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

3.1 Bayesian single sampling multiattribute plans for continuous
prior distribution
3.1.1 Scope
In this chapter it will be assumed that the process average for each attribute has a continuous
prior distribution. We shall examine in particular the problems of choice of a theoretical
distribution as relevant to a multiattribute situation.
In chapter 2.1 we have developed the expression for the generalized cost function at a
given process average. We make use of this to obtain the expression for the average cost
when the process averages follow independent continuous distributions. In particular, we
consider the case when the process average for each attribute can be assumed to follow a
gamma distribution. Further, we demonstrate how this expression can be used to compare
the expected costs of an A kind, D kind and C kind plan.
3.1.2 The cost model for continuous prior
We recall from chapter 2.1 that the average costs for accepted and rejected product at p is
expressed as:
[
]
r∑
r∑
r∑
K(N, n, p) = n(S 0 + S i ) + (N − n) (A 0 + A i p i )P (p) + (R 0 + R i p i )Q(p)
i=1
The notation and interpretation used for the above expression are to be found in section
2.1.4. For our present discussion we rewrite the same as:
[(
) (
r∑
r∑
r∑
) ]
K(N, n, p) = n(S 0 + S i p i ) + (N − n) R 0 + R i p i + (A 1 − R 1 ) d i p i − d 0 P (p)
i=1
i=1
where, d 0 = (R 0 − A 0 )/(A 1 − R 1 ); d i = (A i − R i )/(A 1 − R 1 ) for i = 1, 2, ..., r.
...(3.1.1)
i=1
i=1
Let p i be distributed from lot to lot according to the prior distribution w i (p i ), i = 1, 2, ..., r
and the p i ’s are jointly independent. Then,
∫ ∫ [( r∑
) ]
K(N, n) = nk s +(N−n)k r +(N−n)(A 1 −R 1 ) ... d i p i − d 0 P (p)dw 1 (p 1 )dw 2 (p 2 )...dw r (p r )
∫p 1 p 2 p r i=1
where k s is the average cost of sampling over the prior, i.e.
∫ [
]
r∑
k s = ... (S 0 + S i p i )dw 1 (p 1 )dw 2 (p 2 )...dw r (p r )
∫p 1 p 2
∫p r i=1
i=1
122