Numerical Comparisons of Quality Control Charts f or [630015]

Numerical Comparisons of Quality Control Charts f or
Variables
J.F. Muñoz -Rosas, M.N. Pérez -Aróstegui
University of Granada
Facultad de Ciencias Econ ómicas y Empresariales
Granada, Spain
[anonimizat] ; [anonimizat] E. Álvarez -Verdejo
Universidad Polité cnica de Cartagena
Centro Universitario de la Defensa. Academia General de
Aire
San Javier (Murcia) , Spain
[anonimizat]

Abstract— Control charts for variables can be obtained by using
both sample variances and sample ranges. Control charts can be
affected by various parameters, such as the sample size and the
number of samples used to obtain the control limits. When
calculating a control chart, the gain of using the sample variances
instead the sample ranges is not clear for a specific situ ation.
Assuming different scenarios and Monte Carlo simulations,
control chart for variables based upon samples variances are
empirically compared to control chart based upon samples
ranges.
Keywords – Monte Carlo simulations; quality control; normal
distribution; variance; range.
I. INTRODUCTION
Research on quality involves a range of concerns about
definitions, practices and such specific mechanisms as
statistical quality control (SQC). These techniques are used to
control the quality of the product by ana lyzing one or more
product characteristics. The most widely used techniques in
SQC are control charts. Control chart is a tool used to
determine if a business or manufacturing process is in a state of
statistical control. [3] developed this technique and p rovided a
framework for deciding whether the variation in the result is
due to assignable causes. He created a mathematical model that
assumes the Central Limit Theorem, so it is usual to adopt
mean and range type charts. Recent research shows that the X
control chart is very simple to understand, implement and
design, and may be more suitable in many SQC applications, in
which both the mean and variance of a variable need to be
monitored ( see [4] ). An indispensable assumption for the
correct development of control charts is that the process
parameters are assumed known but, in practice, the process
parameters are rarely known. Many studies do not consider the
effects of in -control parameter estimation on control chart
properties and recommended alternative estimators for mean
and standard deviation. In this sense, [2] and [4] consider
several robust location estimators for X control chart.
Control charts can be affected by various parameters, such
as the sample size and the number of samples used to obtain
the control limits. When calculating a control chart, the gain of
using the sample variances instead the sample ranges is not
clear for a specific situation. Assuming different scenarios and
Monte Carlo simulations, control chart for variables based
upon sa mples variances are empirically compared to control chart based upon samples ranges. The paper proceeds as
follows: the next section summarizes the literature review
about control charts for variables and identifies gaps in the
previous research. Following this, we develop several
numerical comparisons in order to compare numerically
control chart based upon samples variances and sample ranges .
The paper concludes with the main contributions.
II. CONTROL CHARTS FOR VA RIABLES
Control chart is the most commonly u sed tool for
monitoring the production process. This tool is simply a graph
that shows whether a sample of observations falls within the
normal range of variation, which is defined by using the called
control limits. Thus, a process is out of control when a plot of a
set of observations reveals that one or more samples fall
outside the control limits. There exits alternative tests to
analyze out -of-control situations. For example, a process is said
to be out of control when at least eight successive samples fall
on the same side of the center line. However, this study is
based upon the idea of comparing two control charts by
analyzing the percentage of samples which fall outside the
control limits, hence the study of alternative tests to analyze
whether a pr ocess is out of control is a topic which is beyond
the scope of this paper.
The different characteristics that can be measured by
control chart can be divided into variables and attributes. X -bar
chart, R chart and S chart are some examples of control cha rt
for variables. We assume X -bar charts in this paper.
Let
),(Nx be the variable of interest, where
 is
the true process mean and
 is the true process standard
deviation. We assume that the standards
 and
 are
unknown, hence they need to be estimated by using sample
information. Assuming this scenario, control limits are
obtained by selecting m previous samples with size n each one.
It is common t o assume that each sample with size n is selected
from a lot. We assume that each lot has size N. The average of
the observations in the i-th sample, with
m i ,,1 , is given
by
n
i i i x nx11 , and the center line of the X -bar chart can
be defined by
x CL , where
m
i ix mx11 is the grand
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average. Control limits can be obtained by using
m
i iS mS11
or
m
i iR mR11 , where

