Bayesian multiple deferred state (BMDS-1) attribute sampling plan with gamma prior

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Bayesian multiple deferred state (BMDS-1) attribute sampling plan with gamma prior

K.Subbiah ^#1, M.Latha^#2

# 1. Research Scholar, Department of Statistics, Government Arts College, Udumalpet- 642126,Tamil Nadu, India,

Mobile: + 91-8608129595

# 2. Principal, Kamarajar Government Arts College, Surandai- 627859,Tamil Nadu, India, Mobile: + 91- 9566324923

ABSTRACT

Bayesian Acceptance Sampling approach is associated with utilization of prior process history for the selection of Distributions (viz., Gamma Poisson, Beta Binomial) to describe the random fluctuations involved in Acceptance Sampling. In this paper, OC values of BMDS are compared with the conventional values for finding the Multiple Deferred State –I Sampling Plan (C1, C2) involving probability of Acceptance using Gamma Prior Distribution.

Keywords:

Bayesian Multiple Deferred State-I (C₁, C₂), Conventional Plan, Gamma Prior, Operating Characteristic Curve, Probability of Acceptance.

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The 52 year evolution of Acceptance Sampling Plans had its beginning in 1924 with the creation of the inspection engineering department of the Western Electric Company, later called Bell laboratories.

Acceptance Sampling is the methodology that deals with procedures by which decisions to accept or reject the lot of items are based on the results of the inspection of samples. Special purpose of acceptance sampling inspection plans are tailored for special application as against general or Universal use.

For a complete discussion of acceptance sampling four general areas would have to be considered

1. Lot by Lot sampling by the method of attributes in which each unit of a sample is inspected on a go or no go basis for one or more characteristics.

2. Lot by Lot sampling by the methods of variables in which each unit of a sample is measured for a single characteristic such as weight.

3. Continuous sampling of a flow of units by the method of attributes or variables.

4. Special purpose plans including chain sampling, skip lot sampling, dependent stage sampling, deferred state sampling etc.,

Multiple deferred state sampling falls in to those sampling plans defined by (1) and (4) above.

MDS – 1 PLAN

Multiple deferred sate sampling procedures are natural extension of fixed deferred state sampling Plans. These procedures have been found to be appealing to most management in personal in addition to possessing desirable properties with respect to their operating characteristic curves. Multiple deferred state sampling Procedures are designated by MDS (r, b.m)

r = Maximum number of defectives for Unconditional acceptance (r ≥ 0)

b= Maximum number of additional defectives for conditional acceptance (b > 0) m= Number of future lots in which conditional acceptance is based on (m > 0)

• For each lot, draw a sample of size n and observe the number of non conforming units d.

• If d ≤ r, accept the lot; if d > r +b, reject the lot. If r+1 ≤ d≤ r+b, accept the lot if the consecutive m preceding lots were all accepted.

The OC function of the MDS (r,b,m) plan is given by the equation

P_a (p) = P _a,r (p) + [P _a,r+b (p) + P _a,r (p)] [P _a,r (p)]^m GAMMA POISSON DISTRIBUTION

Let x be the number of defects with p as expected number of defects per unit, and let the corresponding Poisson probability be denoted by

P (

n.c)

, (p) = e

^-np

(1)

(3)

convenience approximation by a Gamma density function f(p).

f (p)=

0<p<1, s, t>0 (2)

Thus, the average probability of acceptance is approximately obtained by

= (3)

=

(4)

this can also be written as

(5)

Where µ =s/t is the expectation of the Gamma distribution.

BAYESIAN MDS-1 PLAN

This table presented in this paper relates to the Generalized Bayesian MDS-1 sampling plan using Gamma Prior distribution.

The Average Probability of Acceptance is given by

P = Pc

1

+ [Pc

2

-Pc

1

][Pc

1

]

^m

(6) Where

CONSTRUCTION OF THE TABLE

The (Table 1 and 2) gives the Average Probability of Acceptance for Bayesian multiple deferred sampling plan-1 (BMDS-1) using Gamma Prior in given by the equation (6).

Tables (shown below) give the Average Probabilities of Acceptance for different values of nμ and the figures show the discriminating power of OC Curves for different values of m=1,2 for fixed value of s = 2,3,4,5,6,7,8,9.

