characteristics is set as v ¼ 3. According to the guidelines for measurement system acceptability (see Table 1) suggested by AIAG (2010), the following three different scenarios are adopted in our computer simulation:

3

1 7 7 21v 5

ð32Þ

xa2

According to Anderson (2003), we have:



jS G j E jRG j

21v

  rffiffiffiffiffiffiffiffiffiffiffiffi Qv npi þ 21v i¼1 C 2 2   ¼ n  1 Qv C npi i¼1

ð33Þ

2

Thus, the expected width of confidence interval for P=TR in Eq. (32) can be written as:

  2 rffiffiffiffiffiffiffiffiffiffiffiffi Qv npi 1 C þ 6 i¼1 2 v 2 2   6 P=TR  4 npi n  1 Qv i¼1

C

2

3 1

x1a2

21v  

1 7 7 21v 5

xa2

ð34Þ Fig. 5. Various expected widths of confidence intervals for P=TR at three different P/ T scenarios.

7043

J.-N. Pan et al. / Expert Systems with Applications 42 (2015) 7036–7045 Table 7 The optimal allocations of por parameters under three different scenarios for P=TR . (p, o, r) n ¼ 300

n ¼ 240

Scenario 1. ðP=TR ¼ 0:1Þ

(10, (10, (20, (20, (30, (30,

5, 6, 3, 5, 2, 5,

6) 5) 5)* 3)* 5)* 2)*

(10, (10, (20, (20, (30, (30,

4, 6, 3, 4, 2, 4,

6)* 4)* 4)* 3)* 4)* 2)*

n ¼ 180 (10, (10, (20, (30, (30,

Scenario 2. ðP=TR ¼ 0:3Þ

(10, (10, (20, (20, (30, (30,

5, 6, 3, 5, 2, 5,

6) 5) 5)* 3)* 5)* 2)*

(10, (10, (20, (20, (30, (30,

4, 6, 3, 4, 2, 4,

6)* 4)* 4)* 3)* 4)* 2)*

X

Scenario 3. ðP=TR ¼ 0:5Þ

(10, (10, (20, (20, (30, (30,

5, 6, 3, 5, 2, 5,

6) 5) 5)* 3)* 5)* 2)*

X

3, 6, 3, 2, 3,

6)* 3)* 3)* 3)* 2)*

X

* Denotes the alternative (n, p) combinations other than (300, 10) when considering the limit of inspection cost.

between the precision of measurement system while still maintaining cost-effectiveness in determining the optimal allocation of por parameters, one may select (n, p) = (180, 10) or (240, 10) as alternatives other than (300, 10) combination for the first scenario when considering the limit of inspection cost. Similarly, one may select (n, p) = (240, 10) as an alternative other than (300, 10) combination for the second scenario (P=TR ¼ 0:3Þ as the three lines of p = 30, p = 20, p = 10 are almost coincided when the total number of measurements (n) reaches between 240 and 300. Furthermore, based on the simulation results and using the smallest confidence interval of P=TR as a criterion for the multivariate measurement systems with three quality characteristics, the suggested optimal combinations of por parameters under three (3) different scenarios for P=TR are summarized in Table 7. According to Table 7, one may conclude that (p, o, r) = (10, 3, 6) or (10, 4, 6) [equivalent to (n, p) = (180, 10) or (240, 10)] are the alternative optimal allocations other than (10, 5, 6) [equivalent to (n, p) = (240, 10)] for the first scenario and (p, o, r) = (10, 4, 6) is the alternative optimal allocation for the second scenario, respectively. As for the optimal allocation for the third scenario, (p, o, r) = (10, 5, 6) [equivalent to (n, p) = (300, 10)] is recommended if the limit of inspection cost is considered in conducting a MGRR study. In other words, a minimum of approximately 300 measurements is recommended for rejecting the multivariate measurement systems given the third scenario while a minimum of approximately 180 measurements is recommended for accepting the multivariate measurement systems given the first scenario and a minimum of approximately 240 measurements is recommended for marginally accepting the multivariate measurement systems given the second scenario. Moreover, given the total number of measurements other than 180, 240 and 300 for the first scenario, one may select an optimal allocation of por parameters based on economical consideration. For example, if n = 200 and p = 20 is given, then o = 2 and r = 5 can be one of the options.

