This document presents a model for determining the parameters of a double Bayesian sampling inspection plan to minimize total expected quality control costs. The model considers costs associated with inspecting samples, accepting or rejecting lots, and continuing to a second sample. It defines variables such as sample sizes, acceptance/rejection numbers, and costs. Equations are provided for the average inspection cost of the first and second samples considering the sampling distribution and conditional probability of defective items. The model aims to obtain optimal parameters (n1, a1, r1, n2, a2, r2) of the double sampling plan by minimizing the total expected regret function according to producer and consumer risk levels.

1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME
282
BUILDING A MODEL FOR EXPECTED COST FUNCTION TO
OBTAIN DOUBLE BAYESIAN SAMPLING INSPECTION
1
Dr. Dhwyia Salman Hassan, 2
Nashaat Jaisam Al-Anber, 3
Dr.Suaad Khalaf Salman
1
Department of Statistics, College of Economic and Administration, University of Baghdad,
Baghdad
2
Department of Information Technology, Technical College of Management-Baghdad,
Foundation of Technical Education, Iraq
3
Department of Mathematics, College of Education for Pure Science, University of Karbala,
Iraq
ABSTRACT
This paper deals with building a model for total expected quality control cost for
double Bayesian sampling inspection plan, which consist of drawing a sample (݊ଵ) from a lot
(ܰ), then checking the number of defective (‫ݔ‬ଵ) in (݊ଵ), if (‫ݔ‬ଵ ൑ ܽଵ), were (ܽଵ) is acceptance
number, then accept and stop, if (‫ݔ‬ଵ ൒ ‫ݎ‬ଵ), were (‫ݎ‬ଵ) is rejection number, then take a decision
to reject and stop sampling, if (ܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵ), draw a second sample (݊ଶ) and check the
number of defective (‫ݔ‬ଶ) in (݊ଶ), if (‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ ൑ ܽଶ, accept and stop, if (‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ ൒ ‫ݎ‬ଶ reject
and stop. The model derived assuming that the sampling process is Binomial with (݊, ‫)݌‬ and
( ‫)݌‬ is random variable, varied from lot to lot and have prior distribution ݂ሺ‫݌‬ሻ, here it is
found Beta prior with parameters (ܵ, ܶ).The model is applied on the set of real data obtained
from some industrial companies in Iraq, were the percentage of defective of (140) lots are
studied and tested, ݂ሺ‫݌‬ሻ is found Beta prior (ܵ, ܶ). These parameters are estimated by
moment estimator. After checking the distribution ݂ሺ‫݌‬ሻ and all the parameters of quality
control model, we work first on obtaining single Bayesian sampling inspection plan (݊ଵ, ܿଵ)
from minimizing the standard regret function (ܴଵ), then we find the parameters of double
Bayesian sampling inspection plan from minimizing (ܴଶ), for various values of process
average of quality (‫,)݌‬ with different lot size N.
Keywords: Bayesian sampling Inspection, prior distribution, average of quality, standard
regret function for single sampling plan. standard regret function for double Bayesian
sampling plan.
INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN
ENGINEERING AND TECHNOLOGY (IJARET)
ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 4, Issue 2 March – April 2013, pp. 282-294
© IAEME: www.iaeme.com/ijaret.asp
Journal Impact Factor (2013): 5.8376 (Calculated by GISI)
www.jifactor.com
IJARET
© I A E M E

2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
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1- AIM OF RESEARCH
The aim of this research is to build a model for total cost of double Bayesian sampling
plans, then determine the parameters of this double Bayesian sampling plans
(݊ଵ, ܽଵ, ‫ݎ‬ଵ, ݊ଶ, ܽଶ, ‫ݎ‬ଶ) from minimizing the total expected cost function.
2- NOTATIONS
Here we introduce all necessary notations:
݊ଵ : First sample size drawn from lot N.
‫ݔ‬ଵ : Number of defective units in sample (݊ଵ).
ܽଵ : Number of accepted defective units in (݊ଵ).
‫ݎ‬ଵ : Number of rejected units in (݊ଵ).
݊ଶ : Size of second sample taken from (ܰ െ ݊ଵ).
‫ݔ‬ଶ : Number of defective units in (݊ଶ).
ܺ : Number of defective units in lot (N).
ܽଶ : Number of accepted defectives in (݊ଶ).
‫ݎ‬ଶ : Number of rejected units second sample (݊ଶ).
ܲ : Percentage of defectives in production.
‫݌‬ : percentage of defectives in sample.
ܲ௔
ሺଵሻ
ሺ‫݌‬ሻ : Probability of accepting the product with quality (‫)݌‬ according to the decision from
first sample (݊ଵ).
ܲ௥
ሺଵሻ
ሺ‫݌‬ሻ : Probability of rejecting the product with quality (‫)݌‬ due to the decision of rejecting
from first sample (݊ଵ).
ܲ௔
ሺଶሻ
ሺ‫݌‬ሻ : Probability of accepting the product with quality (‫)݌‬ according to the decision from
second sample (݊ଶ).
ܲ௥
ሺଶሻ
ሺ‫݌‬ሻ : Probability of rejecting the product with quality (‫)݌‬ due to the decision of rejecting
from first sample (݊ଶ).
ܲ௦
ሺଶሻ
ሺ‫݌‬ሻ : Probability of continue sampling after the first sample is chosen and the value of
(ܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵ), so we cannot take a decision to accept or reject according to first sample.
݇‫ݏ‬ሺ‫݌‬ሻ: Average inspection cost from first sample, where;
‫ܧ‬ሺ݊ଵܵଵ ൅ ‫ݔ‬ଵܵଶሻ ൌ ݊ଵሺܵଵ ൅ ܵଶ‫݌‬ሻ ൌ ݊ଵ݇‫ݏ‬ሺ‫݌‬ሻ
ܵଵ : Cost of inspecting item from first sample.
ܵଶ : Cost of repairing or replacement of item in sample (݊ଵ).
ܴଵ : Cost of sampling and testing unit in rejected quantity (ܰ െ ݊).
ܴଶ : Cost of repairing or replacement unit in rejected quantity (ܰ െ ݊).
‫ܣ‬ଵ : Cost of accepting good units which not represents penalty cost.
‫ܣ‬ଶ : Cost of accepting defective units.
3. INTRODUCTION
In rectifying inspection plans, each item produced by a process is inspected according
to some plan and then according to the number of defectives units in the sample, the lot is
either accepted or the lot is rejected. Gunether [1977] explained how to obtain the parameters
(݊, ܿ) for single sampling plan for ordinary sampling and for Bayesian sampling, using the
model of expected total quality control cost depends on cost of inspection and sampling and

3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME
284
cost due to wrong decision, which is accepting bad lot and rejecting good one. Another
researchers gives model for obtaining parameters of single sampling inspection plan (݊, ܿ)
under truncated time to failure, see Epstein, B. [1962]. Goode and Kao [1960]. Gupta, S. S.
