In manufacturing industry there is growing interest in measures of process capability under multivariate setting. While there are many statistical packages to assess univariate capability, a current problem with the multivariate measures of capability is the shortage of user friendly software. In this paper a Visual Basic program has been developed to realize an Excel spreadsheet that may be used to compute two multivariate measures of capability. Our aim is to provide a useful tool for practitioners dealing with multivariate capability assessment problems. The features of the program include easy data entry and clear report formatMultivariate Process capability indices, statistical quality control, Visual Basic, Excel Indici di capacità multivariati, Controllo statistico della qualità

Michele Scagliarini

Raffaele Vermiglio

English

Research Papers in Economics

 
 
*TIM-Telecom Italia Mobile 
 
 
Michele Scagliarini, Raffaele Vermiglio* 
Computing Multivariate Process Capability 
Indices With Microsoft Excel  
 
Dipartimento di Scienze Statistiche “Paolo Fortunati” 
 
Quaderni di Dipartimento 
 
Serie Ricerche 2007, n. 1   1
Abstract 
In manufacturing industry there is growing interest in measures of 
process capability under multivariate setting. While there are many 
statistical packages to assess univariate capability, a current problem with 
the multivariate measures of capability is the shortage of user friendly 
software. In this paper a Visual Basic program has been developed to 
realize an Excel spreadsheet that may be used to compute two 
multivariate measures of capability. Our aim is to provide a useful tool 
for practitioners dealing with multivariate capability assessment 
problems. The features of the program include easy data entry and clear 
report format. 
 
Keywords: Multivariate Process Capability Indices, Statistical Quality 
Control, Visual Basic, Excel 
 
   2
1. Introduction 
 
Process capability indices have been widely used in the manufacturing 
industry providing numerical measures on process performances. Juran et 
al. (1974) first introduced the idea of capability indices (the original 
name was "capability ratios"). The first indices were univariate, measured 
the process capability with regard to a single quality measure and focused 
on the percentage of non-conforming Kotz and Lovelace (1998). 
In recent years multivariate capability indices were developed as a 
natural extension to the univariate concept. Multivariate capability 
indices appeared in the literature during the early 1990s. Most of them 
assumed multivariate normal data, a stable process, and were 
generalizations of their univariate counterparts. 
Wang  et al. (2000) compared three multivariate indices: the 
multivariate capability vector, Shahriari et al. (1995); the multivariate 
capability index MCpm, Taam et al. (1993); the multivariate capability 
index for process potential MCp, Chen (1994). While there are many 
statistical packages for running univariate capability analysis, a current 
problem with the multivariate measures of capability is the shortage of 
user friendly software. Recently, an interesting contribution is the work 
by Phnadnis et al. (2005). The authors proposed a Visual Basic program 
to perform bivariate capability analysis using the MCpm index with Excel. 
In this work we have developed a set of Visual Basic macros that may 
be used to compute multivariate measures of capability using Excel. 
Since Excel is often used by engineers, or generally by non statisticians, 
our purpose is to provide a user-friendly tool to help practitioners in 
performing multivariate capability analysis. We examined two different 
multivariate capability measures: the multivariate capability vector, 
Shahriari et al. (1995) and the multivariate capability index MCpm, Taam 
et al. (1993). In Section 2 we describe the two multivariate capability 
measures. The software description is reported in Section 3. Section 4 
contains the concluding remarks. 
 
 
2. Two multivariate capability indices 
 
We assume that a process can be described by a v-dimensional vector 
of measurements x and we further assume that the joint probability   3
distribution of the v quality characteristics is the multivariate normal 
distribution  
 
() , N xµ Σ ∼                         (1) 
 
2.1. The multivariate capability vector 
 
The multivariate capability vector was proposed by Shahriari et al. 
(1995) based on the original work of Hubele et al. (1991). The vector 
consists of three components and appears as 
 
( ) , , pM CP V L I  
 
The first component of the vector, CpM, is a ratio of areas or volumes, 
analogous to the ratio of lengths of the univariate Cp index. The 
numerator is the area (two-dimensional case) or the volume (three or 
more dimensions) defined by the engineering tolerance region, while the 
denominator is the area or volume of a "modified process region", 
defined as the smallest region similar in shape to the engineering 
tolerance region, circumscribed about a specified probability contour: 
 
