Multivariate nonnormal process capability analysis

Int J Adv Manuf Technol (2009) 44:757–765 DOI 10.1007/s00170-008-1883-9

ORIGINAL ARTICLE

Multivariate nonnormal process capability analysis S. Ahmad · M. Abdollahian · P. Zeephongsekul · B. Abbasi

Received: 24 January 2008 / Accepted: 28 November 2008 / Published online: 11 February 2009 © Springer-Verlag London Limited 2009

Abstract There is a great deal of interest in the manufacturing industry for quantitative measures of process performance with multiple quality characteristics. Unfortunately, multivariate process capability indices that are currently employed, except for a handful of cases, depend intrinsically on the underlying data being normally distributed. In this paper, we propose a general multivariate capability index based on the Mahanalobis distance, which is very easy to use. We also approximate the distribution of these distances by the Burr XII distribution and then estimate its parameters using a simulated annealing search algorithm. Finally, we give an example, based on real manufacturing process data, which demonstrates that the proportion of nonconformance (PNC) using our proposed method is very close to the actual PNC value, which also justifies its adoption in this paper. Keywords Process capability index (PCI) · Nonnormal distributions · Geometric distance (GD) · Covariance distance (CD) · Simulated annealing (SA) · Burr XII distribution · Proportion of nonconformance (PNC)

S. Ahmad (B) · M. Abdollahian · P. Zeephongsekul School of Mathematical and Geospatial Sciences, RMIT University, Melbourne, Victoria, Australia e-mail: [email protected], [email protected] B. Abbasi Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran

1 Introduction 1.1 Background Process capability is a statistical measure used in the manufacturing industry to assess the variability of process outcomes in relation to their engineering specifications. These quantitative measures are especially useful in process development and process improvement endeavors; it helps to identify those process quality characteristics that have high variation when compared with their specification spread. If such a problem exists, then an application of process capability analysis is necessary to identify the proportion of parts being produced outside specification, as well as to prevent further production of unacceptable products. The book by Kotz and Johnson [1] contains a detailed discussion of the most conventional process capability index deployed to evaluate process performance based on a single quality characteristic defined as:

Cp =

USL − LSL , 6σ

(1)

where USL and LSL are the upper and lower engineering specification limits. The basis of the capability index defined by Eq. 1 depends on an implicit assumption that the underlying quality characteristic measurements are normally distributed, for it is only in this case that 6σ may be justifiable as a measure of natural spread; in particular, it is only under the normal assumption that a centered process will have 0.27% of its output outside the μ ± 3σ interval.

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1.2 Nonnormality and multivariate process capability It is an undisputed fact that production processes very often produce nonnormal data. Furthermore, there is always more than one quality characteristics of interest in process outcomes, and very often, these quality characteristics are correlated with each other. For example, in a detailed description of a connecting rod for a combustion engine [2], the crank bore inner diameter, pin bore inner diameter, rod length, bore true location, bore-to-bore parallelism, and other features are specified. To represent how well this connecting rod is made, one may examine numerical summaries of each of these individual characteristics separately or consider all characteristics together to see how they interact with each other. The latter is preferred if one treats the rod as one entity. In situations where the design intention of a product is prescribed by a number of related characteristics, the functionality of this product cannot be represented by individual characteristics separately. Many other such examples are scattered throughout the quality control literature, which points toward the need for multivariate process capability analysis. According to a comprehensive survey given in [3], multivariate process capability assessment can be obtained from: 1. The probability of nonconforming products using distribution of the multivariate process 2. The ratio of a tolerance region to a process region 3. Different approaches using loss functions 4. Geometric distance approach involving principle component analysis Wierda [4], using an analogy from univariate process capability index (PCI), suggested the following index to calculate multivariate PCI for normal processes: 1 MC pk = − −1 (θ) 3 where (.) is the cumulative distribution of the standard normal random variable and θ is the multivariate normal probability that the process lies within a multivariate specification limit. Hubele et al. [5], again assuming that the multivariate characteristics follow a multivariate normal distribution, suggested a PCI that is defined as the ratio of a rectangular tolerance region to a modified process region. The tolerance region is defined as the smallest rectangle around the normal ellipsoid centered at the target value with a type I error of α = 0.0027. The number of quality characteristics p in the process is taken into account by taking the pth root of this ratio.

Int J Adv Manuf Technol (2009) 44:757–765

This index is defined by 

C pm

volume of tolerance region = volume of process region

 1p

.