[1962] who designed sampling inspection plans for normal and lognormal distribution
according to pre – determined level of percentage of defectives. Also Gupta and Kunda
[1999] discuss the properties of generalized exponential distributions and how to use it is
constructing Bayesian sampling. Baklizi, A. [2003] introduced how to obtain acceptance
sampling plans based on truncated life tests in the Pareto distribution of second kind. The
research in this subject were developed when Balakrishnan, N. [2007] designed a group of
inspection sampling for generalized Birnbaum – Saundres distribution. In [2012], Dhwyia S.
Hassun, et, built a model for total expected quality control cost, and compared it with the
model of Schmidt – Taylor [1973]. Also Hald, A. [1968] gives a model for Bayesian single
sampling attribute plans for continuous prior distribution. The parameters of single sampling
inspection plans (݊, ܿ) are obtained from minimizing total expected quality control cost.
Now, in this paper we introduce ( a mixed – Binomial) quality control expected cost
function, to obtain the parameters of double Bayesian sampling inspection plans, from
minimizing total expected regret function, according to conditions of producers and
consumers risk.
Building Model of Total Expected Cost Function
The model comprises:
a- ݇‫ݏ‬ሺ‫݌‬ሻ: Average inspection cost from first sample;
݇‫ݏ‬ሺ‫݌‬ሻ ൌ ‫ܧ‬ሺ݊ଵܵଵ ൅ ‫ݔ‬ଵܵଶሻ
ൌ ݊ଵሺܵଵ ൅ ܵଶ‫݌‬ሻ
ൌ ݊ଵ݇‫ݏ‬ሺ‫݌‬ሻ …………………………………………….. (1)
݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ : Average cost of inspecting units of second sample, under sampling distribution
ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ for number of defectives units in first sample (݊ଵ).
݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൌ ‫ܧ‬ሺ݊ଶܵଵ ൅ ‫ݔ‬ଶܵଶ|‫ݔ‬ଵሻ
ൌ ݊ଶ൫ܵଵ ൅ ܵଶ‫݌‬ሺ‫ݔ‬ଵሻ൯
ൌ ݊ଶ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ……………………………………………… (2)
And;
‫݌‬ሺ‫ݔ‬ଵሻ ൌ ෍ ‫ݎ݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ ൅ ‫ݔ‬ଷ ൌ ‫ݕ‬ሻ
‫׊‬௬
ൌ ݃௡భ
ሺ‫ݔ‬ଵሻ …………………………………………….. (3)
or;
‫݌‬ሺ‫ݔ‬ଵሻ ൌ ෍ ݃௡భ,௡మ
ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ
‫׊‬௫మ
ൌ ݃௡భ
ሺ‫ݔ‬ଵሻ …………………………………………………. (4)
If we assume that [‫ݔ‬ଷ ൌ ሺܺ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶ], then the joint probability distribution of (‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ )
is;
‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷሻ ൌ ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ|‫ݔ‬ሻ‫݌‬ሺ‫ݔ‬ሻ ………………………………………………. (5)
‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ∑ ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷሻ‫׊‬௫య
………………………………………………...(6)
And;
‫݌‬ሺ ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ ݃௡మ
ሺ ‫ݔ‬ଶ|݊ଵ, ‫ݔ‬ଵሻ ……………………………………………….. (7)
The average inspection cost for second sample is equal to:

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6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME
285
‫ܧ‬ሺ݊ଶܵଵ ൅ ‫ݔ‬ଶܵଶ|‫ݔ‬ଵሻ
ൌ ݊ଶ൫ܵଵ ൅ ܵଶ‫݌‬ሺ‫ݔ‬ଵሻ൯
ൌ ݊ଶ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ………………………………………………. (8)
In case of accepting produced lot from first sample, the average cost is;
‫ܧ‬ሼሺܰ െ ݊ଵሻ‫ܣ‬ଵ ൅ ሺܺ െ ‫ݔ‬ଵሻ‫ܣ‬ଶ|‫ݔ‬ଵሽ
ൌ ሺܰ െ ݊ଵሻ‫ܣ‬ଵ ൅ ‫ܧ‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ‫ܣ‬ଶ
ൌ ሺܰ െ ݊ଵሻሼ‫ܣ‬ଵ ൅ ‫ܣ‬ଶ‫݌‬ሺ‫ݔ‬ଵሻሽ
ൌ ሺܰ െ ݊ଵሻ݇ܽሾ‫݌‬ሺ‫ݔ‬ଵሻሿ
The average of acceptance cost for second sample is;
‫ܧ‬ሼሺܰ െ ݊ଵ െ ݊ଶሻ‫ܣ‬ଵ ൅ ሺܺ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶሻ‫ܣ‬ଶ|‫ݔ‬ଵ, ‫ݔ‬ଶሽ
ሺܰ െ ݊ଵ െ ݊ଶሻ݇ܽሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻሿ …………………………………. (9)
After explaining all notations, we can show that the lot size N, in case of double sampling
can divide into;
ܰ ൌ ݊ଵ ൅ ሺܰ െ ݊ଵሻቀ‫݌‬௔
ሺଵሻ
൅ ‫݌‬௥
ሺଵሻ
ቁ ൅ ݊ଶ‫݌‬ௌ
ሺଵሻ
൅ሺܰ െ ݊ଵ െ ݊ଶሻቀ‫݌‬௔
ሺଶሻ
൅ ‫݌‬௥
ሺଶሻ
ቁ … (10)
Assume (݇݉), is the smallest average cost, happened according to results of decisions of
accepting and rejecting, where acceptance happened at (‫݌‬ ൑ ‫,)ݎ݌‬ (i.e ‫,ݎ݌‬ break even quality
level) and rejection when (‫݌‬ ൐ ‫,)ݎ݌‬ (‫)݌‬ is random variable have [݂ሺ‫݌‬ሻ] then (݇݉) is equal to;
݇݉ ൌ ‫׬‬ ሺ‫ܣ‬ଵ ൅ ‫ܣ‬ଶ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ൅
௣௥
଴
‫׬‬ ሺܴଵ ൅ ܴଶ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬
ଵ
௣௥
………………..... (11)
Now the construction of total expected quality control cost [݇ଶሺ‫݌‬ሻ] is given by;
݇ଶሺ‫݌‬ሻ ൌ ݊ଵ݇‫ݏ‬ሺ‫݌‬ሻ ൅ ሺܰ െ ݊ଵሻቄ݇ܽሺ‫݌‬ሻ‫݌‬௔
ሺଵሻ
ሺ‫݌‬ሻ ൅ ݇‫ݎ‬ሺ‫݌‬ሻ‫݌‬௥
ሺଵሻ
ሺ‫݌‬ሻቅ
൅݊ଶ݇‫ݏ‬ሺ‫݌‬ሻ‫݌‬ௌ
ሺଶሻ
ሺ‫݌‬ሻ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻቄ݇ܽሺ‫݌‬ሻ‫݌‬௔
ሺଶሻ
ሺ‫݌‬ሻ ൅ ݇‫ݎ‬ሺ‫݌‬ሻ‫݌‬௥
ሺଶሻ
ሺ‫݌‬ሻቅ ……….…(12)
When ሺ‫݌‬ሻ is random variable have [݂ሺ‫݌‬ሻ], then the average cost is;
݇ଶ ൌ ‫׬‬ ݇ଶሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬
݇ଶ ൌ ݊ଵ݇ܵ ൅ ሺܰ െ ݊ଵሻ ቐ ෍ ݃ሺ‫ݔ‬ଵሻ݇ܽሾ‫݌‬ሺ‫ݔ‬ଵሻሿ
௔భ
௫భୀ଴
ቑ ൅ ෍ ݃ሺ‫ݔ‬ଵሻ݇‫ݎ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ
௡భ
௫భୀ௥భ
൅ ∑ ݃ሺ‫ݔ‬ଵሻൣ݊ଶ݇‫ݏ‬ሾ‫݌‬ሺ‫ݔ‬ଵሻሿ൧
௥భିଵ
௫భୀ௔భାଵ
൅ሺܰ െ ݊ଵ െ ݊ଶሻ ቐ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ݇ܽሼ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻሽሿ
௔మି௫భ
௫మୀ଴
ቑ
൅ ∑ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ‫݌‬௥
ሺଶሻ
ሺ‫݌‬ሻሾ݇‫ݎ‬ሼ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻሽሿ
௡మ
௫మୀ௔మି௫భାଵ ………………………..(13)
Multiplying equation (10) by (݇݉) where ሾܰ݇݉ ൌ ݇݉ሿ and apply
[ሺ݇2 െ ݇݉ሻ|ሺ‫ܣ‬ଶ െ ܴଶሻ], we have obtained the equation of standard cost function (ܴଶ).