test of an electronic product. Ten parts (p = 10) and three operators (o = 3) were taken to conduct this MGRR experiment. Each operator measured all 10 parts in five consecutive trials (r = 5). The measuring conditions are: the insertion speed is 20 mm/s, depth is 3 mm, and time is 5 s. Three values (M 1 ; M 2 ; M 3 ) are recorded during their measuring process, where M 1 denotes the response time (in seconds) when the device begins to solder with Sn and its lower and upper specification limits are set at (0.3, 1.0); M 2 denotes the response time (in seconds) when the solderability reaches to 2/3 maximum force and its lower and upper specification limits are set at (0.5, 1.2); M 3 denotes the maximum force (in milli-newtons, mN) during the measuring process and its lower and upper specification limits are set at (1.0, 1.2). A total of 150 measurements as listed in the Appendix A are collected in their MGRR study. By performing the Mardia (1980)’s normality test, we found that the p-value for Mardia’s SW statistics is 0.309. Thus, the assumption of multivariate normality cannot be rejected at a 95% confidence level. As the 150 randomly collected measurements follow a multivariate normal distribution, the measurement system analysis using MANOVA is then performed. We consider the random-effects MANOVA model with interaction as listed below:

Xijk ¼ l þ Pi þ Oj þ ðPOÞij þ Eijk The correlation coefficient matrix of these three quality characteristics is calculated as shown in Table 8. After performing MANOVA, one can obtain the estimated value of variability of gauge as

2

0:0003355 0:0004691 0:0000523

3

6 7 SG ¼ 4 0:0004691 0:0011773 0:0000906 5 0:0000523

0:0000906

0:0000558

According to Eq. (22), we have

2

0:0086534 0:0004691 0:0005447

3

6 7 A ¼ 4 0:0086461 0:0086534 0:0004823 5 0:0005447 0:0004823

0:0007064

Then, our revised P=TR ratio can be calculated by



21v dR ¼ jSG j ¼ 0:736 P=T  jA j In order to compare our revised P=TR ratio’s performance with Majeske’s, Wang and Yang’s P/T ratios, three different P/T values are summarized in Table 9. Note that both Majeske’s P/T ratio and Wang and Yang’s P/T ratio suggest that the measurement systems are marginally acceptable. According to AIAG (2010), the acceptance of measurement systems should be based upon importance of application measurement, cost of measurement device, costs of rework or repair, and approval by the customer. In contrast with Majeske’s and Wang and Yang’s MGRR results, our revised P/T ratio indicates that the measurement system is considered to be

Table 8 The correlation coefficient matrix of the three quality characteristics.

M1 M2 M3

M1

M2

M3

1 0.9999986 0.9547479

_ 1 0.9552384

_ _ 1

4. Numerical example To compare the appropriateness of our proposed P=TR ratio with Majeske’s P/T and Wang and Yang’s (2007) P/T ratio, a numerical example adopted from Wang and Yang (2007) is used for illustration purposes. There are three quality characteristics in their solderability

Table 9 Comparison of the three P/T ratios using different MGRR methods.

P/T

Wang and Yang (2007)

Majeske (2008)

Revised P/T

0.138

0.208

0.736

7044

J.-N. Pan et al. / Expert Systems with Applications 42 (2015) 7036–7045

unacceptable if the product correlations among three quality characteristics are considered. From Table 8, notice that the three quality characteristics are highly correlated. Therefore, it will be more appropriate to use the revised P/T ratio to evaluate the acceptability of a multivariate measurement system. As the result of this measurement system is considered to be unacceptable, it is worthy to mention that the total number of measurements need to be set as n = 300 according to Table 7 while only 150 measurements are collected in this example. It is suggested that additional 150 measurements be taken to confirm the rejection of this measurement system. Based on our research results, a step-by-step procedure for conducting future MGRR study for the multivariate measurement systems with three quality characteristics is recommended as follows.