(2)

Another method of estimating PCI for multivariate normal process was proposed by Chen [6]. In his paper, a tolerance zone is defined by V = {X ∈ R p : h(X − μ0 ) ≤ r0 }, where r0 is a positive number, μ0 is the target value, and h(x) is a specific positive function. A multivariate process X is deemed capable if Pr(X ∈ V) ≥ 1 − α. Let r = min{c : Pr(h(X − μ0 ) ≤ c) ≥ 1 − α}. If the cumulative distribution function of h(X − μ0 ) is increasing in a neighborhood of r, then r is simply the unique root of the equation Pr(h(X − μ0 ) ≤ c) = 1 − α. Therefore, the process is capable if r ≤ r0 . Note that, from the above definitions, r0 is the half-width of the tolerance interval centered at the target value μ0 and r is the half width of an interval centered on the target value such that the probability of a process realization falling within this interval is equal to 1 − α. Recently, Wang et al. [7] compared the previous two multivariate PCIs and presented some graphical examples to illustrate them. Chen et al. [8] extended Boyles’ work [9] to processes with multivariate characteristics by producing an integrated PCI that can be used for both normal and nonnormal variables. Castagliola et al. [10] extended the univariate PCI developed in Castagliola [11] to bivariate normal quality characteristics and also came up with a novel method based on Green’s formula to compute their PCIs. According to Wang [12], multivariate PCIs that were proposed by many practitioners suffer from the following constraints and limitations: 1. Normality assumption on multivariate data is usually required. 2. Confidence intervals of the multivariate capability indices are difficult to derive. 3. Higher-dimension (more than three quality variables) PCIs are not readily obtainable, except through projection of multivariate data into univariate variables such as the geometric distance approach proposed by Wang and Hubele [12] and Wang and Du [13]. Due to the above restrictions, it is evident that the application of conventional methods is somewhat limited. In order to deal with nonnormal multivariate and correlated quality characteristics data, there is ample opportunity for researchers to develop more suitable

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PCIs that can address the complex situation of multivariate nonnormal and correlated data. 1.3 Proposed methodology In this paper, we propose another approach of dealing with multiple quality characteristics. In this approach, we first cluster correlated quality characteristics and then define a variable, which we refer to as the covariance distance (CD), which is the distance of individual quality characteristics from their respective targets scaled by their variance–covariance matrix. CD is well known in pattern recognition literature as the Mahanalobis distance [14]. The proposed approach is also similar to the geometric distance (GD) approach adopted by Wang [3] but differs in that the scaling factor of the variance–covariance matrix is absent in GD. Furthermore, unlike the approach in [3], we fit Burr XII distribution [15] to the CD data instead of fitting different distributions to GD data, as was done there. The parameters of the fitted Burr XII distribution are obtained using maximum likelihood estimation (MLE) technique through a systematic random search algorithm called simulated annealing (SA). Note that the proposed method is not restricted only to normal multivariate data but can also be applied to nonnormal quality characteristics. In practice, the fundamental objective of process capability analysis is to help engineers and managers decide whether to accept or reject the process outcomes based on conformance to engineering specifications [16]. Keeping this objective in mind, the efficacy of the proposed method will be assessed by the proportion of nonconformance (PNC) criterion, which is frequently used in practice to assess the utility of a PCI method [3]. This paper is organized in the following manner. The proposed multivariate capability index is discussed in Section 2. A review of the Burr XII distribution is given and the SA algorithm is displayed in Section 3. Using the proposed methodology, an application example with real data from manufacturing is presented in Section 4. Finally, the paper is concluded with some suggestions for future research in Section 5.

2 Multivariate capability index based on CD approach Although several methods have been proposed to handle the issues related to nonnormality for univariate quality characteristic data in PCI studies, there has not been much research on this topic devoted to similar issues in multivariate nonnormal data. In this section,

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we will discuss our proposed methodology based on CD variable. Our approach is closely related to the GD approach proposed by Wang and Hubele [12], which reduces the dimension of the multivariate process data and renders them more tractable for a statistical analysis. The GD approach utilizes the Euclidean distance (or L2 norm), which is defined as follows: let X = (X1 , X2 , . . . , Xn ) be a point from the sample space and let T = (T1 , T2 , . . . , Tn ) be the corresponding target value. Then, the GD variable G is defined by  (X − T) (X − T)  = (X1 − T1 )2 + (X2 − T2 )2 + . . . + (Xn − Tn )2 .

G=

(3) A comprehensive study of the distribution of G when the underlying variables have a multivariate normal distribution was undertaken in [12]. When the underlying distribution is nonnormal, Wang [3] combined correlated quality characteristics to form G and determined the distribution that best fit G by using a best-fit statistical software. Instead of GD, we propose using the CD defined by C=

 (X − T)  −1 (X − T),

(4)

where  −1 refers to the variance–covariance matrix of X. Note that G differs from C in that a scaling factor based on the variance–covariance matrix is present in the latter but absent from the former. The reason for proposing CD in this paper is motivated by the fact that C2 is equal to the maximum value of the squared weighted distance between the point X and its target T, scaled by its variance–covariance matrix. Mathematically, this is expressed as

max a=0

((X − T) a)2 = (X − T)  −1 (X − T) a  −1 a

(5)

where the maximum is achieved when a = c  −1 (X − T) for any c = 0. (The proof follows easily from the maximization lemma, c.f. p.81 of Johnson and Wichern [17].) Note that G2 corresponds to the case when  = I, the identity matrix, which corresponds to the case when the variables are statistically independent of each other. However, this is not appropriate for most multivariate quality characteristics due to the variables showing some degree of dependency, and for this reason, we have chosen CD over GD.