ܴଶ ൌ ݊ଵ݀‫ݏ‬ ൅ ሺܰ െ ݊ଵሻ ቊන ሺ‫ݎ݌‬ െ ‫݌‬ሻ‫݌‬௥
ሺଵሻ
ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬
௣௥
଴
ቋ
൅ ቊන ሺ‫݌‬ െ ‫ݎ݌‬ሻ‫݌‬௔
ሺଵሻ
ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬
ଵ
௣௥
ቋ
൅݊ଶ ቄ‫׬‬ ሺ‫ݎ݌‬ െ ‫݌‬ሻ‫݌‬ௌ
ሺଵሻ
ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ൅ ‫׬‬ ߜሺ‫݌‬ሻ‫݌‬ௌ
ሺଵሻ
ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬
ଵ
଴
௣௥
଴
ቅ
൅ሺܰ െ ݊ଵ െ ݊ଶሻ ቊන ሺ‫ݎ݌‬ െ ‫݌‬ሻ‫݌‬௥
ሺଶሻ
ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬ ൅ න ሺ‫݌‬ െ ‫ݎ݌‬ሻ‫݌‬௔
ሺଶሻ
ሺ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬
ଵ
௣௥
௣௥
଴
ቋ
… (14)

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6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME
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‫ݎ݌‬ ൌ break even quality level;
‫ݎ݌‬ ൌ
஺మିோమ
஺భିோభ
Equation (14), now transformed to standard cost function in terms of conditional distribution
of percentage of defectives, which is [݂ሺ‫|݌‬ሺܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵሻ] especially these parts obtained
from second sampling, where;
݂ሺ‫|݌‬ሺܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵሻ ൌ
௣௥ሺ௔భழ௫భழ௥భ|௣ሻ௙ሺ௣ሻ
∑ ௕௪ሺ௫భ,௡భሻ
ೝభషభ
ೌభశభ
ൌ
ቄ௣ೌ
ሺమሻ
ሺ௣ሻା௣ೝ
ሺమሻ
ሺ௣ሻቅ௙ሺ௣ሻ
‫׬‬ ቄ௣ೌ
ሺమሻ
ሺ௣ሻା௣ೝ
ሺమሻ
ሺ௣ሻቅ௙ሺ௣ሻௗ௣
భ
బ
……… (15)
This because when [ܽଵ ൏ ‫ݔ‬ଵ ൏ ‫ݎ‬ଵ] a second sample (݊ଶ) is drawn, and according to this
sample either we accept or reject.
The final formula for standard regret function (ܴଶ) used to obtain the parameters of second
sampling plan is;
ܴଶ
‫כ‬
ൌ ܴଵ
‫כ‬ሺܽଵ, ݊ଵ, ܰሻ െ ሺܰ െ ݊ଵሻ ෍ ݃ሺ‫ݔ‬ଵሻ݀‫ݎ‬ሺ‫ݔ‬ଵሻ
௥భିଵ
௫భୀ௔భାଵ
൅ ∑ ݃ሺ‫ݔ‬ଵሻܴଵሺܽଶ െ ‫ݔ‬ଵ, ݊ଶ, ܰ െ ݊ଵ|‫ݔ‬ଵ ሻ௥భିଵ
௫భୀ௔భାଵ ……………………………….… (16)
Where;
݀‫ݎ‬ሺ‫ݔ‬ଵሻ ൌ න ሺ‫ݎ݌‬ െ ‫݌‬ሻ݂ሺ‫ݔ|݌‬ଵሻ݀‫݌‬
௣௥
଴
ܴଵ
‫כ‬ሺܽଵ, ݊ଵ, ܰሻ ൌ
௄ሺே,௡,௖ሻ ି ே௞௠
௞௦ ି ௞௠
………………………………… (17)
Where;
‫ܭ‬ሺܰ, ݊, ܿሻ ൌ ݊݇‫ݏ‬ ൅ ቂሺ‫ܣ‬ଶ െ ܴଶሻ
ௌ
ௌା்
‫ܨ‬ଵሺܿ, ܵ ൅ 1, ‫݌‬ሻቃ
൅ሾሺ‫ܣ‬ଵ െ ܴଵሻ‫ܨ‬ሺܿ, ܵ, ‫݌‬ሻሿ ൅ ቂሺܴଵ ൅ ܴଶሻ
ௌ
ௌା்
ቃ
݇‫ݏ‬ ൌ ܵଵ ൅ ܵଶ‫݌‬
݇‫ݏ‬ ൌ ܵଵ ൅ ܵଶ ቀ
ௌ
ௌା்
ቁ , ‫݌‬ ൌ
்
௡ା்
݇ሺܰ, ݊, ܿሻ ൌ ݊݇‫ݏ‬ ൅ ሺܰ െ ݊ሻሾሺ‫ܣ‬ଶ െ ܴଶሻ‫ܨ݌‬ଵሺܿ, ܵ ൅ 1, ‫݌‬ሻሿ
൅ሾሺ‫ܣ‬ଵ െ ܴଵሻ‫ܨ‬ሺܿ, ܵ, ‫݌‬ሻሿ ൅ ሺܰ െ ݊ሻሺ‫ܣ‬ଵ ൅ ܴଶ‫݌‬ሻ ൅ ሺܰ െ ݊ሻሺܴଵ െ ‫ܣ‬ଵሻ
Which can be simplified to;
݇ሺܰ, ݊, ܿሻ ൌ ݊ሺ݇‫ݏ‬ െ ݇݉ሻ ൅ ሺܰ െ ݊ሻሾሺ‫ܣ‬ଶ െ ܴଶሻ‫ܨ݌‬ଵሺܿ, ܵ ൅ 1, ‫݌‬ሻሿ
൅ሾሺܴଵ െ ‫ܣ‬ଵሻሾ1 െ ‫ܨ‬ሺܿ, ܵ, ‫݌‬ሻሿሿ ൅ ܰ݇݉ ……………... (18)
Therefore the equation of standard expected cost (Regret function ܴଵ) is;
ܴଵሺܰ, ݊, ܿሻ ൌ
௞ሺே,௡,௖ሻ ି ே௞௠
௞௦ ି ௞௠
Which is the first part of equation (ܴଶ
‫כ‬
) defined in equation (16), so first of all and
after determining the parameters of single Bayesian sampling plan [given in table (1)] we
work on minimizing (16) to find the parameters of double Bayesian sampling inspection plan
given in table (4).
