Qual Quant (2010) 44:905–914
Constructing a fuzzy Shewhart control chart for variables
when uncertainty and randomness are combined
Alireza Faraz · R. Baradaran Kazemzadeh ·
M. Bameni Moghadam · Aliasghar Bazdar
Published online: 28 May 2009
© Springer Science+Business Media B.V. 2009
Abstract In this paper we introduce a fuzzy chart for variables which is used in situa-
tions when uncertainty and randomness are combined. It is showed that the Shewhart chart’s
control limits must be adjusted in these situations. However, this chart is based on a fuzzy
acceptance region and this method arises when a decision should be made by referring to the
grade of a sample statistic belonging to the fuzzy acceptance region.
As is well known, one of the aims of statistics is to infer about the population, based on
the sample data obtained. Two important areas in this regard are parameter estimation and
hypothesis testing. The problem of testing of hypotheses is used in quality control frequently.
In this paper, ﬁrst we summarize the classical method which is associated with a crisp method.
Then we consider the fuzzy approach of this problem which seems to have more applications
in reality. Then based on the fuzzy conﬁdence region, we will deﬁne the fuzzy Shewhart
chart for controlling the process mean. Finally, we illustrate the method with a real data in
order to understand the usefulness of our proposed fuzzy chart.
2 Classical method
We start this section with an example. Let x
be an observed random sample from
normal distribution N(μ, σ
) where the standard deviation σ is known. The object is to test
A. Faraz (
) · R. B. Kazemzadeh
Industrial Engineering Department, School of Engineering, Tarbiat Modares University,
P.O. Box 14115–179, Tehran, Iran
M. B. Moghadam
Department of Statistics, Allameh Tabatabaee University, Tehran, Iran
School of Mathematical Science, Isfahan University of Technology, Isfahan, Iran