1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME
82
COST BASED INDUSTRIAL RECTIFYING SAMPLING INSPECTION
Faris M. Al-Athari
Professor, Department of Mathematics, Faculty of Science and
Information Technology, Zarqa University, Jordan
Dhwyia Salman Hassan
Professor, Department of Statistics, college of Economic and
Administration, University of Baghdad, Iraq
ABSTRACT
This paper aimed to develop a quality cost model which is considered to be a generalization
or extension to the model of Schmidt and Taylor (1973). The new model is built by using a more
general formula for p (t), the average proportion defective of a lot where line failure occurred at time
t after the start of production of a lot. The proposed formula for p (t) fits all kinds of the time failure
distributions not just the uniform time failure distribution. In this paper we assumed that the average
proportion defective p (t) depends upon the failure time which varies exponentially, and the number
of defective items in a sample of size n has a Poisson distribution with mean n p (t). The expected
total cost has been built and a sample plan for rectifying inspection which minimizes the expected
total cost is done by determining the optimum sample size, acceptance number and inter inspection
period. A real- life data show that the percentage reduction in the total cost by using the proposed
model instead of Schmidt's and Taylor's model is about 65.9%.
Keywords: Rectifying sampling inspection, generalized cost model, Percentage reduction, Poisson
distribution
1. INTRODUCTION
The quality of products has been the focus of producers in different areas of industrial
process control. The control chart is one of the most powerful tools for monitoring industrial process.
Shewhart (1931) introduced control charts, which play an important role in statistical quality
control to detect out-of-control signals and to improve the quality of product. Burr (1967), Schilling
and Nelson (1976), Champ and Woodall (1987), Ryan (1989), Quesenberry (1997), Deming (2000),
Woodall et al. (2004), Montgomery (2005), Albers et al. (2006), and several other researchers
modified and popularized Shewhart control charts. Dodge (1964) asserted that distinguishing
INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN
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ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
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2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME
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between sampling inspection schemes and quality control charts was clear since World War II. The
usage of acceptance sampling plans may belong to the date of establishing the department of
inspection and quality control of Bell laboratories in United States of America in the year of 1924.
The minimum cost determinations that satisfy the required quality of the products are the main
objectives to industrial engineering and to production engineering as well as to different areas of
research sectors.
Several methods for obtaining ordinary sampling plans, Bayesian sampling plans, double
sampling and sequential sampling plans which minimize the total quality control cost function have
been proposed in the literatures by several researchers, among them Barnett (1972), Guenther (1977,
1981), Bain (1978), Hald (1981), Schilling (1982), Baklizi (2003), and Hassan et al. (2013).
The objective of this paper is to build a quality control cost model used in developing a single
sampling plan for rectifying inspection which minimizes the expected total cost by determining the
sample size n, acceptance number c and time period, between successive inspections. This paper
derives the expected total cost model under the same assumptions given by Schmidt and Taylor
(1973) except that the average proportion defective of the lot, p (t), is given by,
)|()|()( 10 τττ <<<+≤≤= TTtPpTtTPptP (1)
The proposed formula of equation (1) fits all kinds of time failure distributions not just the uniform
time failure distribution which reduces the formula (1) to the special case
ττ /))(()( 10 ptpttp −+= (2)
that has been given by equation (4) of Schmidt and Taylor (1973). The proposed model is compared
with the model of Schmidt and Taylor (1973) by using a real- life data from an industrial company in
Iraq which produces pure corn oil. The results from using the multivariate and a partial enumeration
show that the percentage reduction in the expected total cost by using the proposed model instead of
Schmidt's and Taylor's model is about 65.9%.
2. DEVELOPMENT OF THE COST MODEL
The main objective of the rectifying inspection is to determine the sample size n, the
acceptance number c, and the inter inspection time that minimizes the following expected total
cost T.C of quality control,
T.C=E(inspection cost) + E (rejection cost) + E (acceptance cost) + E (line failure cost)
under the following assumptions:
1. The probability density function ),,( λtf of the time until failure T is exponential of mean ./1 λ
2. The probability mass function p(x, n, p), of the number of defectives X, in a sample of size n
drawn from a lot of proportion defective p, is a Poisson of mean n p.
3. A decision to reject any given lot causes a line shutdown and adjustment at a cost CF.
4. C1, C2, and C3 are respectively represented the unit cost of inspection, the unit cost of rejection,
and the unit cost of acceptance.
In order to accomplish this task, the probability B that the system is in a failed state after any
given inspection is
B=
1
1
)(1 QcXP
Q
+≤−
and hence,

3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME
84
where
And
Q1 is the probability that the system is in a failed state after the first inspection and may be given as:
(4)
and the new average proportion defective p(t) of a lot is given by
and hence the expected total cost of the quality control system will be
Simplifying the results we get,
where
(7)

4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 5, July – August (2013), © IAEME
85
3. APPLICATION
After the expected total cost function is built, it is necessary to apply the proposed model
given in equation (6) and compare it with model that has been given by Schmidt and Taylor (1973).
In this paper we considered a production facility from an industrial company in Iraq which
produces pure corn oil at a constant rate 700=ψ units/hr. The historical data about this production
facility show that the time between two successive failures is
Exponentially distributed with rate ./1667.0 hr=τ Moreover, the following data concerning the
costs are collected by the authors:
C1=$ 0.009/unit p0=0.004265 units
C2=$ 0.260/unit p1=0.05 units
C3=$ 0.742/unit 700=ψ units/hr
CF =$ 288/shutdown .λ =0.1667/hr
A multivariate optimization technique and a partial enumeration with a one- factor- at- a time
was employed on the proposed model and gave the following optimum sampling plan
(c=8 units, n=240 units, ,.5.2 hours=τ T.C. =$ 131.576).
But the optimum sampling plan for the model of Schmidt and Taylor (1973) using the same
data was
(c=7 units, n=400 units, ,.5.8 hours=τ T.C. =$ 386.29).
To find the percentage reduction R in the expected total cost by using the proposed model
instead of Schmidt's and Taylor's model, we get
100*
29.386
576.13129.386 −
=R
=65.9%
This means that the expected total cost can be reduced by 65.9% if one uses the proposed
model instead of the model of Schmidt and Taylor (1973). This is because of using the general
formula of p (t) with exponentially distributed time failure instead of uniformly distributed time
failures.
CONCLUSIONS
This research paper developed a new cost- based rectifying sampling inspection scheme
which is considered to be a generalization of the model of Schmidt and Taylor (1973) which to our
knowledge has not been considered before in the literature. The results of the optimum proposed
sampling plan were compared with the optimum sampling plan for the model of Schmidt and Taylor
(1973) through considering a real- life data. It appears that the proposed model is more flexible and
more accurate than the model of Schmidt and Taylor (1973) when the distribution of the time until
failure is not uniformly distributed. The results show that the percentage reduction in expected cost
1s 65.9% when using the proposed model.

5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN
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86
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