1
Volume of engineering tolerance region
Volume of modified process region
v
pM C
⎡⎤
= ⎢⎥
⎣⎦
              (2) 
 
where v is the number of characteristics of the process. 
The volume of the engineering tolerance region is 
 
()
1
ii
i
USL LSL
ν
=
− ∏                       (3) 
 
where USLi and LSLi are the upper and lower limits respectively, relative 
to the characteristic i (i=1,2,...,v). 
To compute the volume of the modified process region it is worth 
reminding that under the hypothesis of multivariate normality the statistic 
   4
() () ( ) g ′ −− = x µ Σ x µ x                    (4) 
 
follows a 
2 χ  distribution with v degree of freedom. Therefore, the 
borders of the process region UPLi, the upper process limit, and LPLi, the 
lower process limit (i=1,2,...,v) are determined by solving the systems of 
equations of first derivative, with respect to each xi of the quadratic form 
 
() ()
2
(,) ν α χ
′
−− = xµΣ xµ                     (5) 
 
where 
2
(, ) v α χ  is the upper 100(α) of a 
2 χ  distribution with v degrees of 
freedom associated with the probability contour. Usually, in analogy with 
the "6σ" in the denominator of the univariate indices, α=0.0027. 
The solutions (two for each dimension i) of equation (5) are Wang et 
al. (2000): 
 
( )
()
21
(,)
1
det
det
i
ii UPL
να χ
µ
−
− =+
Σ
Σ
                 (6) 
 
( )
()
21
(,)
1
det
det
i
ii LPL
να χ
µ
−
− =−
Σ
Σ
                  (7) 
 
where i=1,2,...,v and  ( )
1 det i
− Σ  is the determinant of a matrix obtained 
from Σ
-1 by deleting the i
th row and column. Thus, the volume of the 
modified process region is 
 
()
1
ii
i
UPL LPL
ν
=
− ∏                       (8) 
 
In practice µ and Σ are unknown and their sample estimates x and S 
can be used: 
   5
∑
=
=
n
i
i n 1
1
x x                         (9) 
 
() ()
'
1
1
1
n
ii
i n =
=− −
− ∑ Sx x x x                 (10) 
 
Values of CpM higher than one indicate that the modified process 
region is smaller than the engineering tolerance region, therefore we have 
high probability that the produced items will be classified as conform. 
The second component of the vector is defined as the significance 
level of a Hotelling T
2 statistic computed under the assumption that the 
center of the engineering specifications is considered to be the true 
underlying mean of the process: 
 
()
()
2
,
1
Pr (ν n ν)
vn -
PV T F
n-ν
−
⎛⎞
=> ⎜⎟ ⎜⎟
⎝⎠
                  (11) 
 
where 
 
() () µ x S µ x − ′ − =
−1 2   n T                    (12) 
 
and F(v,n-v) is the F distribution with v and n-v degrees of freedom. 
Values of PV close to zero indicate that the center of the process is far 
from the engineering target value. 
The third component of the vector summarizes a comparison of the 
location of the modified process region and the tolerance region. It 
indicates whether any part of the modified process region falls outside the 
engineering specifications. It has a value of 1 if the entire modified 
process region is contained within the tolerance region and, otherwise, a 
value of 0: 
 
1  if modified process region is contained within the tolerance region
0  otherwise
LI
⎧
= ⎨
⎩
 
(13) 
   6
2.2. The multivariate capability index MCpm 
 
The index MCpm was proposed by Taam et al. (1993) and is defined as 
a ratio of two volumes. The numerator is the volume of the modified 
tolerance region R1 and the denominator is the volume of the scaled 99.73 
percent process region R2. Under the multinormality hypothesis we have 
an elliptical process region, while the modified tolerance region is the 
largest ellipsoid that is centerd at the target completely within the original 
tolerance region. 
In the general case of v characteristics R1 is an hyperellipsoid and the 
volume is given by Kendall (1961) 
 
()
/2
1
1
2
2
Vol.( )
v
v
i i
v
a
R
v
π = =
Γ
∏                 (14) 
 
where the  i a  ( i=1,2,...v) are the lengths of the semi-axes. Then the 
multivariate capability index is written as 
 
