Designing effectively a set of X quality control charts
Francisco Aparisi
Marco A. de Luna
Eugenio Epprecht
Universidad Politécnica de Valencia
ITESM
Departamento de Engenharia
Departamento de Estadística e I. O.
Campus Guadalajara
Industrial
Aplicadas y Calidad
Departamento de Ingeniería Mecánica Pontificia Universidade Católica de Río
e Industrial.
Camino de Vera s/n. 46022 Valencia
de Janeiro
Spain
+34-963877490
Mexico
+52-3336693000
faparisi@eio.upv.es
mdeluna@itesm.mx
ABSTRACT
Nowadays it is a common practice in industry to control
simultaneously p variables of a unit or a productive process. There
are two possibilities: first, to employ a set of univariate charts
(like the X chart) or, second, to use a multivariate chart (like the
T2 control chart). The design and performance of the T2 control
chart has been widely studied in the bibliography. However, the
effective design of a set of
depth.
X charts has not been researched in
In this paper we show how it is possible to find the best
parameters of a set of X charts. The parameters are found
solving an optimization problem employing Genetic Algorithms
(GAs). For example, in the case of two variables (p = 2) the
variables to find are: upper and lower control limits of the two
charts (UCLX1, LCLX1, UCLX2, and LCLX2) and the two sample
sizes (n1 and n2). The objectives are: 1.- to detect as soon as
possible (minimum ARL) a shift of size A = (A1, A2). 2.- not to
detect a shift of size B = (B1, B2). 3.- The in-control ARL has to
be the specified value ARL0. After the optimization is carried out
we make a comparison against the performance of the T2 control
chart.
Categories and Subject Descriptors
J.2 [Physical Sciences and Engineering]: Engineering.
G.3 [Probability and Statistics]: Multivariate Statistics.
General Terms
Algorithms
Keywords
Statistical Process Control, Genetic Algorithms, multivariate
quality control.
1. INTRODUCTION
In industrial situations quality control charts are often used to
observe whether a process is in control. When there is one quality
characteristic Shewhart control charts ( X charts) are usually
applied to monitor process shifts. But there are many situations in
which the simultaneous control of two or more related quality
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characteristics is necessary. For example, the quality control of
some dimensions of a part. In these cases, it is still possible the
design of a statistical process control for all quality characteristics
employing Shewhart control charts but there are another feasible
multivariate schemes.
As example of how to control several variables with X charts, let
us consider a part which has two quality characteristics,
represented by the variables X1 and X2 normally distributed. First
we consider the case where both characteristics are independent.
If the two variables are monitored separately a univariate X chart
with, for example, 3-sigma limits can be constructed for each
characteristic. Each chart has a probability of exceeding 3-sigma
control limits  = 0.0027 (the type I error). The probability that
both variables fall inside control limit when the process is in
control is (1-0.0027)(1-0.0027) = 0.994607. So the overall type I
error for this case is ' = 1-0.994607 = 0.0054. If there are p
statistically independent quality characteristics and charts with α
type I error are used, the real type I error α’ is
 = 1- (1- ) p
(1)
Therefore, if we want to fix the type I error for the process as a
whole, having p independent variables, equation (1) can be used
to calculate the suitable type error for each chart, and then, to
obtain the correct control limits.
If the variables are not independent, which is the most common
case, a more complex process are to be employed to obtain the
control limits to have an ' value. For simplicity, let us continue
with the bivariate case supposing that the variables follow a
bivariate normal distribution with mean vector  0 '  (  0,1 ,  0,2 )
and covariance matrix
2
  0,1
 0  

 0,1 0,2
 0,1 0,2 

2
 0,2

(2)
and X charts are used. Then an a value has to be calculate such
that

 '  1  P  a 


X1   0,1
X   0,2
a  a  2
 a 
 0,1 / n
 0, 2 / n

small or moderate process shifts. (Crosier [4], Pignatello and
Runger [5] and Lowry, Woodall, Champ and Rigdon [6]).
 1  P a  Z1  a   a  Z2  a  
 1 
a