the others in terms of MSE and MAPE. In the numerical example, it further indicates that both Majeske’s and Wang and Yang’s P/T ratios are underestimated when the quality characteristics are correlated. Therefore, our proposed new P=TR is more appropriate in performing the measurement system analysis for correlated characteristics. As the optimal allocation of por parameters is essential to determine the adequacy of a multivariate measurement system, we have found that two parameters (number of parts p and a total number of measurements n) significantly affect the expected length of confidence interval of P=TR based on the simulation results shown in Fig. 5. A reference table with the optimal allocation of por parameters is constructed accordingly if the inspection cost/budget is limited. In order to show the practical use of our findings in the optimal allocation of por parameters, a numerical example with a step-by-step procedure for conducting a MGRR study is given to illustrate the appropriateness of our proposed P/T ratio. Hopefully, it can be served as a useful guideline for quality practitioners when performing a multivariate measurement system analysis in industries. Due to the time limitation, only the random-effect MANOVA models with and without interaction are considered in this paper. Moreover, the number of quality characteristics for computer simulation for determining the optimal allocation of por parameters is set as v ¼ 3. Future researchers may consider other numbers of multiple quality characteristics besides v ¼ 3 and further extending the simulation and discussions from random-effect to mixed-effect MANOVA models in case the number of operators or inspectors are fixed. The other approval criteria, such as %R&R, number of distinct categories (ndc) as stated in AIAG (2010) need to be further explored in performing the MGRR analysis for correlated quality characteristics. Then, a comparative study can also be conducted to explore the differences among these approval criteria.

Step 1: Decide the gauges for measurement, and the specifications of multiple quality characteristics. Step 2: Use Table 7 as a reference to determine the optimal allocation of measurement parameters, i.e. sample size of parts (p), number of operators (o) and replications (r). Step 3: Perform actual measurements and collect data for multivariate quality characteristics. Step 4: Perform Mardia test to check the multivariate normality of collected data and estimate the revised P/T ratio of the measuring process by performing MANOVA. Step 5: Evaluate the adequacy of multivariate measurement systems based on the revised P/T ratio. The precision of a multivariate measurement system is achieved with the required total number of measurement and its associated optimal allocation of por parameters under three different scenarios. 5. Conclusions and future research areas An adequate multivariate measurement system plays an important role in quality improvement for manufacturing industries. Recently, He et al. (2011) proposed a modified principal component analysis (PCA) method to transform multivariate measurement data with dependent variables into those with independent principal components. Although He’s and the other existing PCA methods can be served as an alternative approach to multivariate MSA, these methods cannot provide and remove the causes of production problems. Hence, in this paper, we modify Majeske’s (2008) P/T ratio by taking correlation coefficients among tolerances into account. Then, the simulation method is used to compare the performance of the revised P/T ratio with that of the existing ratios. The simulation results show that our revised P/T ratio outperforms

Acknowledgements We are grateful for the two anonymous reviewers for their constructive comments. The first author would like to gratefully acknowledge financial support from the National Science Council of Taiwan, ROC. Appendix A A.1. measurement data with p = 10, o = 3, r = 5 adopted from Wang and Yang (2007)

Operators A Parts

1

2

3

Responses

M1 M2 M3 M1 M2 M3 M1 M2

B

C

Replicates 1

2

3

4

5

1

2

3

4

5

1

2

3

4

5

0.45 0.79 1.079 0.61 0.97 1.103 0.55 0.9

0.41 0.75 1.072 0.62 0.99 1.105 0.53 0.83

0.38 0.73 1.075 0.61 0.96 1.096 0.54 0.89

0.48 0.85 1.073 0.61 0.9 1.103 0.55 0.92

0.46 0.84 1.081 0.6 0.97 1.109 0.52 0.86

0.46 0.77 1.082 0.61 0.92 1.103 0.54 0.86

0.42 0.74 1.071 0.62 0.94 1.102 0.55 0.88

0.45 0.79 1.068 0.63 0.97 1.112 0.54 0.92

0.39 0.71 1.075 0.61 0.94 1.108 0.52 0.82

0.48 0.85 1.081 0.59 0.88 1.093 0.56 0.91

0.41 0.78 1.065 0.61 0.93 1.093 0.51 0.85

0.49 0.89 1.085 0.6 0.95 1.105 0.54 0.88

0.45 0.8 1.079 0.62 0.97 1.109 0.54 0.9

0.47 0.88 1.073 0.61 0.96 1.101 0.58 0.95

0.44 0.85 1.071 0.65 1.02 1.119 0.53 0.91

7045

J.-N. Pan et al. / Expert Systems with Applications 42 (2015) 7036–7045 Appendix A (continued) Operators A Parts