()()
1
1
Vol.( )
Vol. ( )
pm
T
R
MC
Kv
−
=
′ ⎛⎞ −− ≤ ⎜⎟
⎝⎠
xµΣ xµ
            (15) 
 
where x is the vector (v×1) of measurements from a multivariate normal 
distribution with mean vector µ  and covariance matrix Σ, 
() () T E
′ ⎡⎤ =−− ⎢⎥ ⎣⎦
Σ xTxT  is the mean square error matrix from the 
process, T is a vector of target values, and K(v) is a 99.73
th percentile of a 
2 χ  with v degrees of freedom. 
The denominator of MCpm can be also expressed as a product of two 
terms:    7
() ()
() ()
() ( ) ( )
1 /2 1/2
2
1/2
1
1/2
1
3
Vol.( ) ( ) /2 1
1
Vol. 1
v
RK v v
R
π
−
−
−
⎡⎤ =Γ + × ⎣⎦
⎡⎤ ′ ×+ − − = ⎢⎥ ⎣⎦
⎡⎤ ′ =× + − − ⎢⎥ ⎣⎦
Σ
µTΣ µT
µTΣ µT
            (16) 
 
where R3 is the region in which 99.73% of the process values fall within. 
Therefore MCpm can be rewritten as: 
 
() ()
1
1/2
1 3
Vol.( ) 1
Vol.( )
1
p
pm
C R
MC
R D −
==
⎡⎤ ′ +− − ⎢⎥ ⎣⎦
µTΣ µT
        (17) 
 
The MCpm index is a function of two components: Cp which represents 
the process variability relative to the modified tolerance region; D which 
detects the process deviation from the target. Given a random sample of n 
measurements, x1, x2, ...,xn each of dimension v, the estimator for MCpm is 
given by 
 
() ( )
() ()
1
1 /2 1/2
1/2
1
Vol.( ) ˆ
/2 1
1
1
1
ˆ
ˆ
pm v
p
R
MC
Kv
n
n
C
D
π
−
−
=×
⎡⎤ Γ+ ⎣⎦
=
⎡⎤ ′ +− − ⎢⎥ − ⎣⎦
=
S
xTS xT
            (18) 
 
When the process the process mean vector equals the target vector, 
and the index has the value 1, then 99.73% of the process values lie 
within the modified tolerance region. The numerator Cp is analogous to 
the univariate Cp, that is, a value greater than 1 implies that the process 
has smaller variation than allowed by the specification limits with a 
certain confidence level; a value less than 1 implies more variation. Also,   8
ˆ 01 1 D <<  measures the closeness between the process mean and the 
target; a larger  ˆ 1 D indicates that the mean is close to target. 
 
 
3. Software description  
 
The macros are stored in the MultiCap.xls file. The user can directly 
open this file to perform the capability analysis. To illustrate the software 
we consider a simulated example. We generate a sample of 100 
observation from a multivariate normal process of dimension v=3 with 
mean vector and covariance matrix given by  
 
[ ] 40 60 15 ′ = µ  
and  
1.100 0.483 0.308
0.483 0.4 0.185
0.308 0.185 0.600
⎡ ⎤
⎢ ⎥ = ⎢ ⎥
⎢ ⎥ ⎣ ⎦
Σ  
 
respectively. 
The target values coincide with the means and the specification limits 
are reported in Table (1). 
 
Characteristic  LSLi  LSLi 
1 33  47 
2 52  68 
3 12  18 
Table 1. Specification limits 
 
The user interface is the worksheet "INPUT" (Figure 1) where the 
main parameters of the analysis can be specified. 
   9
 
Figure 1. The worksheet INPUT 
 
Clicking the button "PARAMETERS" the form "ANALYSIS 
PARAMETERS" will appear, as shown in Figure 2, thus can be 
specified: the number v of quality characteristics i.e. the dimensions of 
the process (for this version the maximum number is 5); the α value to 
define the size of the tolerance region (usually α =0.0027); the sample 
size. 
 