Equation (16) shows that the average of standard cost function for double sampling
plan, is represented by standard cost function for single sampling plan and also standard
conditional cost function weighted by the probability distribution of number of defective
units [݃ሺ‫ݔ‬ଵሻ], from first sample, it is obtained from;
‫݌‬ሺ‫ݔ‬ଵሻ ൌ ݃௡భ
ሺ‫ݔ‬ଵሻ ൌ ෍ ݃௡భ,௡మ
ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ
‫׊‬௫మ

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287
To satisfy the aim of research which is to find the parameters ሺܽଵ, ‫ݎ‬ଵ, ܽଶሻ from
minimizing (ܴଶ), we apply difference equation since the distribution of number of defectives
in production is of discrete type, and also we consider;
∆ ܴଶ ൌ
∆ ௞మ
஺మିோమ
The differences equations according to ሺ‫ݎ‬ଵ, ܽଵ, ܽଶሻ, required to rewrite (݇ଶ) as in equation
(19) follows;
Since;
݇ଶ ൌ
݇ଵሺܽଵ, ݊ଵ, ܰሻሺ‫ܣ‬ଶܴଶሻ ∑ ݃ሺ‫ݔ‬ଵሻ൛݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ
௥భିଵ
௫భୀ௔భାଵ
݊ଶሻ ∑ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ
௔మି௫భ
௫మୀ଴ ൟ …………………………………….. … (19)
Then using;
∆ ܴଶ ൌ
∆ ௞మ
஺మିோమ
ൌ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ
௔మି௫భ
௫మୀ଴
ቑ
௥భିଵ
௫భୀ௔భାଵ
െ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ
௔మି௫భ
௫మୀ଴
ቑ
௥భିଵ
௫భୀ௔భାଵ
… (20)
We can find [ ∆௥భ
ܴଶሺ‫ݎ‬ଵሻ] as;
∆
௥భ
ܴଶሺ‫ݎ‬ଵሻ ൌ ݃௡భ
ሺ‫ݎ‬ଵሻቐ݊ଶߜቂ‫݌‬௡భ
ሺ௥భሻ
ቃ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃௡మ
ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻቂ‫݌‬௡భା௡మ
ሺ௥భ, ௫మሻ
െ ‫ݎ݌‬ቃ
௔మି௥భ
௫మୀ଴
ቑ
… (21)
And also;
∆
௔భ
ܴଶሺܽଵሻ ൌ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ
௔మି௫భ
௫మୀ଴
ቑ
௥భିଵ
௫భୀ௔భାଶ
െ ෍ ݃ሺ‫ݔ‬ଵሻ ቐ݊ଶߜሾ‫݌‬ሺ‫ݔ‬ଵሻሿ ൅ ሺܰ െ ݊ଵ െ ݊ଶሻ ෍ ݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻሾ‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ െ ‫ݎ݌‬ሿ
௔మି௫భ
௫మୀ଴
ቑ
௥భିଵ
௫భୀ௔భାଵ
… (22)
And also the difference of [ܴଶሺܽଵሻ] with respect to ሺܽଵሻ;
∆௔భ
ܴଶሺܽଵሻ ൌ ݃௡భ
ሺܽଵ ൅ 1ሻ ቄ݊ଶሺ‫ܣ‬ଶ െ ܵଵሻቂ‫݌‬௡భ
ሺ௔భାଵሻ
ቃ െ ሺܵଵ െ ‫ܣ‬ଵሻ|ሺ‫ܣ‬ଶ െ ܴଶሻ ൅ ݃௡భ
ሺܽଵ ൅
1ሻሺܰ െ ݊ଵ െ ݊ଶሻ ∑ ݃௡మ
ሺ‫ݔ‬ଶ|ܽଵ ൅ 1ሻቂ‫݌‬௡భା௡మ
ሺ௔భାଵା ௫మሻ
െ ‫ݎ݌‬ቃ
௡మ
௫మୀ௔మି௔భ
ቅ …………. … (23)
From ∆௔మ
ܴଶሺܽଶሻ we can find the value of parameter (ܽଶ) for given (݊ଵ, ݊ଶ ) which satisfy
the inequality (24) follows.
‫̂݌‬௡భା௡మ
ሺܽොଶሻ ൑ ‫ݎ݌‬ ൑ ‫̂݌‬௡భା௡మ
ሺܽොଶ ൅ 1ሻ …………………………………….. (24)
Then;
‫̂ݎ‬ଵ ൌ ܽොଶ ൅ 1 ‫݄݊݁ݓ‬ ݇‫ݏ‬ ൌ ݇‫ݎ‬
‫̂ݎ‬ଵ ൑ ܽොଶ ൅ 1 ݂‫ݎ݋‬ ݇‫ݏ‬ ൐ ݇‫ݎ‬

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And when [ܽଶ ൌ ܽଵ], the second term of ∆௔భ
ܴଶሺܽଵሻ is positive, so we assume that
[ܽଵ ൌ ܽଵ
‫כ‬
] is the value at which the first term of ∆௔మ
ܴଶሺܽଶሻ is positive therefore
∆ ܴଶሺܽଵ
‫כ‬ሻ ൐ 0 and when ݇‫ݏ‬ ൌ ݇‫ݎ‬ ฺ ሺ‫̂ݎ‬ଵ ൌ ܽොଶ ൅ 1ሻ, and we see that the average cost of
inspection equal to average cost of reject, therefore we need only to find the parameters
(ܽଵ, ܽଶ ) and from ∆௔భ
ܴଶሺܽଵሻ we obtain ‫݌‬௡భ
ሺܽොଵ ൅ 1ሻ ൑ ‫ݎ݌‬ subject to ‫ݎ‬ଵ ൑ ݊ଵ ൅ 1.
