Multiattribute acceptance sampling plans - Library(ISI Kolkata ...

g(X i , N ¯p i , s i ) = Γ(s i + X i )
θ s i
i .(1 − θ i ) X i
s i
; θ i =
X i !Γs i
(s i + N ¯p i )
and X i non-negative integer.
...(3.1.6)
[Note that we use g(x, θ) to denote the Poisson distribution term and g(x, θ 1 , θ 2 , ) to denote
the gamma-Poisson term.]
It follows that the distribution of the number of defectives or defects x i in the sample
folows gamma-Poisson law with probability mass function given by g(x i , n ¯p i , s i ). Further
there is stochastic independence of x 1 , x 2 , ...., x r . This is the framework for development of
cost models in the present chapter. In the sections which follow, we use gamma prior distribution
which work either approximately accurately or as exact distributions under different
situations as explained.
3.1.4 The verification for the appropriate prior distribution
Table 3.1.1 presents the inspection data for 86 lots containing about 25500 pieces of filled
vials of an eye drop produced by an established pharmaceutical company based at Kolkata.
Each vial is inspected for six attributes. From the criticality point of view, however, the
defects can be grouped in two categories. The first category of defects is due to the presence
of foreign matters viz. glass, fiber or impurities, which are critical from the user’s point
of view. The other category of defects consists of breakage, defective sealing and leakage,
reasonably obviously detectable and can be easily discarded by the user.
Since the lot size ( 25500 ) is quite large, the observed lot quality variation approximates
almost exactly the distribution of process average. We have therefore, instead of a gamma-
Poisson distribution, fitted a gamma distribution with mean and the variance estimated from
the observed data for both types of defects.The results are presented in Table 3.1.2 and table
3.1.3.
Note that for both the attributes the ¯p i ’s are less than 0.1 ( ¯p 1 = 0.01708; ¯p 2 = 0.02302).
The estimated s i = ¯p i 2 /V ar(p i ) are 3.4532 and 0.6229 respectively so that ¯p i /s i are much
less than 0.2 justifying the use of gamma distribution as a substitute of beta distribution.
The tables also present the usual χ 2 goodness of fit analysis. It can be seen that the
computed χ 2 as a measure of the goodness of fit values are small (they are not statistically
significant) enough for both types of attributes to justify the assumption that the prior
distributions follow the assumed theoretical gamma distributions.
Further, the scatter plot (see figure 3.1.1) of the observed numbers of defects of the second
category against those of the first category exhibits no specific pattern. It would be therefore
125