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Int J Adv Manuf Technol (2009) 44:757–765

To illustrate the calculation of C, we use the values of variables X1 , X2 , and X3 given in Table 3, which gives ⎛



−1

118.502 −5.565 = ⎝ −5.565 9.314 0.036 0.002

⎞ 0.036 0.002 ⎠ . 0.006

PNCTotal = 1 −

X − T = (0.1165 − 0.10, 0.0614 − 0, 10.7824 − 11), and applying Eq. 4 gives C = 0.2368. Furthermore, the maximum radial distance (MRD), used as upper specification of the GD variable, is the distance from the target to the perimeter of the tolerance region. For multivariate characteristics, MRD is defined by MRD = (Tol X1 )2 + (Tol X2 )2 + . . . + (Tol Xn )2 ,

(6)

where Tol Xi is the tolerance of the quality characteristic Xi . For the previous example, MRD is equal to MRD = (Tol X1 )2 + Tol X2 )2 + Tol X3 )2  = (0.04)2 + (0.50)2 + (5.0)2 = 5.025 mm. One criterion, proposed by many researchers, such as Liu and Chen [18], Wierda [4], and Singpunvalla [16], for assessing the efficacy of a proposed CPI method for univariate characteristics, is to determine the PNC. Using MRD as upper bound, the estimated PNC is given by: MRD

1 − F(MRD) = 1 −

f (x)dx,

(7)

0

where f (x) is the density function of the GD variable and F(x) represents its cumulative distribution Table 1 A flowchart of the proposed methodology

1. 2. 3. 4. 5. 6. 7. 8.

k

 i=1

Suppose we observe values of X1 , X2 , and X3 to be 0.1165, 0.0614, and 10.7824, and assume that the target values are within the specification limits 0.10 ± 0.04, 0 ± 0.50, and 11 ± 5 mm, where 0.04, 0.5, and 5 are the tolerances of X1 , X2 , and X3 , respectively. Then,



function. For manufactured product with multiple quality characteristics, the estimated PNC is given by MRDi

fi (y) dy,

(8)

0

where fi (y) is the density function of the ith GD variable, based on grouping subsets of correlated multivariate variables, and MRDi is its corresponding MRD, i = 1, 2, . . . , k. In this paper, we apply PNC to CD variables; also, Burr XII distribution will be used to fit fi (y) and this will be discussed in the next section. The flowchart of the proposed methodology is presented in Table 1.

3 Fitting Burr XII distribution to CD data using a SA algorithm Burr XII distribution is a versatile distribution that has been applied extensively in the area of quality control, reliability analysis, and failure time modeling [19]. It has been observed by several researchers that many quality characteristics have positively skewed data, and the Burr distribution can be suitably employed to fit the underlying distribution of these data. The distribution was introduced by Burr [15] in 1942 as one member of a family of twelve cumulative distribution functions having simple functional forms that could be used to fit data by matching sample to population moments. Castagliola [11] used Burr’s general approach to compute the proportion of nonconforming items based on the process empirical distribution function. Rodriguez [20] demonstrated that the Weibull distribution is a limiting distribution of the Burr XII distribution and the two-parameter Burr XII distribution can be used to describe data in the real world. Liu and Chen [18] proposed a modified Clements method based on Burr XII distribution to evaluate PCI for nonnormal univariate data. In their paper, they demonstrated that the accuracy of the estimated PCI for nonnormal

Identify the desired quality characteristics along with their respective engineering specifications from a manufacturing process Collect measurements of these quality characteristics data from a manufacturing process Determine the correlated and uncorrelated quality characteristics Compute CD variables for correlated and uncorrelated quality characteristics using Eq. 4 Compute MRDs from the target value using Eq. 6 Use SA search algorithm to find out parameters of the fitted univariate Burr distribution to each CD variable Compute the PNC for each CD variable using Eq. 7 Compute total PNC (PNCTotal ) value using Eq. 8

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data can be improved using Burr distribution. Various estimation techniques involving Burr XII distributions have been proposed by Wingo [21], Jaheen [22], and others. Burr XII distribution has also been extended to the multivariate case by Takahashi [23]. The cumulative distribution function and probability density function of the Burr XII distribution are defined, respectively, by

the log-likelihood function of the Burr XII distribution is [21]: l(c, k; x1 , x2 , . . . , xn ) = n log(c) + n log(k) − (1 + k)

n

 log 1 + xic

i=1

+ (c − 1)

n

log xi ).

(11)

i=1

F(x) = 1 −

1 (1 + xc )k

(9)

The first order condition for obtaining optimal c and k values gives rise to the following differential equations: xc log(xi ) ∂l n i log(xi ) − (1 + k) = 0 (12) = + ∂c c 1 + xic i=1 i=1 n

and f (x) =

ckxc−1 , (1 + xc )k+1

(10)

and ∂l n = − log(1 + xic ) = 0. ∂k k i=1 n

x ≥ 0, c, k ≥ 1. The parameters c and k are associated with the skewness and kurtosis of the Burr XII, respectively. Burr [24] showed that various probability density functions can be approximated by a Burr distribution using different values for c and k. For example, the normal density function can be estimated by a Burr distribution with c = 4.85437 and k = 6.22665, a Gamma distribution with shape parameter of 16 can be approximated by a Burr XII distribution with c = 3 and k = 6, and the log-logistic distribution is a special case of the Burr XII distribution. Hence, the two-parameter Burr XII distribution can be used to describe many important distributions that occur in the real world. In this paper, we fit the Burr XII distribution to CD variable defined by Eq. 4. We will use the method of MLE to estimate parameters c and k. Given a random sample of size n, x1 , x2 , . . . , xn from the population,

Table 2 SA algorithm

n

1. 2.

