Open Journal of Applied Sciences, 2012, 2, 236-240
doi:10.4236/ojapps.2012.24035 Published Online December 2012 (http://www.SciRP.org/journal/ojapps)
The Principle of Mathematical Induction Applied to the
Generalized Model for the Economic Design of
-Control Charts
Mohamed E. Seliaman1, Salih O. Duffuaa2
1Department of Information Systems, King Faisal University, Al-Ahsa, Saudi Arabia
2Department of Systems Engineering, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia
Email: mseliaman@gmail.com
Received August 4, 2012; revised September 6, 2012; accepted September 25, 2012
ABSTRACT
Rahim and Banerjee [1] developed a general model for the optimal design of
-control charts. The model minimizes
the expected cost per unit time. The heart of the model is a theorem that derives the expected total cost and the expected
cycle length. In this paper an alternative simple proof for the theorem is provided based on mathematical induction.
Keywords: Discrete Mathematics; Mathematical Induction; Statistical Quality Control; Design of
-Control Charts;
Hazard Rate Function
1. Introduction
Quality control charts are graphical statistical tools used
for process control. The first control chart was developed
by Walter A. Shewart [2]. Since then, these charts have
been widely applied in industry and also received inten-
sive attention from researchers. The
-control charts are
the most used statistical control charts when continuous
variables are used to measure quality characteristics [3].
Generally, control charts are designed using four ap-
proaches: heuristics, statistical design, economic design
and economic statistical design [4]. For more information
on the design of control charts, interested readers are re-
ferred to [1-7].
The economic design of
-control charts was origi-
nated by Duncan [8]. In his model, Duncan considers a
production process that is subject to an occurrence of an
assignable cause that will drive the process out of control.
The output of the process is measurable on a certain scale
and is normally distributed with mean 0
and standard
deviation 0
. The assignable cause is assumed to occur
according to a Possion process with intensity
and
causes a shift in the process mean to 0
, where
is a positive parameter. The control limits of the
-control chart are set at 00
k
, where 0
is the
standard deviation of the process. A sample of size n is
taken from the output of the process every h hour, and
used to decide whether the process is in control or not.
Banerjee and Rahim [9] generalized Duncan’s model [8]
for the economic design of
-control charts by relaxing
Duncan’s assumption that the in-control period follows an
exponential distribution. Instead, they assumed it follows
a general probability distribution having an increasing
hazard rate function. The increasing hazard rate assump-
tion resulted in the modification of the fixed sampling
interval to a variable sampling interval, dependent on the
process age. In both models, if the sample falls outside the
control limits, a search is initiated to locate the assignable
cause. If the search indicates that there is a false alarm,
the process continues. On the other hand, if the alarm is
true, repair or replacement is undertaken to bring the
process in control. Later, Rahim and Banerjee [1] ex-
tended the model in Banerjee and Rahim [9] by assuming
a general distribution of in-control periods with increasing
failure rate and considering an age-dependent repair be-
fore failure. The objective of Rahim and Banerjee’s
model [1] is to find the parameters n, hj, and k. Under the
assumptions described for this generalized model, they
derived expressions for the expected cycle length and the
total expected cost per cycle. They also presented proofs
for these expressions using a recursive argument. In this
paper a simpler and shorter mathematical induction proof
of these results is presented. Thus, our alternative proof
may make it easier for students and professors interested
in the topic within graduate coursework or further re-
search on extending the model. In this paper, we adopt the
same notations and model descriptions as defined in Ra-
him and Banerjee [1].
In order to make the paper self-contained, we describe
the main elements involved in the design of control charts,
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