After completing model for double Bayesian sampling inspection plan, this model is
applied to distribution of Beta – Binomial.
From ∆௔మ
ܴଶሺܽଶሻ we can find (ܽଶ) that satisfy the inequality (24).
‫݌‬௡భା௡మ
ሺܽොଶሻ ൑ ‫ݎ݌‬ ൑ ‫݌‬௡భା௡మ
ሺܽොଶ ൅ 1ሻ ………………………………………. (25)
When (ܽଵ ൌ ܽଶ), the second term of ∆௔భ
ܴଶሺܽଵሻ is positive, so we assume that (ܽଵ ൌ ܽଵ
‫כ‬
) is
the value of which the first term of ∆௔మ
ܴଶሺܽଶሻ is positive therefore ∆ ܴଶሺܽଵ
‫כ‬
ሻ ൐ 0.
When;
݇‫ݏ‬ ൌ ݇‫ݎ‬
Then;
∆ ܴଶሺ‫ݎ‬ଵሻ ൑ 0 ฺ ‫̂ݎ‬ଵ ൌ ܽොଶ ൅ 1
And for ݇‫ݏ‬ ൐ ݇‫ݎ‬ ฺ ‫̂ݎ‬ଵ ൑ ܽොଶ ൅ 1
And when ݇‫ݏ‬ ൌ ݇‫,ݎ‬ we saw that average cost of inspection equal average cost of reject,
therefore we need only to find the parameters (ܽଵ, ܽଶ), and from equation ∆௔భ
ܴଶሺܽଵሻ, we
obtain ‫݌‬௡భ
ሺܽොଵ ൅ 1ሻ ൑ ‫ݎ݌‬ subject to (‫ݎ‬ଵ ൑ ݊ଵ ൅ 1).
4 - APPLICATION OF PROPOSED MODEL
After the total cost function are built, it is necessary to apply this model for sampling
distribution to show how the parameters of double sampling plans are obtained, to take a
decision for accept or reject, assume (‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ ) are three independent random variables
follow Binomial distribution as:
‫ݔ‬ଵ~ܾሺ݊ଵ, ‫݌‬ሻ
‫ݔ‬ଶ~ܾሺ݊ଶ, ‫݌‬ሻ
‫ݔ‬ଷ~ܾሺ݊ଷ, ‫݌‬ሻ
And (‫)݌‬ is random variable varied from lot to lot, and it's prior distribution [݂ሺ‫݌‬ሻ] is
assumed here ‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ, (Beta prior), so the joint probability distribution of (‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ ) is;
‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶ, ‫ݔ‬ଷ ሻ ൌ න ‫ܥ‬௫భ
௡భ
ଵ
଴
‫݌‬௫భ‫ݍ‬௡భ ି ௫భ‫ܥ‬௫మ
௡మ
‫݌‬௫మ‫ݍ‬௡మ ି ௫మ‫ܥ‬௫య
௡య
‫݌‬௫య‫ݍ‬௡య ି ௫య
1
‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ
‫݌‬ௌିଵ
‫ݍ‬௧ି ଵ
݀‫݌‬
ൌ
஼ೣభ
೙భ஼ೣమ
೙మ஼ೣయ
೙య
஻௘௧௔ ሺ௦,௧ሻ
‫׬‬ ‫݌‬௫భା௫మା௫యାௌିଵଵ
଴
‫ݍ‬௡భ ା௡మ ା௡య ି ௫భି௫మି௫యା௧ିଵ
݀‫݌‬
ൌ
஼ೣభ
೙భ஼ೣమ
೙మ஼ೣయ
೙య
஻௘௧௔ ሺ௦,௧ሻ
‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ൅ ‫ݔ‬ଷ ൅ ܵ, ݊ଵ ൅ ݊ଶ ൅ ݊ଷ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶ െ ‫ݔ‬ଷ ൅ ‫ݐ‬ሻ …. (26)
Also we can find the posterior distribution ݂ሺ‫ݔ|݌‬ଵሻ and ݂ሺ‫ݔ|݌‬ଵ, ‫ݔ‬ଶሻ ൌ ݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶሻ, and;
‫݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ ‫׬‬ ܾሺ‫ݔ‬ଶ, ݊ଶ, ‫݌‬ሻ݂ሺ‫ݔ|݌‬ଵሻ ݀‫݌‬
ଵ
଴
Since;
݂ሺ‫ݔ|݌‬ଵሻ ൌ
݆‫ݐ݊݅݋‬ ሺ‫,݌‬ ‫ݔ‬ଵሻ
݃ሺ‫ݔ‬ଵሻ
݃ሺ‫ݔ‬ଵሻ ൌ න ‫ܥ‬௫భ
௡భ
ଵ
଴
‫݌‬௫భ‫ݍ‬௡భ ି ௫భ
1
‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ
‫݌‬ௌିଵ
‫ݍ‬௧ି ଵ
݀‫݌‬

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݃ሺ‫ݔ‬ଵሻ ൌ
‫ܥ‬௫భ
௡భ
‫ܽݐ݁ܤ‬ ሺ‫,ݏ‬ ‫ݐ‬ሻ
‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଵ ൅ ܵ, ݊ଵ െ ‫ݔ‬ଵ ൅ ‫ݐ‬ሻ
And from equation below which is Beta distribution;
݂ሺ‫ݔ|݌‬ଵሻ ൌ
௚ሺ௣,௫భ ሻ
௚ሺ௫భሻ
The conditional distribution of (‫)݌‬ given (‫ݔ‬ଵ) is;
݂ሺ‫ݔ|݌‬ଵሻ ൌ
ଵ
஻௘௧௔ሺ௫భାௌ,௡భ ି ௫భା௧ሻ
‫݌‬௫భାௌିଵ
‫ݍ‬௡భ ି ௫భା௧ିଵ
…………………….. (27)
And;
‫ܧ‬ሺ‫ݔ|݌‬ଵሻ ൌ
‫ݔ‬ଵ ൅ ܵ
݊ଵ ൅ ܵ ൅ ‫ݐ‬