3. 4. 5. 6.

7.

(13)

To determine the MLE estimators of c and k, we solve Eqs. 12 and 13 using a SA search algorithm. As its name implies, SA exploits an analogy between the way in which a metal cools and freezes into a minimum energy crystalline structure (the annealing process) and the search for a minimum in a more general system. The algorithm is based on that of Metropolis et al. [25], which was originally proposed as a means of finding the equilibrium configuration of a collection of atoms at a given temperature. The connection between this algorithm and mathematical minimization was first observed by Pincus [26], but it was Kirkpatrick et al. [27] who proposed it as an optimization technique for combinatorial and other optimization problems. Ease of use and provision of good solutions to realworld problems makes this algorithm one of the most

Obtain a CD variable data Decide control parameters of SA, i.e., initial temperature (T0 ), freezing temperature (T f ), and number of repetitions allowed at each temperature level (C), temperature reducing constant (I); typically, 0.75 ≤ I ≤ 0.95 Generate random values c, k parameters of Burr distribution Compute the likelihood function, L, at this randomly generated value While T > To , T = CT For I = 1 to I 6.1. Generate neighboring values, say c1 , k1 , for c, k 6.2. Compute the likelihood function at this new solution, Lo 6.3. Evaluate parameters 6.3.1. if Lo > L, then c = c1 , k = k1 , and L = Lo 6.3.2. else generate a random value, u, of Uni(0,1) 6.3.2.1 if u < e− FoT−F then c = c1 , k = k1 Print c, k and L

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Table 3 Manufacturing data and the generalized CD variables X1

X2

X3

X4

X5

X6

X7

G1

G2

G3

G4

C1

C2

C3

C4

0.1165 0.1259 0.1265 0.1185 0.1414 0.092 0.0804 0.1103 0.1022 0.1103 0.1069 0.1075 0.117 0.1276 0.1039 0.1251 0.1153 0.0961 0.117 0.1158 0.1134 0.1356 0.1222 0.1124 0.1155 0.1223 0.113 0.1137 0.1112 0.1114 0.0978 0.1209 0.1092 0.1161 0.1131 0.1193 0.1233 0.089 0.1074 0.0947 0.1048 0.1192 0.1143 0.1102 0.1084 0.1071 0.1106 0.1051 0.1167 0.1062 0.1091 0.1138 0.1309 0.1075 0.1157 0.1193 0.1239

0.0614 0.0277 0.0762 0.0957 0.1319 0.0476 0.044 0.0901 0.0918 0.0823 0.0943 0.095 0.1177 0.1378 0.0645 0.0988 0.1553 0.0142 0.0808 0.078 0.1511 0.0407 0.0654 0.1516 0.0695 0.0853 0.0869 0.0652 0.1357 0.0909 0.097 0.0939 0.0419 0.1088 0.0698 0.1445 0.09 0.1102 0.0507 0.0845 0.0661 0.0448 0.1009 0.0815 0.1617 0.0423 0.0585 0.1151 0.0827 0.0566 0.0221 0.119 0.0699 0.0417 0.0377 0.0752 0.0849

10.7824 10.8395 10.9538 10.5824 10.7395 10.7109 10.4824 10.7966 10.8681 10.7824 10.6681 10.6109 10.9538 10.7681 10.6109 10.6395 11.2966 10.9657 11.0657 10.249 10.8324 11.0824 10.8157 7.399 6.7824 7.2538 10.5681 6.5109 6.8824 10.799 10.4324 10.6657 11.4324 11.199 10.3824 11.949 12.2681 12.3824 12.3681 12.4538 12.2824 11.8395 12.2252 12.5109 12.0681 12.0252 11.8252 12.3824 12.3395 12.2538 12.2824 12.4395 12.6324 11.4824 11.9157 12.4157 11.8324

0.009 0.0091 0.0153 0.0216 0.0088 0.0226 0.0104 0.0064 0.0782 0.031 0.0265 0.0564 0.0069 0.1162 0.0341 0.0671 0.0208 0.0019 0.0343 0.0631 0.075 0.0138 0.0021 0.0406 0.0111 0.018 0.0195 0.0073 0.0293 0.0034 0.0408 0.0211 0.0451 0.0032 0.0137 0.0745 0.0253 0.0674 0.0104 0.0024 0.0003 0.0476 0.0215 0.0309 0.0883 0.0129 0.0382 0.0349 0.0166 0.03 0.0077 0.0093 0.0219 0.0081 0.0056 0.0351 0.0553

0.5536 0.529 0.5444 0.5747 0.5621 0.5196 0.5175 0.5653 0.5245 0.5242 0.5321 0.5214 0.5451 0.5302 0.5326 0.531 0.5622 0.5131 0.5281 0.5532 0.546 0.5225 0.5472 0.5788 0.5322 0.5333 0.5268 0.4953 0.5301 0.5269 0.5198 0.5107 0.526 0.5353 0.5331 0.5393 0.5294 0.5253 0.5245 0.5212 0.5305 0.5334 0.5203 0.5224 0.5304 0.5262 0.5352 0.5089 0.5439 0.513 0.5208 0.5402 0.5271 0.5178 0.5173 0.5527 0.5278

