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Journal of Industrial Engineering International Islamic Azad University, South Tehran Branch
June 2009, Vol. 5, No. 9, 27-36
S. Raissi*
Dep. of Industrial Engineering, Islamic Azad University, South Tehran Branch, Tehran, Iran
Abstract
Multivariate Process Capability Indices (MPCI) show how well a manufacturing process can meet specifica-
tion limits when quality characteristics enclose a relative correlation. Process capability is an important and
commonly used metric for assessing and improving the quality of a production process. When quality charac-
teristics of a product are correlated then an attractive comes close to MPCI methods, which are not usually an
easy task to carry out. In this investigation after a full reviewing of the MPCI, a simple method to estimate
product capability indices based on ridge regression models in the presence of priority for quality characteris-
tics is presented. The technique is demonstrated for evaluation of product capability through the use of an ex-
ample which shows performance of the proposed method.
Keywords: Multivariate process capability indices; Product capability; Ridge regression method; Quality
characteristic priority; Multivariate statistical process control
1. Introduction which are effective tools for quality assurance.
These indices are defined as:
Process produces products according to a certain
quality characteristic, for example weight, length, USL−LSL
C = i
hardness, viscosity, etc. The degree a process is P 6σ
producing data within tolerance limits, can be
measured using Process Capability Indices (PCIs). USL −µ µ−LSL
C =Min ,
PCIs are generally used in industry to measure cha- PK 3σ 3σ
racteristics that are independence of each others. A
standard practice in Statistical Process Control
(SPC) programs is to ensure that the process is un- C = USL−LSL i =1, 2, ..., p (1)
PM 2 2
der statistical control prior to conducting a process 6 σ +(µ−T)
capability analysis. Unfortunately, it is a fairly
common practice to perform capability analysis
using a sample of historical process data without C =Min USL−µ , µ−LSL
PMK
any consideration of whether or not the process is in 3 σ2+(µ−T)2 3 σ2+(µ−T)2
statistical control. As Montgomery [25] stated, if
the process is not in control then its parameters are
unstable and the value of these parameters in the where USL and LSL are the upper and the lower
future is uncertain. Hence, the predictive aspects of specification limits, respectively, µ is the process
the process capability indices regarding the number mean, σ is the process standard deviation and T is
of nonconforming items produced are lost. target amount anywhere within the specification
The most frequently used univariate PCIs includ- interval.
ing C , C , C , and C have been proposed in PCIs have recently received a considerable
P PK PM PMK
the manufacturing industry to provide numerical amount of attention in the literature of SPC. Nu-
measures on process capability and performance, merous authors including Kane [20], Marcucci and
* Corresponding author. E-mail: raissi@azad.ac.ir
Beazley [24],Chan and Cheng [3], Choi and Owen 2. Some techniques in MPCIs evaluations
[8], Spiring [37],Koons [21],Wheeler and Cham-
ber [43], Pearn [32], Bissel [2], Wright [45], Pearn Various authors have proposed alternative ap-
and Chen [29], Stoumbos [38], Pearn and Shu [33], proaches to assess process capability in multivariate
Chen and Chen [6], Perakis and Xekalaki [34] and environment. Taam et al. [39] recommend using a
Chou et al. [10] have discussed theories and appli- multivariate capability index that is defined as a
cations of univariate PCIs, when process normally ratio of two volumes:
distributed.
Extensive studies have also been conducted to de- Vol.(R )
MC = 1 (2)
termine the effects of non-normality on the various PM Vol.(R )
PCIs. Gunter [13,14,15,16] in a series of articles 2
pointed out many flaws of the indices particularly
where R is a modified tolerance region and R is a
C when applied to non-normal data. Interested 1 2
PK
readers are referred to Munechika [26], Clemets scaled 99.73 percent process region. In particular, if
[11], Wright [44], Somerville and Montgomery the process data are multivariate normal, then R2 is
[36], Bai and Choi [1], Chen and Ding [7] and an elliptical region. A process region and modified
Chou et al. [9] for more discussions on the univa- tolerance region is shown in Figure 1 at appendix.
riate process capability indices when normality as- The modified tolerance region is defined as the
sumption is violated. largest ellipsoid that is centered at the target com-
During the past decade, there has been a growing pletely located inside the original tolerance region.
The estimate for MC is given by:
concern about the normality and independence as- PM
sumptions required to compute univariate capability
indices. In practice, it is common to use two or ˆ
ˆ CP
more related quality characteristics of a product to MCPM = ˆ (3)
evaluate the performance of a manufacturing D
process. Since the early work of Hotelling [17], it
has become evident that such problems, due to the where,
correlation that exists among quality characteristics,
need to be addressed in multivariate context to en- ˆ Vol.(tolerance region)
CP =
sure proper evaluation. Similar to the univariate Vol.(estimated 99.73% of process region)
case, the Multivariate Process Capability Indices Vol.(tolerance region)
(MPCIs) have also captured attentions of many re- = ν −1 (4)
1 ν
searchers including Hubele et al. [19], Chan et al. S 2(πK) 2 Γ( +1)
[4], Taam et al. [39], Nickerson [27], Chen [14],
2
Karl et al. [23], Niverthi and Dey [28], Shahriari
et al. [35], Wang et al. [42], Wang and Du [40], and,
Frey et al. [12], Wang and Hubele [41], Pearn et al.