Also we can find ‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ;
‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ ‫׬‬ ܾሺ‫ݔ‬ଶ, ݊ଶ, ‫݌‬ ሻ
ଵ
଴
݂ሺ‫ݔ|݌‬ଵሻ݀‫݌‬ ………………………………….. (28)
ൌ
ଵ
௞
‫׬‬ ‫ܥ‬௫మ
௡మଵ
଴
‫݌‬௫మ‫ݍ‬௡మ ି ௫మ‫݌‬௫భାௌିଵ
‫ݍ‬௡భ ି ௫భା௧ିଵ
݀‫݌‬
‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ
ଵ
௞
‫ܥ‬௫మ
௡మ
‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶ ൅ ‫ݐ‬ሻ
‫ݎ݌‬ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ ൌ
ௌା௫భ
ௌା௫భା௫మ
஼ೣమ
೙మ஼ೄశೣభ
೙భశ೟శೄషభ
஼ೄశೣభశೣమ
೙భశ೙మశ೟శೄషభ ………………………………………. (29)
Which is [݃ሺ‫ݔ‬ଶ|‫ݔ‬ଵሻ]
Assume ‫ݔ‬ଷ ൌ ܺ െ ‫ݔ‬ଵ െ ‫ݔ‬ଶ
‫ݔ‬ଷ~ ‫݈ܽ݅݉݋݊݅ܤ‬ ሺ݊ଷ, ‫݌‬ሻ
Then the conditional distribution of (‫ݔ‬ଷ) given (‫ݔ‬ଵ, ‫ݔ‬ଶ) is;
‫ݎ݌‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ‫׬‬ ܾሺ‫ݔ‬ଷ, ݊ଷ, ‫݌‬ሻ
ଵ
଴
݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶሻ݀‫݌‬ ……………………….. (30)
To solve equation (30), we need to find ݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶሻ;
Let ‫ݕ‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ
‫׶‬ ‫ݔ‬ଵ~ ܾሺ݊ଵ, ‫݌‬ሻ
‫ݔ‬ଶ~ ܾሺ݊ଶ, ‫݌‬ሻ
And are independent, then;
‫ݕ‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶ ~ ‫ܤ‬ሺ݊ଵ ൅ ݊ଶ, ‫݌‬ሻ
݃ሺ‫,݌‬ ‫ݕ‬ሻ ൌ ‫ܥ‬௬
௡భା௡మ
‫݌‬௬
‫ݍ‬௡భା௡మି௬
1
‫ܤ‬ሺܵ, ‫ݐ‬ሻ
‫݌‬ௌିଵ
‫ݍ‬௧ିଵ
‫׵‬ ݃ሺ‫݌|ݕ‬ሻ ൌ
1
‫ܤ‬ሺܵ, ‫ݐ‬ሻ
න ‫ܥ‬௬
௡భା௡మ
‫݌‬௬ାௌିଵ
‫ݍ‬௡భା௡మି௬ା௧ିଵ
݀‫݌‬
ଵ
଴
݃ሺ‫݌|ݕ‬ሻ ൌ
1
‫ܤ‬ሺܵ, ‫ݐ‬ሻ
‫ܥ‬௬
௡భା௡మ
‫ܽݐ݁ܤ‬ ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݕ‬ ൅ ‫ݐ‬ሻ
‫׵‬ ݂ሺ‫ݕ|݌‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ ൌ
݆‫ݐ݊݅݋‬
݉ܽ‫݈ܽ݊݅݃ݎ‬
ൌ
భ
ಳሺೄ,೟ሻ
஼೤
೙భశ೙మ௣೤శೄషభ௤೙భశ೙మష೤శ೟షభ
భ
ಳሺೄ,೟ሻ
஼೤
೙భశ೙మ஻௘௧௔ ሺ௬ାௌ,௡భା௡మି௬ା௧ሻ
ൌ
ଵ
஻௘௧௔ ሺ௬ାௌ,௡భା௡మି௬ା௧ሻ
‫݌‬௬ାௌିଵ
‫ݍ‬௡భା௡మି௬ା௧ିଵ
0 ൑ ‫݌‬ ൑ 1 …………. (31)
Then ݂ሺ‫ݕ|݌‬ ൌ ‫ݔ‬ଵ ൅ ‫ݔ‬ଶሻ is distributed as Beta distribution with parameters (‫ݕ‬ ൅ ܵ, ݊ଵ ൅
݊ଶ െ ‫ݕ‬ ൅ ‫.)ݐ‬ This distribution is necessary to find the distribution of number of defectives
ሺ‫ݔ‬ଷ) after drawing the first and second sample, i.e to find the distribution of ‫݌‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ.

9. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
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‫݌‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ න ܾሺ‫ݔ‬ଷ, ݊ଷ, ‫݌‬ሻ݂ሺ‫ݔ|݌‬ଵ ൅ ‫ݔ‬ଶ ൌ ‫ݕ‬ሻ݀‫݌‬
ଵ
଴
ൌ න ‫ܥ‬௫య
௡య
ଵ
଴
‫݌‬௫య‫ݍ‬௡యି ௫య
1
‫ܽݐ݁ܤ‬ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ ൅ ‫ݐ‬ െ ‫ݕ‬ሻ
‫݌‬௬ାௌିଵ
‫ݍ‬௡భା௡మା௧ି௬ିଵ
݀‫݌‬
ൌ
‫ܥ‬௫య
௡య
‫ܽݐ݁ܤ‬ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݕ‬ ൅ ‫ݐ‬ሻ
න ‫݌‬௫యା௬ାௌିଵ
ଵ
଴
‫ݍ‬௡భା௡మା௡యି௫యା௧ି௬ିଵ
݀‫݌‬
ൌ
‫ܥ‬௫య
௡య
‫ܽݐ݁ܤ‬ሺ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ െ ‫ݕ‬ ൅ ‫ݐ‬ሻ
‫ܽݐ݁ܤ‬ ሺ‫ݔ‬ଷ ൅ ‫ݕ‬ ൅ ܵ, ݊ଵ ൅ ݊ଶ ൅ ݊ଷ െ ‫ݔ‬ଷ ൅ ‫ݐ‬ െ ‫ݕ‬ሻ
ൌ
‫ܥ‬௫య
௡య
Γሺ‫ݔ‬ଷ ൅ ‫ݕ‬ ൅ ܵሻ Γ ሺ݊ଵ ൅ ݊ଶ ൅ ݊ଷ െ ‫ݔ‬ଷ െ ‫ݕ‬ ൅ ‫ݐ‬ሻ
Γ ሺ‫ݕ‬ ൅ ܵሻΓ ሺ݊ଵ ൅ ݊ଶ ൅ ‫ݐ‬ െ ‫ݕ‬ሻ
Γ ሺ݊ଵ ൅ ݊ଶ ൅ ‫ݐ‬ ൅ ܵሻ
Γ ሺܵ ൅ ‫ݐ‬ ൅ ݊ଵ ൅ ݊ଶ ൅ ݊ଷሻ
ൌ
஼ೣయ
೙యሺ௡భା௡మା௧ାୗିଵሻ!
ሺ௬ାௌିଵሻ!ሺ௡భା௡మା௧ି୷ିଵሻ!
ൈ
ሺ௫యା௬ାௌିଵሻ!ሺ௡భା௡మା௡యି௫యି௬ା௧ିଵሻ!
ሺௌା௧ା௡భା௡మା௡యିଵሻ!