0.0642 0.0994 0.098 0.0856 0.0803 0.0804 0.0616 0.0573 0.065 0.0992 0.0564 0.1035 0.0965 0.0997 0.0785 0.0945 0.0724 0.081 0.0782 0.0854 0.0672 0.0377 0.1091 0.0473 0.076 0.0967 0.0945 0.0463 0.0757 0.0879 0.0935 0.0971 0.0679 0.0856 0.0748 0.1064 0.0688 0.0763 0.0879 0.0733 0.1004 0.1066 0.0968 0.1038 0.0855 0.0861 0.086 0.0963 0.0938 0.0875 0.0923 0.09 0.0839 0.0683 0.0718 0.0714 0.0553

0.0585 0.0889 0.0974 0.0837 0.087 0.0743 0.0534 0.062 0.0523 0.0804 0.0718 0.092 0.0867 0.0797 0.0785 0.077 0.0626 0.0577 0.0785 0.0821 0.0621 0.0601 0.0921 0.0552 0.0639 0.0992 0.0932 0.1148 0.0919 0.0727 0.0832 0.1126 0.0651 0.0874 0.0812 0.1008 0.085 0.0825 0.0822 0.0307 0.098 0.1053 0.1012 0.1113 0.0891 0.0837 0.106 0.1085 0.0897 0.0923 0.0965 0.0939 0.0908 0.0743 0.0681 0.0655 0.041

0.2267 0.1649 0.0930 0.4289 0.2949 0.2931 0.5199 0.2227 0.1607 0.2329 0.3451 0.4006 0.1276 0.2712 0.3944 0.3746 0.3352 0.3373 0.1055 0.7552 0.2261 0.0985 0.1968 3.6042 4.2182 3.7473 0.4408 4.4896 4.1199 0.2209 0.5759 0.3479 0.4345 0.2274 0.6217 0.9602 1.2715 1.3868 1.3690 1.4562 1.2841 0.8409 1.2294 1.5132 1.0803 1.0261 0.8273 1.3871 1.3421 1.2551 1.2826 1.4445 1.6341 0.4842 0.9166 1.4178 0.8370

0.009 0.0091 0.0153 0.0216 0.0088 0.0226 0.0104 0.0064 0.0782 0.031 0.0265 0.0564 0.0069 0.1162 0.0341 0.0671 0.0208 0.0019 0.0343 0.0631 0.075 0.0138 0.0021 0.0406 0.0111 0.018 0.0195 0.0073 0.0293 0.0034 0.0408 0.0211 0.0451 0.0032 0.0137 0.0745 0.0253 0.0674 0.0104 0.0024 0.0003 0.0476 0.0215 0.0309 0.0883 0.0129 0.0382 0.0349 0.0166 0.03 0.0077 0.0093 0.0219 0.0081 0.0056 0.0351 0.0553

0.0036 0.021 0.0056 0.0247 0.0121 0.0304 0.0325 0.0153 0.0255 0.0258 0.0179 0.0286 0.0049 0.0198 0.0174 0.019 0.0122 0.0369 0.0219 0.0032 0.004 0.0275 0.0028 0.0288 0.0178 0.0167 0.0232 0.0547 0.0199 0.0231 0.0302 0.0393 0.024 0.0147 0.0169 0.0107 0.0206 0.0247 0.0255 0.0288 0.0195 0.0166 0.0297 0.0276 0.0196 0.0238 0.0148 0.0411 0.0061 0.037 0.0292 0.0098 0.0229 0.0322 0.0327 0.0027 0.0222

0.0129 0.0350 0.0392 0.0208 0.0199 0.0113 0.0186 0.0150 0.0184 0.0310 0.0137 0.0401 0.0313 0.0312 0.0120 0.0255 0.0078 0.0165 0.0118 0.0196 0.0084 0.0338 0.0449 0.0271 0.0086 0.0396 0.0337 0.0507 0.0226 0.0181 0.0270 0.0505 0.0053 0.0234 0.0122 0.0477 0.0150 0.0140 0.0217 0.0394 0.0413 0.0508 0.0411 0.0534 0.0246 0.0211 0.0394 0.0466 0.0309 0.0283 0.0346 0.0312 0.0250 0.0046 0.0026 0.0047 0.0325

0.47613 0.40608 0.30496 0.65490 0.54305 0.54139 0.72104 0.47191 0.40087 0.48260 0.58745 0.63293 0.35721 0.52077 0.62801 0.61205 0.57896 0.58078 0.32481 0.86902 0.47550 0.31385 0.44362 1.89847 2.05383 1.93579 0.66393 2.11887 2.02975 0.47000 0.75888 0.58983 0.65917 0.47686 0.78848 0.97990 1.12761 1.17762 1.17004 1.20673 1.13318 0.91701 1.10878 1.23012 1.03937 1.01297 0.90956 1.17775 1.15849 1.12031 1.13252 1.20187 1.27832 0.69584 0.95739 1.19071 0.91488