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nu\s 2 3 4 5 6 7 8 9

0.02 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.03 0.99998 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.04 0.99996 0.99997 0.99998 0.99998 0.99998 0.99998 0.99998 0.99998 0.5 0.96481 0.96899 0.97126 0.97267 0.97364 0.97434 0.97488 0.97530 0.6 0.94657 0.95178 0.95469 0.95654 0.95783 0.95877 0.95949 0.96005 0.7 0.92539 0.93122 0.93459 0.93677 0.93829 0.93947 0.94029 0.94098 0.8 0.90192 0.90787 0.91142 0.91377 0.91543 0.91666 0.91762 0.91838 0.9 0.87677 0.88230 0.88573 0.88805 0.88971 0.89095 0.89192 0.89270 1 0.85048 0.85508 0.85807 0.86014 0.86165 0.86280 0.86370 0.86443 1.5 0.71552 0.70964 0.70668 0.70489 0.70371 0.70286 0.70223 0.70173 2 0.59375 0.57373 0.56206 0.55436 0.54888 0.54478 0.54159 0.53903 2.5 0.49328 0.46074 0.44127 0.42823 0.41887 0.41181 0.40629 0.39003 3 0.41282 0.37109 0.34607 0.32933 0.31732 0.30828 0.30122 0.29555 4 0.29766 0.24661 0.21679 0.19726 0.18349 0.17327 0.16539 0.15913 5 0.22302 0.17035 0.14086 0.12219 0.10937 0.10007 0.09303 0.08752 6 0.17272 0.12208 0.09506 0.07858 0.06762 0.05987 0.05412 0.04972 7 0.13751 0.09034 0.06638 0.05234 0.04330 0.03707 0.03257 0.02918 8 0.11198 0.06870 0.04775 0.03595 0.02860 0.02369 0.02021 0.01766 10 0.07836 0.04242 0.02657 0.01832 0.01351 0.01047 0.00842 0.00699 15 0.03950 0.01651 0.00826 0.00467 0.00289 0.00191 0.00133 0.00097 25 0.01586 0.00455 0.00161 0.00066 0.00030 0.00015 8.4247E-05 4.8613E-05

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0 5 10 15 20 25

0.2 0.4 0.6 0.8 1.0 1.2

nm

s=2 s=3 s=4 s=5 s=6 s=7 s=8 s=9

` ` R

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nu\s 2 3 4 5 6 7 8 9

0.02 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.03 0.99998 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.04 0.99996 0.99997 0.99979 0.99981 0.99998 0.99998 0.99998 0.99998 0.5 0.95765 0.96203 0.96443 0.96594 0.96698 0.96774 0.96831 0.96877 0.6 0.93551 0.94073 0.94369 0.94559 0.94692 0.94789 0.94864 0.94923 0.7 0.91001 0.91549 0.91872 0.92084 0.92233 0.92344 0.92429 0.92497 0.8 0.88206 0.88716 0.89029 0.89239 0.89390 0.89502 0.89590 0.89660 0.9 0.85249 0.85657 0.85923 0.86107 0.86241 0.86342 0.86422 0.86486 1 0.82202 0.82451 0.82634 0.82767 0.82867 0.82944 0.83006 0.83056 1.5 0.67257 0.66195 0.65616 0.65249 0.64995 0.64809 0.64667 0.64555 2 0.54687 0.52202 0.50753 0.49797 0.49119 0.48611 0.48217 0.47902 2.5 0.44881 0.41318 0.39216 0.37823 0.36830 0.36085 0.35506 0.34012 3 0.37341 0.33081 0.30580 0.28930 0.27758 0.26882 0.26202 0.25660 4 0.26921 0.22070 0.19299 0.17508 0.16257 0.15334 0.14626 0.14065 5 0.20316 0.15445 0.12763 0.11077 0.09926 0.09024 0.08462 0.07969 6 0.15882 0.11233 0.08773 0.07277 0.06282 0.05576 0.05053 0.04651 7 0.12762 0.08425 0.06226 0.04934 0.04099 0.03522 0.03103 0.02787 8 0.10483 0.06481 0.04538 0.03437 0.02747 0.02283 0.01954 0.01711 10 0.07439 0.04072 0.02572 0.01784 0.01321 0.01027 0.00829 0.00688 15 0.03831 0.01620 0.00816 0.00463 0.00287 0.00190 0.00133 0.00097 25 0.01565 0.00452 0.00161 0.00066 0.00030 0.00015 8.4226E-05 4.8606E-05

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0 5 10 15 20 25

0.0 0.2 0.4 0.6 0.8

1.0

s=2

s=3 s=4 s=5 s=6 s=7 s=8 s=9

nm

` R

CONCLUSION:

In order to study the effect of acceptance s, some sets of curves are drawn for different combinations of n and p which are given in figure 1 and figure 2. In each figure, there are eight curves which represent the case of s=2 to 9. For moderately large values of s=7, 8, 9, the steepness of the OC Curves are increasing.

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©2016 RS Publication, [email protected] Page 115 A slight deviation from the situation (s=2, 3, 4, 5, 6) may be observed for higher values of s=7, 8, 9, which gives protection to the producer and consumer.

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