a
a a
f ( z1, z2 )dz1dz2
(3)
where n is the sample size and f ( z1 , z2 ) is the joint density
function of a bivariate standard normal distribution with
correlation,
.
So
the
control
limits
are:
 0,1
 0, 2
 0,1  a
for X1 and  0,2  a
for X 2
n
n
.
Another solution to the problem of controlling several variables
was provided by Hotelling (1947). Consider p correlated
characteristics are being measured simultaneously and these
characteristics follow a multivariate normal distribution with
 '  (  0,1 ,  0,2 , ... ,  0, p ) and covariance matrix Σ
mean vector 0
0
when the process is in control. When a ith sample of size n is
taken we have n values of each characteristic and it is possible to
calculate the X i vector, which represents the ith sample average
vector for the p characteristics.
The charting statistic
Ti 2  n( X i   0 ) '  1 ( X i   0 )
(4)
2
is called Hotelling´s T2 statistic. Ti is distributed as a chi-square
2
variate with p degrees of freedom. Notice that Ti  0 .
When the process is in control,  i   0 , there is a probability α
2
that this statisitic exceeds a critical point p, , so that the overall
error rate (type I) can be maintained exactly at the level α by
T 2   2p,
triggering a signal when i
. It is common practice to
suggest the use of a Shewhart type
2 control chart (figure 1)
2
with an upper control limit (CL) of  p, (see, e.g., Jackson [1],
2
Jackson [2] and Alt [3]). If  i   0 Ti is distributed as a non-
central chi-squared distribution with p degrees of freedom and
' 1
with non-centrality parameter   n(i  0 )  (i  0 ) . This
scheme corresponds to the likelihood ratio test for
H0: i  0 vs. H1: i  0 .
Recently multivariate CUSUM and multivariate EWMA schemes
has been defined showing better power than T2 chart specially for
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2. DESIGNING A SET OF
CHARTS
X
CONTROL
The objective of a X control chart, or Shewhart chart, is to detect
that the mean of the process has shifted. The following statistic
hypothesis test is posed:
H 0 : m  m0
H1 : m  m0
where m is the present mean of the process and m0 is the mean
when the process was in an in-control state. The procedure
consists of taking a random sample from the process and
computing the sample average, X . The value of this statistic is
plotted in a chart with control limits UCL (Upper Control Limit)
and LCL (Lower Control Limit). If the point exceeds one of the
control limits then the alternative hypothesis, H1, must be
accepted, therefore, we accept that there is a shift in the process
mean, the process is out of control. Therefore, an out-of-control
signal is obtained when a point plots outside control limits
In order to measure the performance of a control chart the ARL is
the most used parameter. The ARL (Average Run Length) is the
average number of points (samples) until an out-of-control signal
is shown. When the process is in-control, the value of ARL
should be as large as possible (no false alarms). However, when
there is a shift in the process mean, we would like to detect it
quickly, and, therefore, the value of ARL should be as low as
possible. The “ideal” chart should have an in-control ARL of
infinity and out-of-control ARL equals to 1. Nevertheless, it is not
possible to obtain this ideal chart, and the in-control and out-ofcontrol ARLs values must be balanced.
When p variables are monitored using a set of X control charts, p
means are monitored simultaneously, following the previous
scheme. The whole scheme (the p charts) must have the in-control
ARL that the users desires and must have an out-of-control ARL
as low as possible, i.e. minimum. Therefore, the following
parameters of the p charts are to be found:
1.- Control limits: UCLX1, LCLX1, UCLX2, LCLX2 …
UCLXp, LCLXp.
2.- Sample sizes: n1 , n2,
np
Therefore, the design of a set of X control charts consists of
finding the previous parameters to fulfill the ARL requirements of
the user.
Figure 1. Software solving the example of application.
Figure 2. Comparison of ARLs.
3. SOLVING THE OPTIMIZATION
PROBLEM
The design of a set of X control charts can be posed as an
optimization problem.
Given:

Magnitude of the shift to be detected, d.

In-control ARL, ARL(d = 0), ARL0.
Find:
Control limits and sample sizes: UCLX1, LCLX1, UCLX2,
LCLX2 … UCLXp, LCLXp, n1 , n2, n p
that minimizes ARL(d) with the restriction ARL(d = 0) = ARL0.

Number of variables to monitor, p.
The ARL function to minimize is:
1
1
ARL 
1 
UCLX1
LCLX1

UCLXp
LCLXp
(2 )
p/2

1/ 2
e
1
( 1  0 )T 1 ( 1  0 )
2

n = 5 (for both variables).

UCLX1 = 2.81; LCLX1 = 4.25.

UCLX2 = 3.81; LCLX2 = 4.43.

ARL(d = 0) = ARL0 = 400.02
·
(5)
where 1 is the vector of out-of-control means, and  0 in the
vector of in-control means and Σ is the covariance matrix of the
variables..
In addition, more restrictions are added taking into account the
real application in industry. Normally, there is a maximum value
of sample size that can be taken form the process, nmax. On the
other hand, in some cases the user may specify that all sample
sizes must be equal. This optimization problem is not easy, as
integers and real number are mixed. On the other hand, the ARL
function is not linear. Therefore, it is a suitable problem to be
solved using Genetic Algorithms (GAs).
The use of GAs to find the optimum parameters of quality control
charts is quite new. Some examples can be found in Aparisi and
García-Díaz [7], Champ and Aparisi [8] and He and Gregorian
[9].
4. SOFTWARE AND EXAMPLE OF
APPLICATION
On of the objectives of this work is to help the final user in the
industry to solve the optimization problem posed here. For that
reason, friendly software has been developed. Figure 1 shows this
software, that, at the moment, solves the problem when two or
three variables are monitored, the most common cases. The user
has to input the “Model Parameters”, i.e, the number of variables
(means) to be controlled, the desired in-control ARL, the
maximum sample size that can be used, the option to obtain all
sample sizes equal, the specification of the shift that must be
detected, and the correlation coefficients of variables.
The user can specify the parameters of the GA (number of
generations, population size, …). But the default values shown by
the software are been sought to optimize the performance of the
GA.
An example of application is solved using the software. Figure 1
shows the solution to this problem. Two variables are monitored.
When the process is in an in-control state the means are:
 2.2 

