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

From (2.1.3) and (2.1.4), it can be easily shown that:
(
K(N, n, p) = n S 0 + ∑ ) [(
S i p i + (N − n) A 0 + ∑ ) (
A i p i P (p) + R 0 + ∑
i
i
i
R i p i
)
Q(p)
]
,
where P (p) denotes the average probability of acceptance at p.
P (p) = ∑ ( )
r∏ n x
p i i (1 − p i ) n−x i
,
x∈A i=1
x i
and Q(p) = 1 − P (p).
...(2.1.5)
B) When the defect occurrences are mutually exclusive.
In this situation the expression for the probability of observing (x 1 , x 2 , ..., x r ) defective in
a sample of size n from a lot of size N, containing (X 1 , X 2 , ..., X r ) defectives of types
i = 1, 2, ..., r, will be multivariate hypergeometric as
( )( ) ( )( ) ( )
X1 X2 Xr N − X1 − X 2 ... − X r N
P r(x 1 , x 2 , ..., x r | X 1 , ..., X r ) =
...
/ .
x 1 x 2 x r n − x 1 − x 2 ... − x r n
...(2.1.6)
At any process average the joint probability distribution of (X 1 , X 2 , ..., X r ) can be assumed
to be multinomial such that
( )( ) ( )
N N − X(1) N − X(r−1)
( )
P r (X 1 , X 2 , ..., X r | p 1 , p 2 , ..., p r ) =
...
p X (N−X(r)
1
1 ...p Xr
)
r 1 − p(r) .
X 1 X 2 X r
.
X (i) = X 1 + X 2 + ... + X i and p (i) = p 1 + p 2 + ... + p i ; i = 1, 2, ..., r.
...(2.1.7)
From (2.1.6) and (2.1.7) it follows that average cost at p can be expressed as (2.1.5)
replacing P (p) by
P (p) =
∑
x 1 ,x 2 ,...,x r∈A
x (i) = x 1 + x 2 + ... + x i for i = 1, 2, ..., r.
( )( ) ( )
n n − x(1) n − x(r−1)
( )
...
p x (n−x(r)
1
1 ...p xr
r 1 − p(r) .
x 1 x 2 x r
...(2.1.8)
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