2/1
12
11




n
ji ij i xxnS
and
) min() max(i i i x x R  are, respectively, the
standard deviation and the range for the i-th sample. Control
limits based on the standard deviation (S) are given by
SAx LCLS 3
and
SAx UCLS 3 , whereas control
limits based on the range (R) are given by
RAx LCLR 2
and
RAx UCLR 2 , where
2A and
3A are constants
based on n. Tables for
2A and
3A constants and for various
values of n can be seen by Montgomery (2009) Appendix A
VI.
The gain of using the X -bar chart based on S in comparison
to the X -bar chart based on R may depend on the parameters n,
N, m or
 , which are previously defined.
III. NUMERICAL COMPARISONS
Assuming different scenarios, Monte Carlo simulations are
now carried out to compare the gain of using X-bar charts
based on S instead using X-bar chart based on R. For this
reason, we generated observations from a Normal distribution
with mean
10 and st andard deviation
2 ,1 ,5.0 .
At the first iteration run, control limits were obtained by
selecting m samples with size n. In practice, it is common to
consider that samples with size n are selected from a lot with N
observations, hence this si tuation is also considered in our
empirical comparisons. We considered the values
50 ,20 ,5m
,
 0100 ,010 ,25N and values of n
between 3 and 25. When control limits are obtained,
1000nD
samples with size n are selected from lots with
size N, and it is studied whether such samples falls outside the
control limits. At the second iteration run, m new samples with
size n are obtained in order to calculate the new control limits
and
nD samples are selecte d to analyze if they are out of
control. This process was repeated
1000mD times. The
criterium to compare the different control charts is the
empirical probability ( EP) of the various samples of falling
inside the control limits, i.e.,

  
     m n m n m D
iD
jij
nmD
iD
jij
n mD
ii
mPDDPD DEPDEP
1 1 1 1 11 1 1 1
,
where
1ijP if the sample in the i-th iteration run and the
j-th sample is inside the control limits and
0ijP otherwise.
Note that EP should be close to 0.9973, which is the critical value for control chart based on 3 sigma criterium. Note also
that EP is related to the empirical Type I error (ETI), which is
defined as the empirical probability of samples falls outside the
control limits even though no special cause are operating. In
partic ular, EP=1 -ETI. Note that the Type I error is fixed at
0.027 for the 3 sigma criterium.
Figure 1 gives the values of EP for the various control
charts with
5.0 and under different scenarios. We observe
that the value of N does not an impact on the performance of
the various control charts. This was expected because samples
are selected from the same distribution. We also observe that
control charts generally have a similar performance when n is
smaller than 15. When
5m and
15n , control charts
obtain values of EP smaller than the critical value 0.9973,
which indicates that the empirical Type I error is larger than the
fixed Type I error. However, the control chart b ased on S is
better than the con trol chart based on R in this situation. When
20m
and
15n , the control chart for S has a very good
performance, with values of EP close to 99 .73%. When
50m
and
15n , the values of EP for the control chart
based on S are slightly larger than 99 .73%, whereas the values
of EP for the control chart based on R are slightly smaller than
99.73%. Results derived from the simulation study are very
similar when the true process standard d eviation takes other
values (Figures 2 and 3), hence the value of
 does not a
relevant impact on the performance of the various control
charts.
IV. CONCLUSIONS
When standards
 and
 are unkn own, control limits for
X-bar charts can be obtained by using the sample variances or
the sample ranges. The performance of such control chart may
depend on various parameters, such as m, n or
 .
Assuming Monte Carlo simulations and different scenarios,
control charts based on S and R have been compared, and some
relevant conclusions have been obtained. First, empirical
results indicate that values of n smaller than 15 can give a
larger Type I error for both control charts. Second, r esults
indicate that a value of m close to 20 can be a good choice,
specially for the control chart based on S, which achieves
values of EP close to the required 99 .73%. Third, we observed
that the control chart based on S generally has a better
performanc e that the control chart based on R. Finally, we also
concluded that the values of N and
 does not a relevant
impact on the performance of the various X -bar charts.

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Figure 1. Values of EP for X -bar charts based on S and R. The true process standard deviation is
5.0 .

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Figure 2. Values of EP for X -bar charts based on S and R. The true process standard deviation is
1 .

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Figure 3. Values of EP for X -bar charts based on S and R. The t rue process standard deviation is
2 .

REFERENCES
[1] D.C. Montgomery, Statistical quality control . A modern introduction.
6th ed. New York, Wiley, 2009 .
[2] M. Schoonhoven , H. Nazir , M. Hafiz , R. Does , “Robust Location
Estimators for the X Control Chart ”, J. Qual. Tech. Vol. 43, No. 4, pp.
363-379, 2011.

[3] W.A. Shewhart, W.A, Economic control of quality of manufactured
product. Milwaukee, ASQC Quality Press, 1931.
[4] M. Yang, Z. Wu, K. Lee and M. Khoo , “The X control chart for
monitoring proc ess shifts in mean and variance ”. Intern . J. Prod .
Resear . Vol 50, No 3, pp. 893 -907, 2012 .
[5] Y. Zhang and P. Castagliola , “Run rules X charts when process
parameters are unknown,” Intern . J. Reliab . Qual . Saf. Eng. Vol. 17,
No. 4 , pp. 381–399, 2010 .

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