ൌ
஼ೣయ
೙య஼೤శೄ
೙భశ೙మశ೟శೄషభ
ሺ௫యାௌା௬ሻ
஼ೣయశೄశ೤
೙భశ೙మశ೙యశ೟శೄషభ
ሺ௬ାௌሻ
Which is mixed hypergeometric distribution.We can obtain the conditional mean of ሺ‫ݔ‬ଷሻ
given ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ;
‫ܧ‬ሺ‫ݔ‬ଷ|‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ݊ଷ‫݌‬௡భା௡మ
ሺ௫భ,௫మሻ
Where,
‫݌‬ሺ‫ݔ‬ଵ, ‫ݔ‬ଶሻ ൌ ‫݌‬௡భା௡మ
ሺ௫భା௫మሻ
5- CASE STUDY
The following data represents the percentage of defectives of (140) lots from some
industrial companies in Iraq, these percentages are grouped in frequency table as follows;
Table (1): The frequency table for percentage of defectives of 140 lots
Percentage of
defectives
Observed
frequency ሺ݂௜ሻ
Class mark
‫݌‬௜ ‫݌‬௜݂௜ ‫݌‬௜
ଶ
݂௜
0.0 - 6 0.01 0.06 0.0116
0.02 - 13 0.03 0.39 0.0117
0.04 - 17 0.05 0.85 0.0425
0.06 - 24 0.07 1.68 0.1176
0.08 - 18 0.09 1.62 0.1456
0.10 - 16 0.11 1.76 0.1936
0.12 - 15 0.13 1.95 0.2535
0.14 - 10 0.15 1.50 0.2250
0.16 - 9 0.17 1.53 0.2601
0.18 - 7 0.19 0.33 0.2527
0.20 - 3 0.21 0.63 0.1323
0.22 – 0.24 2 0.23 0.46 0.1058
Total 140 13.699 1.7413

10. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME
291
Then we draw the histogram and frequency curve, the curve indicate it is Beta distribution
with parameters (S, T), which are estimated by moments method using the equations;
ܵመ ൌ ‫ݔ‬௣ሾ ‫ݔ‬௣‫ݔ‬௤ െ ܵ௣
ଶ
ሿหܵ௣
ଶ
ܶ෠ ൌ ‫ݔ‬௤ሾ ‫ݔ‬௣‫ݔ‬௤ െ ܵ௣
ଶ
ሿหܵ௣
ଶ
Where;
‫ݔ‬௣ ൌ
∑ ௣೔௙೔
∑ ௙೔
ൌ 0.09785
ܵ௣
ଶ
ൌ
∑ ௣೔
మ
௙೔
∑ ௙೔
െ ‫݌‬
ଶ
ൌ 0.00286
Then;
ܵመ௠௢௠ ൌ 2.9 ؆ 3 ܶ෠௠௢௠ ൌ 26.9 ؆ 27
Then the distribution of (‫݌‬௜ሻ is tested using ( ߯ଶ
) test, where the cumulative probabilities
are computed from;
‫ܨ‬ሺ‫݌‬ሻ ൌ ‫׬‬
ଵ
஻ሺௌ,ఛሻ
௣
଴
‫ݑ‬ௌିଵ
ሺ1 െ ‫ݑ‬ሻ்ିଵ
݀‫ݑ‬
ൌ ‫ܧ‬ሺܵ, ܵ ൅ ܶ െ 1, ‫݌‬ሻ ൌ ‫ݎ݌‬ሺ‫ݔ‬ ൐ ܵሻ ൌ ∑ ‫ܥ‬௫
ௌା்ିଵௌା்ିଵ
௫ୀௌ ‫݌‬௫
‫ݍ‬ௌା்ି௫ିଵ
Where;
ܵመ ൌ 3 , ܶ෠ ൌ 27
The hypothesis tested is;
‫ܪ‬௢: ‫݌‬ ~ ‫ܽݐ݁ܤ‬ ሺܵ, ܶሻ
‫ܪ‬ଵ: ‫݌‬ ‫؂‬ ‫ܽݐ݁ܤ‬ ሺܵ, ܶሻ
Using Chi – square test, which depend on the following formula;
߯ଶ
ൌ ෍
ሺܱ௜ െ ‫ܧ‬௜ሻଶ
‫ܧ‬௜
௡
௜ୀଵ
The following table represents the necessary computations for goodness of fit.
Table (2): The necessary computations for goodness of fit
Proportion of
defectives
Observed
frequency (ܱ௜ሻ
‫ݎ݌‬ሺ‫݌‬ଵ ൏ ‫݌‬ ൏ ‫݌‬ଶሻ
‫ܧ‬௜
ሺܱ௜ െ ‫ܧ‬௜ሻଶ
‫ܧ‬௜
0.0 - 6 0.0231 3.234 2.3657
0.02 - 13 0.0974 13.636 0.0296
0.04 - 17 0.1662 23.268 1.6384
0.06 - 24 0.1612 22.568 0.0908
0.08 - 18 0.1500 21.00 0.4285
0.10 - 16 0.1252 17.528 0.1332
0.12 - 15 0.0827 11.578 1.0114
0.14 - 10 0.0715 10.01 0.00001
0.16 - 9 0.0435 6.09 1.2904
0.18 - 7 0.0373 5.222 0.0605
0.20 - 3 0.0252 3.528 0.0790
0.22 – 0.24 2 0.0167 2.338 0.0488
Total 140 1 140 7.1931

11. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
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292
Comparing [ ߯௖௔௟
ଶ
ൌ 7.1931ሿ with ሾ߯௧௔௕ሺ଴.଴ହ,଼ሻ
ଶ
ൌ 15.507ሿ, we find that, ߯௖௔௟
ଶ
൏ ߯௧௔௕
ଶ
, we
accept the hypothesis that, ‫݌‬௜ ~ ‫ܽݐ݁ܤ‬ distribution with parameters ሺS,Tሻ. Then the prior
distribution of percentage of defective is Beta with ܵመ ൌ 3 , ܶ෠ ൌ 27.
After we know that the prior distribution of percentage of defective ሾ݂ሺ‫݌‬ሻሿ is Beta –
prior with the estimated parameters ሺܵመ, ܶ෠ ሻ we also obtain the parameters of quality
control cost for each produced units, which are;
‫ܣ‬ଵ ൌ 0, ܴଵ ൌ 10.000 ‫ܦܫ‬
ܵଵ ൌ 10,000 ‫ܦܫ‬
ܴଶ ൌ 28,000 ‫ܦܫ‬
ܵଶ ൌ 28,000 ‫ܦܫ‬
‫ܣ‬ଶ ൌ 153.000 ‫ܦܫ‬
‫ݎ݌‬ ൌ
ோభି஺భ
஺మିோమ
ൌ 0.08, which is the break even quality control point.