 7.8 
0  
When the process is in-control an ARL of 400 is desired, ARL0 =
400. The maximum sample size that can be used is nmax = 5, and it
is required that the two variables will be sample with the same
sample size. The correlation coefficient between the variables is r
= 0.8. The user want to obtain the best parameters for the two
Shewhart charts to minimize the ARL to detect an increment of
the first variable equals to 0.7 and an increment of the second
variable equals to 0.5.
After running the software (less than one minute) the solution
found is:
 0.7 
ARL(d   )  9.38
 0.5 
Therefore, this scheme will need an average of 9.38 samples to
detect a shift with new mean
 2.9 
1   
 8.3 
5. COMPARISON WITH THE T2
CONTROL CHART
As it was commented in the introduction, the alternative to
monitor p variables is the use of the T2 control chart. The software
helps the user to make a comparison between the performances of
using p Shewhart charts versus employing a T2 control chart.
In Figure 1, it is possible to see how the software shows the ARL
of the T2 control chart to detect the specified shift, having the
same in-control ARL, ARL0 = 400. In this case, the ARL of the
multivariate chart is 20.89, in comparison against the ARL of the
two Shewhart charts, 9.38. Therefore, the set of two Shewhart
charts takes less than the half to detect the out-of-control state.
In this first approach, it seems that the use of a set of X charts
will be a better option. However, the chart on the right of Figure 2
has to be considered. This chart shows an ARL comparison
between both schemes. It is possible to see that the set of X
charts produces a lower out-of-control ARL moving the maximum
values of ARL from the in-control point. That means that the
maximum values of ARL are not located for the in-control state
(no shift). Therefore, for some shifts the set of X charts will
obtain very large out-of-control ARLs, i.e, these shifts will be
very difficult to detect. However, the T2 control chart will always
show lower ARL for any given shift, in comparison with its incontrol ARL.
The fact that some shifts are not easily detected by the set of X
charts, but at the same time, it shows a quite lower out-of-control
ARL for a given shift, can be utilized by the user. As Woodall
[X] states, in some situations it is required to not detect some
shifts, keeping the performance to detect other shifts.
6. CONCLUSIONS
In this work we have posed the optimization of a set of X quality
control charts for the case that the user specify the in-control ARL
with the objective of finding the parameters that minimizes the
out-of-control ARL for a given shift magnitude. Friendly software
has been developed to help users in industry to find these optimal
parameters using GAs.
The results of the optimization shows that the set of X charts
will detect before the out-of-control state (lower out-of-control
ARL) in comparison versus the use of a T2 control chart.
[4] Crosier, R. B., 1988, Multivariate Generalizations of
cumulative sum quality-control schemes. Technometrics, 30,
291-303.
However, the set of X charts optimized for a given shift
magnitude will not detect easily other shifts. These shifts are
shown by the software, and as commented by Woodall [X], this
behavior solves a need found in some industries.
[5] Pignatello, J. J., Jr. and Runger, G. C., 1990, Comparisons of
multivariate CUSUM charts. Journal of Quality Technology,
22, 173- 186.
7. ACKNOWLEDGMENTS
[6] Lowry, C.A., Woodall, W. H., Champ, C. W. and Rigdon,
S.E., 1992, A multivariate exponentially weighted moving
average control chart. Technometrics, 34, 46-53.
This work has been supported by the Ministry of Education and
Science of Spain, research project number DPI2006-06124
including European FEDER funding, and the support of the
ITESM-Foundation Carolina agreement.
[7] Aparisi, F. and García-Díaz, J. C., 2004, Optimization of
Univariate and Multivariate Exponentially Weighted Moving
Average Control Charts using Genetic Algorithms,
Computers and Operations Research, 31 (9), 1437-1454.
8. REFERENCES
[8] Champ, C. W. and Aparisi, F., 2007, Hotelling’s T2 Double
Sampling Charts. Quality and Reliability Engineering
International, accepted
[1] Jackson, J. E., 1959, Quality Control Methods for Several
Related Variables, Technometrics, 1 (4), 359-377.
[2] Jackson, J. E., 1985, Multivariate Quality Control,
Communications in Statistics, 14 (11), pp. 2657-2688.
[3] Alt, F.B., 1985, Multivariate Control Charts, Encyclopedia
of Statistical Sciences, vol 6. (S. Kotz and N. L. Johson, Eds.
Wiley, New York), pp. 110-122.
[9] He, D. and Grigoryan, A., 2002, .Construction of Double
Sampling S- Control Charts for Agile Manufacturing,
Quality and Reliability Engineering International 18, 343355.