0.37829 0.38250 0.64310 0.90790 0.36989 0.94994 0.43714 0.26901 3.28695 1.30301 1.11386 2.37064 0.29003 4.88419 1.43331 2.82039 0.87428 0.07986 1.44172 2.65226 3.15245 0.58005 0.08827 1.70652 0.46656 0.75659 0.81964 0.30684 1.23156 0.14291 1.71493 0.88689 1.89567 0.13450 0.57585 3.13143 1.06343 2.83300 0.43714 0.10088 0.01261 2.00075 0.90370 1.29881 3.71148 0.54222 1.60565 1.46694 0.69774 1.26098 0.32365 0.39090 0.92051 0.34046 0.23538 1.47535 2.32440

0.32980 1.92383 0.51302 2.26279 1.10849 2.78497 2.97735 1.40164 2.33607 2.36356 1.63983 2.62007 0.44889 1.81389 1.59403 1.74060 1.11765 3.38044 2.00628 0.29315 0.36644 2.51930 0.25651 2.63839 1.63067 1.52990 2.12537 5.01111 1.82305 2.11621 2.76665 3.60030 2.19866 1.34668 1.54822 0.98024 1.88718 2.26279 2.33607 2.63839 1.78641 1.52074 2.72084 2.52846 1.79557 2.18034 1.35584 3.76520 0.55883 3.38960 2.67503 0.89779 2.09789 2.94987 2.99567 0.24735 2.03376

0.11358 0.18708 0.19799 0.14422 0.14107 0.10630 0.13638 0.12247 0.13565 0.17607 0.11705 0.20025 0.17692 0.17664 0.10954 0.15969 0.08832 0.12845 0.10863 0.14000 0.09165 0.18385 0.21190 0.16462 0.09274 0.19900 0.18358 0.22517 0.15033 0.13454 0.16432 0.22472 0.07280 0.15297 0.11045 0.21840 0.12247 0.11832 0.14731 0.19849 0.20322 0.22539 0.20273 0.23108 0.15684 0.14526 0.19849 0.21587 0.17578 0.16823 0.18601 0.17664 0.15811 0.06782 0.05099 0.06856 0.18028

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Table 3 (continued) X1

X2

X3

X4

X5

X6

X7

G1

G2

G3

G4

C1

C2

C3

C4

0.1133 0.1206 0.122 0.1161 0.1186 0.102 0.1116 0.1111 0.0986 0.1166 0.1187 0.1206 0.1139 0.1103 0.1089 0.1043 0.1056 0.1145 0.1277 0.112 0.1134 0.1237 0.127 0.1154 0.0967 0.1056 0.1116 0.1113 0.1149 0.1091 0.112 0.1074 0.1112 0.1097 0.1085 0.1009 0.1071 0.1157 0.1127 0.1124 0.1017 0.121 0.1107

0.0418 0.0818 0.0733 0.117 0.1044 0.0529 0.0563 0.0352 0.0437 0.0651 0.0535 0.1073 0.0956 0.0332 0.0843 0.0558 0.0651 0.121 0.0945 0.1285 0.0854 0.0752 0.0933 0.1247 0.0685 0.0575 0.1168 0.0839 0.1019 0.017 0.0734 0.0685 0.0807 0.0993 0.0882 0.1252 0.0686 0.0518 0.1203 0.1029 0.0753 0.0299 0.1671

11.8157 9.9824 10.3538 10.1109 11.7824 9.2966 9.4657 11.8824 12.1824 11.9157 12.349 12.1157 11.549 12.0324 12.1966 12.0252 11.9395 11.9538 11.9252 12.0538 12.0109 12.3681 12.2109 12.2395 11.3252 11.6681 11.3395 11.7538 11.9966 11.2252 11.7157 11.7324 11.8324 12.6657 12.249 12.149 9.3157 10.2824 10.1824 11.7824 9.3681 9.3824 12.549

0.0048 0.0287 0.0024 0.0622 0.0495 0.0217 0.0321 0.0268 0.016 0.0259 0.0281 0.0551 0.0173 0.0055 0.0097 0.0372 0.0205 0.0258 0.0383 0.0172 0.0117 0.0021 0.0055 0.0728 0.0149 0.0003 0.0116 0.0102 0.0592 0.0056 0.0228 0.011 0.0403 0.0444 0.0429 0.0778 0.0069 0.0247 0.0431 0.0574 0.0516 0.0255 0.0845

0.5204 0.5361 0.5361 0.5565 0.5499 0.5247 0.5281 0.5166 0.5417 0.5192 0.5433 0.5514 0.5441 0.5189 0.5395 0.5469 0.5438 0.5823 0.5384 0.541 0.5636 0.5483 0.5493 0.5581 0.5417 0.5653 0.5664 0.5581 0.5486 0.5341 0.5401 0.53 0.5239 0.5407 0.5473 0.5281 0.5632 0.5346 0.5545 0.5394 0.5634 0.5554 0.5398