According to the above parameters we find the parameters of single Bayesian sampling
inspection plan from;
݊ଵ
‫כ‬
ൌ ߣଵ√ܰ ൅ ߣଶ
‫ܥ‬ଵ
‫כ‬
ൌ ݊ଵ
‫כ‬
‫ݎ݌‬ ൅ ‫ܤ‬ଵ
‫ܤ‬ଵ ൌ ܶ෠‫ݎ݌‬ െ ܵመ‫ݎݍ‬ െ
ଵ
ଶ
‫݌‬ ൌ
௣
௣௥
ൌ 1.2
‫ݎ݌‬ ൌ 0.08
And;
ߣଵ
ଶ
ൌ
௣ೝ
ೄ ௤ೝ
೅ሺ஺మିோమሻ
ଶ஻ሺௌ,்ሻሺ௞௦ି௞௠ሻ
݇‫ݏ‬ ൌ ܵଵ ൅ ܵଶ‫݌‬
݇݉ ൌ ݇‫ݎ‬ െ ሺ‫ܣ‬ଶ െ ܴଶሻ ‫׬‬ ሺ‫ݎ݌‬ െ ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬
௣௥
଴
ߣଶ is a constant obtained according to the prior distribution of percentage of defective
as;
ߣଶ ൌ ሾ3ሺܵ ൅ ܶሻଶ
െ 11ሺܵ ൅ ܶሻ െ 2 െ ሺ3ܵ െ 1ሻܵ|‫ݎ݌‬ െ ሺ3ܶ െ 1ሻܶ|‫ݎݍ‬ െ 1| ‫ݎݍݎ݌‬ሿ 12⁄
And
݇݉ ൌ ݇‫ݎ‬ െ ሺ‫ܣ‬ଶ െ ܴଶሻ ‫׬‬ ሺ‫ݎ݌‬ െ ‫݌‬ሻ݂ሺ‫݌‬ሻ݀‫݌‬
௣௥
଴
Since ܵመ ൌ 3 , ܶ෠ ൌ 27
ߣଵ ൌ 4.663, ߣଶ ൌ െ27, ܰ ൌ 200
The parameters of single Bayesian sampling inspection plan are;
݊‫כ‬
ൌ ߣଵ√ܰ ൅ ߣଶ ൌ 39
‫ܥ‬‫כ‬
ൌ ݊‫כ‬
‫ݎ݌‬ ൅ ‫ܤ‬ଵ ൌ 2
‫׵‬ parameters of single Bayesian sampling plan is; ሺ݊‫כ‬
ൌ 39, ‫ܥ‬‫כ‬
ൌ 2 ሻ, for different
values of ሾ ‫݌‬ ൌ 0.07ሺ0.05ሻ0.09ሿ for ሺܰ ൌ 200, 225, 250,275,300ሻ, the following table
summarize single sampling plans.

12. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME
293
Table (3): Single Bayesian sampling inspection plans
P 0.07 0.075 0.08 0.085 0.09
N ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ ݊ଵ ܿଵ ܴଵ
150 10 0 71.6 15 0 80.16 23 1 102.4 17 0 70.9 20 1 72.1
175 13 0 81.8 19 1 94.37 27 1 116.0 22 1 87.9 23 1 88.9
200 17 0 95.4 23 1 107.9 30 2 126.2 27 1 104.9 27 1 105.7
225 20 0 105.65 26 1 118.2 35 2 143.2 32 2 120.3 35 2 122.3
250 23 1 115.82 29 1 128.4 38 2 153.6 36 2 135.6 38 3 137.1
275 26 1 126.02 32 2 138.6 41 3 163. 6 40 2 149. 2 41 3 150.2
300 29 1 135.22 35 2 148.8 45 3 177.2 43 2 159.4 45 3 161.4
After we find the single Bayesian sampling inspection plans, from minimizing (ܴଵ), then
minimizing (ܴଶሻ gives the double Bayesian sampling inspection plans, which works on
minimizing total expected quality control cost, derived in equation (10) and then transformed
to standard cost function (ܴ2) in equation (14).
Table (4): Double Bayesian sampling inspection plans
P 0.07 0.075 0.08 0.085 0.09
N ݊ଶ ܽଶ ܴଶ ܴଵ⁄ ݊ଶ ܽଶ ܴଶ ܴଵ⁄ ݊ଶ ܽଶ ܴଶ ܴଵ⁄ ݊ଶ ܽଶ ܴଶ ܴଵ⁄ ݊ଶ ܽଶ ܴଶ ܴଵ⁄
150 7 1 1.00 17 1 0.996 20 2 1.008 18 2 0.996 20 2 0.94
175 13 2 0.9 21 2 0.986 25 2 1.06 21 2 0.986 22 2 0.96
200 16 1 0.88 23 2 0.97 28 2 0.99 26 2 0.977 25 3 0.97
225 19 2 0.85 25 2 0.85 32 2 0.97 30 3 0.967 30 3 0.99
250 20 2 0.80 29 2 0.83 36 2 0.88 32 3 0.933 39 3 1.04
275 23 1 0.75 31 3 0.79 40 3 0.86 36 3 0.942 40 3 1.65
300 30 1 0.74 36 3 0.75 42 3 0.85 38 4 0.931 45 4 1.85
From the data we find the mean of percentage of defective is ‫݌‬ ൌ 0.09785, ‫ݎ݌‬ ൌ
0.08, ܰ ൌ 200. The single Bayesian sampling inspection plan is found ሺ݊ଵ
‫כ‬
ൌ 39, ‫ܥ‬ଵ
‫כ‬
ൌ 2ሻ,
which defined as drawing at random from lot (ܰ ൌ 200ሻ, a sample of size ሺ݊ଵ
‫כ‬
ൌ 39ሻ, and
testing it's units, if the number of defective is found (‫ܥ‬ଵ
‫כ‬
ൌ 2ሻ, then accept from first sample
(‫ݔ‬ଵ ൑ 2ሻ but if ሺ‫ݔ‬ଵ ൒ 2) then reject from first sample and also stop, but according to the
model, we find the single Bayesian sampling inspection plans (݊ଵ, ‫ܥ‬ଵ, ܴଵሻ as tabulated in
table (5), then the second Bayesian sampling inspection plans are found and put in table (4).
CONCLUSION
1- Finding plan which minimize the linear combination of two expected cost function
for first sampling and second sampling required a procedure carried out by means of
difference equation, since the cost function is of discrete type, and the minimization is done
in two stages to find (ܴଵሻ and ሺܴଶ).
2- We try also to find a family of optimum plans through maximizing the OC curve, but it
depend on probability of sampling under double sampling plans, and do not consider the
parameters of cost function.
3- The model was built under Beta – Binomial distribution since the distribution of process is
Binomial (݊, ‫݌‬ሻ, and ሺ‫)݌‬ is random variable have [݂ሺ‫݌‬ሻሿ, and it found to be Beta (ܵ, ܶ).
4- The parameters of [݂ሺ‫݌‬ሻ], (ܵ, ܶሻ are estimated by moments method and it can be estimated
also by any other statistical method.

13. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME
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5- The decision to accept or reject obtained from second sample is more effective than the
decision just obtained from first sampling especially for expensive units.
6- The producer risk for the studied company was (‫ܤ‬ ൌ 0.10), but it is found greater than this
number, because of bad station of production in Iraq, but in spite of this the company work to
satisfy the required quality control level.
7- The percentage of defectives in 140 lot are varied from to lot, so the distribution of these
percentage is considered, and it is found to be Beta prior with estimated parameters (ܵመ, ܶ෠),
therefore the model is build for Bayesian sampling plan.
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