[31] and Pearn and Chien-Wei [30]. For a quick 1
ˆ n ′ −1 2
survey and interpretations on univariate and multi- ( ) ( )
D= 1+ X−0 S X−0 (5)
n−1
variate process capability indices see Kotz and
Johnson [22].
An existing serious problem in multivariate quali- where K is the 99.73 percent quantile of a χ 2 dis-
ty control is in complexity of methodology for as- S
sessing MPCIs. The purpose of this paper is to pro- tribution and denotes the determinant of sample
vide a relatively simple method for estimating the variance-covariance matrix.
most well-known relevant PCIs (priory showed Shahriari et al. [35] proposed a vector consisting
through set of Eq. (1) in multivariate environment of three components. The first two components use
which define by MC , MC , MC , and MC the assumption that the process data is from a mul-
P PK PM PMK
respectively. A brief discussion to some MPCIs is tivariate normal distribution with elliptical contours
presented in Section 2 and the proposed methodol- defining probability regions and the third compo-
ogy to estimate MPCIs are offered in Section 3. The nent is based on a geometric understanding of
fourth part discusses a numerical example. Conclu- process relative to the engineering specifications.
sions are provided in the final section. The first component is defined as:
1 F denotes the value of F distribution with
vol. of engineering tolerance region ν (ν ,n−ν )
MCPM =
ν and n-ν degrees of freedom. It should be pointed
vol. of modified process region out that large values of PV indicate the closeness of
1 the center of the process to the pre-specified target
ν ν
∏(USL −LSL ) value.
i i The third component of the vector that is referred
=
i =1 i =1,2,...,ν (6)
ν to as location index (LI) compares the location of
∏(UPLi −LPLi) the modified process region to the tolerance region.
i =1 This index has a value of one if the entire modified
process region is contained within the tolerance re-
Figure 2 at appendix illustrates their method for a gion indicating that all the manufactured products
product in which the engineering specifications de- conform to the specification limits, otherwise it will
fine a rectangular tolerance region but bi-variate take a value of zero.
normal process variables define an elliptical proba- Chen [5] proposed a multivariate process capabil-
bility contour referred to as process region. Their ity index based on a multiple bilateral tolerance
proposed method forms a modified process region zone defined by:
by drawing the smallest rectangle around the ellip-
tical process region. The edges of the modified { V }
V = X∈R : X − ≤r,i=1,2,....,v (11)
process region are defined as the lower and upper i 0 i
process limits (LPLi and UPLi, respectively, where
i=1,2,…ν) and are given by: where µ is the specification limit and r is a con-
0i i
stant. The multivariate process capability index is
χ2 −1 given by MC = 1 , where r is defined such that:
UPL =µ + ν,α i P r
i i −1 (7)
X −
i 0
i
2 −1 P Max r ,i =1,2,...,v ≤r =1−α (12)
χ i
ν,α i
LPL =µ − ; i = 1,2,...,ν (8)
i i −1
Let F be the cumulative distribution function of,
2 α
where χν,α denotes the upper 100( ) % of X −
i 0
aχ2distribution with ν degrees of freedom and h(X− )=Max i ≤r,i=1,2,....,v (13)
0 r i
i
−1
i is the determinant of the variance-covariance
matrix with its ith row and column deleted. −1 MC
Then r = F (1−α). If the value of P is
The second component of the proposed vector is greater than or equal to 1, the process is capable
based on the assumption that the center of the speci- with a certain confidence level.
fication limits denotes the process mean. This com- Frey [12] proposed a matrix of dimensionless pa-
ponent is defined as the significance level of a Ho- rameters (C), which represents a linear mapping of
telling’s T 2 statistic, which is computed as follows:
noise variables (nj ; j=1,2,…,n) to quality characte-
ristics (qi ; i=1,2,…,m), for evaluating process ca-
2 ν(n−1) pability in multivariate environment. The elements
PV = PT ≥ F (9)
n−ν (ν ,n−ν ) of this matrix, C , are defined as:
ij
where, ∂q
6Sj
∂n
′ −1 n=t ij
T2 = n(X − ) S (X − ) (10) C = (14)
0 0 ij USL −LSL
i i
Figure 1. Typical modified tolerance region (R1) versus estimated 99.73 % process region (R2) in a bivariate case.
Figure 2. Rectangular Tolerance Region versus Modified Process Region.
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