0.0858 0.0868 0.0947 0.0663 0.0956 0.0948 0.085 0.104 0.0926 0.0767 0.089 0.0925 0.1027 0.0687 0.0925 0.0711 0.0873 0.091 0.0949 0.0721 0.1001 0.0661 0.078 0.0733 0.0891 0.0961 0.09 0.0824 0.0803 0.1045 0.0826 0.0781 0.0982 0.0782 0.1172 0.0721 0.103 0.0865 0.1016 0.0671 0.0873 0.0647 0.0656

0.0981 0.0964 0.0882 0.0617 0.0846 0.1156 0.0863 0.1013 0.1137 0.0891 0.0985 0.1016 0.1062 0.0445 0.075 0.0882 0.084 0.0907 0.0944 0.0794 0.0927 0.0669 0.0917 0.0786 0.0899 0.0852 0.0886 0.0891 0.0861 0.1097 0.0799 0.1002 0.1118 0.0798 0.1031 0.0942 0.0958 0.0746 0.114 0.107 0.1069 0.0999 0.0669

0.8169 1.0211 0.6507 0.8969 0.7895 1.7042 1.5354 0.8831 1.1832 0.9181 1.3502 1.1210 0.5575 1.0329 1.1996 1.0267 0.9418 0.9615 0.9304 1.0617 1.0146 1.3703 1.2148 1.2458 0.3324 0.6706 0.3592 0.7585 1.0019 0.2260 0.7195 0.7356 0.8363 1.6687 1.2522 1.1558 1.6857 0.7197 0.8265 0.7892 1.6337 1.6181 1.5580

0.0048 0.0287 0.0024 0.0622 0.0495 0.0217 0.0321 0.0268 0.016 0.0259 0.0281 0.0551 0.0173 0.0055 0.0097 0.0372 0.0205 0.0258 0.0383 0.0172 0.0117 0.0021 0.0055 0.0728 0.0149 0.0003 0.0116 0.0102 0.0592 0.0056 0.0228 0.011 0.0403 0.0444 0.0429 0.0778 0.0069 0.0247 0.0431 0.0574 0.0516 0.0255 0.0845

0.0296 0.0139 0.0139 0.0065 0.0001 0.0253 0.0219 0.0334 0.0083 0.0308 0.0067 0.0014 0.0059 0.0311 0.0105 0.0031 0.0062 0.0323 0.0116 0.009 0.0136 0.0017 0.0007 0.0081 0.0083 0.0153 0.0164 0.0081 0.0014 0.0159 0.0099 0.02 0.0261 0.0093 0.0027 0.0219 0.0132 0.0154 0.0045 0.0106 0.0134 0.0054 0.0102

0.0322 0.0313 0.0307 0.0091 0.0295 0.0519 0.0222 0.0462 0.0492 0.0202 0.0343 0.0388 0.0488 0.0255 0.0230 0.0182 0.0223 0.0295 0.0349 0.0096 0.0377 0.0050 0.0231 0.0092 0.0276 0.0302 0.0273 0.0228 0.0191 0.0526 0.0160 0.0313 0.0504 0.0128 0.0576 0.0243 0.0419 0.0171 0.0542 0.0371 0.0408 0.0304 0.0054

0.90383 1.01049 0.80666 0.94705 0.88854 1.30545 1.23911 0.93973 1.08775 0.95818 1.16198 1.05877 0.74666 1.01632 1.09526 1.01326 0.97046 0.98056 0.96457 1.03039 1.00727 1.17060 1.10218 1.11615 0.57654 0.81890 0.59933 0.87092 1.00095 0.47539 0.84823 0.85767 0.91449 1.29178 1.11902 1.07508 1.29835 0.84835 0.90912 0.88837 1.27816 1.27205 1.24820

0.20176 1.20634 0.10088 2.61443 2.08062 0.91211 1.34925 1.12647 0.67252 1.08865 1.18112 2.31600 0.72716 0.23118 0.40772 1.56361 0.86167 1.08444 1.60985 0.72296 0.49178 0.08827 0.23118 3.05998 0.62629 0.01261 0.48758 0.42873 2.48833 0.23538 0.95834 0.46236 1.69391 1.86625 1.80320 3.27014 0.29003 1.03821 1.81161 2.41267 2.16888 1.07183 3.55176

2.71168 1.27339 1.27339 0.59547 0.00916 2.31775 2.00628 3.05980 0.76037 2.82161 0.61379 0.12826 0.54050 2.84909 0.96191 0.28399 0.56799 2.95903 1.06268 0.82450 1.24591 0.15574 0.06413 0.74205 0.76037 1.40164 1.50242 0.74205 0.12826 1.45661 0.90695 1.83222 2.39104 0.85198 0.24735 2.00628 1.20926 1.41081 0.41225 0.97107 1.22758 0.49470 0.93443

0.17944 0.17692 0.17521 0.09539 0.17176 0.22782 0.14900 0.21494 0.22181 0.14213 0.18520 0.19698 0.22091 0.15969 0.15166 0.13491 0.14933 0.17176 0.18682 0.09798 0.19416 0.07071 0.15199 0.09592 0.16613 0.17378 0.16523 0.15100 0.13820 0.22935 0.12649 0.17692 0.22450 0.11314 0.24000 0.15588 0.20469 0.13077 0.23281 0.19261 0.20199 0.17436 0.07348

powerful and popular meta-heuristics to solve many optimization problems (refer to Abbasi et al. [28] for a recent application). Detail for implementing SA algorithm in c and k is presented in Table 2.