4

5

6

7

8

9

10

Responses

M3 M1 M2 M3 M1 M2 M3 M1 M2 M3 M1 M2 M3 M1 M2 M3 M1 M2 M3 M1 M2 M3

B

C

Replicates 1

2

3

4

5

1

2

3

4

5

1

2

3

4

5

1.036 0.64 1.01 1.078 0.68 1.04 1.083 0.57 0.88 1.124 0.61 0.96 1.063 0.49 0.84 1.074 0.58 0.92 1.067 0.71 1.05 1.085

1.027 0.63 1 1.082 0.67 1.02 1.078 0.56 0.91 1.098 0.62 1.01 1.054 0.51 0.85 1.089 0.59 0.94 1.075 0.75 1.06 1.098

1.041 0.64 0.96 1.063 0.67 1.07 1.099 0.58 0.92 1.129 0.63 0.98 1.057 0.5 0.83 1.09 0.6 0.94 1.061 0.73 1.1 1.097

1.051 0.65 1.05 1.082 0.66 1.01 1.091 0.56 0.87 1.113 0.62 0.97 1.06 0.49 0.81 1.087 0.6 0.98 1.068 0.74 1.12 1.098

1.024 0.64 1.02 1.079 0.69 1.08 1.088 0.57 0.93 1.121 0.63 1.03 1.048 0.51 0.86 1.078 0.59 0.95 1.057 0.7 1.06 1.088

1.015 0.63 0.96 1.09 0.68 1.05 1.091 0.58 0.94 1.135 0.61 0.94 1.069 0.52 0.86 1.085 0.57 0.9 1.058 0.71 1.03 1.086

1.051 0.66 0.99 1.079 0.67 1 1.094 0.56 0.87 1.115 0.65 1.03 1.054 0.5 0.8 1.081 0.6 0.95 1.061 0.75 1.07 1.097

1.037 0.64 0.98 1.082 0.68 1.03 1.098 0.57 0.89 1.128 0.63 0.99 1.061 0.51 0.85 1.09 0.59 0.94 1.068 0.72 1.06 1.088

1.046 0.65 0.97 1.068 0.66 0.99 1.086 0.56 0.92 1.121 0.62 0.98 1.058 0.53 0.88 1.094 0.61 0.99 1.076 0.75 1.12 1.098

1.034 0.61 0.92 1.079 0.69 1.07 1.106 0.59 0.95 1.112 0.63 0.97 1.048 0.49 0.81 1.087 0.59 0.93 1.065 0.73 1.09 1.095

1.035 0.66 1 1.091 0.72 1.09 1.098 0.56 0.92 1.116 0.62 0.95 1.046 0.49 0.82 1.076 0.59 0.94 1.076 0.75 1.09 1.098

1.026 0.64 0.97 1.08 0.67 1.03 1.09 0.58 0.91 1.124 0.63 0.96 1.058 0.52 0.87 1.093 0.6 0.93 1.069 0.72 1.04 1.086

1.046 0.64 0.99 1.079 0.68 1.01 1.094 0.57 0.94 1.13 0.61 0.95 1.061 0.51 0.85 1.088 0.58 0.92 1.056 0.74 1.08 1.096

1.038 0.67 1 1.082 0.69 1.02 1.096 0.55 0.91 1.128 0.62 0.97 1.052 0.51 0.86 1.09 0.6 0.94 1.067 0.71 1.06 1.091

1.043 0.62 0.97 1.076 0.66 0.98 1.083 0.59 0.97 1.131 0.67 1.03 1.069 0.53 0.9 1.095 0.62 0.97 1.061 0.78 1.11 1.106

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