Figure 2. User form "ANALYSIS PARAMETERS"   10
Therefore, these values will be displayed in the Table "GENERAL 
PARAMETERS" and the Tables "SPECIFICATIONS LIMITS AND 
TARGET" and "SAMPLE DATA" will be automatically modified on the 
basis of the values of v and n. 
Clicking the button "LIMITS-TARGET" the corresponding form 
(Figure 3) will appear, thus for each quality characteristic can be entered: 
the specification limits; the target value. 
 
 
Figure 3. User form "LIMITS AND TARGET" 
 
Clicking the button "SAMPLE DATA" the form "INPUT SAMPLE 
DATA" (Figure 4) will appear, in this way the sampling observations can 
be inserted. If the sampling observations are already available, then the 
data can be directly pasted in the Table "SAMPLE DATA". 
   11
 
Figure 4. User form "INPUT SAMPLE DATA" 
 
Using the simulated data the worksheet appears as shown in Figure 5. 
 
 
Figure 5. A portion of the worksheet "INPUT" with the example's data 
 
The calculations can be performed clicking the button "COMPUTE". 
The procedure is splitted in two separate steps. The first step consists in 
the estimation of the mean vector and covariance matrix. The results of 
the computations are displayed in the worksheet "MEANS AND 
COVARIANCES" as shown in Figure 6.   12
 
Figure 6. Worksheet MEANS AND COVARIANCES  
 
In the second step the user can choose between the two capability 
measures (Figure 7). The results will be displayed, together with brief 
report, in a suitable form (Figures 8 and 9). 
 
 
Figure 7. Index choice 
   13
 
Figure 8. Results and report 
 
 
Figure 9. Results and report 
   14
In the report box we introduced some comments on the results. The 
messages reported here are only examples and can be modified if 
necessary. 
In the worksheet "VECTOR" (Figure 10 and Figure 11) are reported 
the details of the calculations for the multivariate capability vector. 
 
 
Figure 10. The upper portion of the worksheet "VECTOR"  
 
   15
 
Figure 11. The lower portion of the worksheet "VECTOR"  
 
Similarly, in the worksheet "INDEX MCpm" (Figure 12 and 13) are 
reported the details of the calculations for the MCpm index. 
   16
 
Figure 12. The upper portion of the worksheet "INDEX MCpm" 
   17
 
Figure 13. The lower portion of the worksheet "INDEX MCpm" 
 
Results concerning both capability measures are summarized in the 
worksheet "OUTPUT" as shown in Figure 14. 
   18
 
Figure 14. Worksheet "OUTPUT" 
 
3.1. Some details 
 
The Visual Basic code for the calculation management is stored in the 
Macro CALCOLO1. The inversion of the matrices and the computation 
of the determinants are performed using the Excel functions MInverse 
and MDeterm respectively. Moreover, we used the function INV.CHI() to 
calculate the quantiles of the chi-square distributions in the computations 
of the process regions and the function DISTRIB.F() to compute the PV 
component of the multivariate vector. As pointed out by Knusel (1998) 
and McCullog and Wilson (2002), it is important to take into account the 
accuracy problems when using these Excel functions. Keeping this 
caution in mind, in agreement with the purposes of the present work, we 
retain the degree of precision acceptable. 
   19
4. Concluding remarks 
 
In this article we have developed an Excel spreadsheet which can be 
used to calculate two multivariate capability measures: the multivariate 
capability vector Shahriari et al. (1995) and the multivariate capability 
index  MCpm Taam et al. (1993). The proposed software requires no 
installation, since the user can directly open the .xls file. The spreadsheet 
interface is easy to use, moreover a set of instructions can be visualized 
clicking the button "INSTRUCTIONS". Since a problem with the 
multivariate measures of capability is the shortage of and user-friendly 
programs we hope that this tool can help practitioners in performing 
multivariate capability analyses. The software has been validated using 
several data set. However, the user should understand that there may be 
undetected bugs and problems and will be grateful for any feedback with 
relevant comments and suggestions for improvements. 
 
 
Availability 
 
The file MultiCap.xls file can be obtained freely from the author writing 
to Michele Scagliarini (scagliarini@stat.unibo.it). 
 
 
ACKNOWLEDGMENTS 
 
This work has been partly funded by a 2006 MIUR (Italian Ministry 
for University and Research) grant. 
 
 
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Computing Multivariate Process Capability Indices With Microsoft Excel

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