4 A manufacturing example In this section, the proposed methodology will be demonstrated using real data from Wang [3]. The data set is from a manufacturing process with multivari-

ate quality characteristics. It contains a sample of 100 parts that were tested on seven quality characteristics of interest to the manufacturer. These characteristics are related to the connector associated with desktop personal computers. The full data set is given in Table 3. The specification limits for these seven quality characteristics can be two-sided or one-sided, and they are 0.10 ± 0.04, 0 + 0.50, 11 ± 5, 0 + 0.2, 0.55 ± 0.06, 0.07 ± 0.05, and 0.07 ± 0.05 mm for X1 to X7 , respectively. Based on the precise nature of the quality characteristics and manufacturing processes implemented,

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Table 4 Burr distribution parameter (c, k) estimation using SA Generalized covariance distance variables

MRDs

c

k

C1 C2 C3 C4

5.0251 0.2000 0.0600 0.0707

3.4041 0.6040 1.2047 1.1003

41.1290 39.7236 132.4337 202.8647

it was found that the clusters of correlated variables are {X1 , X2 , X3 }, {X4 }, {X5 }, and {X6 , X7 }. Therefore, applying the CD approach, the following four new univariate variables based on the four clusters are: C1 = (X1 − T1 ) 1 −1 (X1 − T1 ) C2 =

X4 s4

X5 − 0.55 s5 C4 = (X2 − T2 ) 2 −1 (X2 − T2 ), C3 =

where X1 − T1 = (X1 − 0.1, X2 , X3 − 11), X2 − T2 = (X6 −0.07, X2 , X7 −0.07); 1 −1 , 2 −1 are the variance– covariance matrices of {X1 , X2 , X3 } and {X6 , X7 }, respectively; s4 and s5 are the standard deviations of X4 and X5 , respectively. The values of C1 , C2 , C3 , and C4 are tabulated in the last four columns of Table 3. As a check, we also determine from the correlation matrix of these four CD variables that they do not have any significant correlation with each other. Burr distribution is fitted to each of these variables and the results are displayed in Table 4. Using Eq. 6, the MRDs for the four CD variables are calculated as MRD1 = 5.025, MRD2 = 0.2, MRD3 = 0.06 and MRD4 = 0.0707, respectively. Finally, using Eq. 7 and the fitted Burr distributions for the four CD variables, we obtain their PNC values, as well as those obtained using the GD method. These are displayed in Table 5. The results in Table 5 indicate that the proposed CD method yields a PNCTotal value (given by Eq. 8) that is closer to the true proportion of nonconformTable 5 PNC for CD data

ing items falling outside their respective specifications than the GD method (0.238 vs 63.72, respectively). The results also show that C3 has a significantly larger PNC value than the other CD variables. Consequently, quality characteristic X5 would be the first candidate for improvement in contrast with the other four quality characteristics.

5 Conclusion This paper introduces a novel approach, the CD approach, to evaluate process capability for nonnormal multivariate quality characteristics. This approach is based on the idea of reducing the dimension of multivariate data by transforming correlated variables into univariate ones through a metric function. Unlike the GD approach [3], our approach takes into account the scaling effect of the variance–covariance matrix and produces a CD that is based on the Mahanalobis distance. As is demonstrated in this paper, another justification for using our approach is the fact that it does not assume that the variables are mutually independent, which is implicitly assumed in the GD method. In contrast to the approach adopted in [3], where different distributions are used to fit different sets of GD data, a single distribution, the Burr XII distribution [15], is fitted to the CD data using SA. We provide an example of an application using real data with several nonnormal quality characteristics from the manufacturing industry. To evaluate the performance of our approach, we compare the PNC values obtained using our approach and the GD approach. The results in Table 5 indicate that the CD approach gives a total PNC value that is much closer to the true value. Quantitative measure of process performance for multivariate quality characteristics is of great interest to quality control practitioners and has a huge potential of expanding its application to other multivariate industrial quality research areas. Hence, there is much scope in extending the present work. For example, instead of using univariate Burr, a multivariate Burr distribution

Generalized covariance distance variables

MRD’s

Actual PNC value

Probability of the product conforming GD method CD method

C1 C2 C3 C4

5.0251 0.2000 0.0600 0.0707

0.0100

1.0000 0.6123 0.9999 0.5932 0.36282

1.000000 0.999997 0.987643 0.999978 0.987618

PNC using GD 0.6372

PNC using CD 0.012382

Total (PNC)

0.0100

Int J Adv Manuf Technol (2009) 44:757–765

[23] could be directly employed to fit multivariate CD data. However, it is anticipated that the numerical work involved in estimating the parameters of the distribution could prove to be extremely laborious. This leads to the consideration of other search algorithms, such as genetic algorithms, neural networks, etc., for their potential in estimating parameters of multivariate distributions using the MLE procedure. Finally, since our approach does produce significant improvement over an existing method using our chosen data set, we recommend that the proposed method be applied to other nonnormal multivariate PCI studies for further comparisons.

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