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

where, p = (p 1 , p 2 , ..., p r ) and p ′ = (p ′ 1, p ′ 2, ..., p ′ r) are the process average vectors at state
1( satisfacory) and state 2 (unsatisfactory) respectively for the two point prior distribution
of the process average. P () and Q() are probabilitiy of acceptance and the probability of
rejection respectively for the C plan for which the sample size is n and acceptance numbers
are (c 1 , c 2 , ..., c r ) with acceptance criteria as defined.
We now consider the optimal C kind plan obtained by minimizing the regret function value
for given γ 1 , γ 2 , p, p ′ and N.
Having obtained an optimal C kind Plan we construct two moment equivalent single samling
plans (SSP) (a 0 , n 0 ) and (a ′ 0, n ′ 0) such that, P (p) ≃ G(a 0 , n 0 p) and P (p ′ ) ≃ G(a ′ 0, n ′ 0p ′ )
where, p = p 1 + p 2 + ... + p r and p ′ = p ′ 1 + p ′ 2 + ... + p ′ r.
Note that n 0 < n and n ′ 0 < n from the results of section 2.2.5.
Case I
a 0 > a ′ 0, n 0 < n ′ 0
In this case suppose we choose SSP(a 0 , n 0 ).
Then Q(p) ≃ 1 − G(a 0 , n 0 p) and P (p) ≃ G(a ′ 0, n ′ 0) > G(a 0 , n 0 p ′ )
Given γ 1 , γ 2 , p, p ′ and N, let RE and RC denote the regret function values for the chosen
SSP and the optimal C plan respectively. Then
RC = n + (N − n)[γ 1 Q(p) + γ 2 P (p ′ )] = n + (N − n)γ 1 d
where,
and,
d = Q(p) + (γ 2 /γ 1 )P (p ′ )
RE = n 0 + (N − n 0 )[γ 1 (1 − G(a 0 , n 0 p)) + γ 2 G(a 0 , n 0 p ′ )] = n 0 + (N − n 0 )γ 1 d 1
where
d 1 = 1 − G(a 0 , n 0 p) + (γ 2 /γ 1 )G(a 0 , n 0 p ′ ).
Then d − d 1 > 0 in the region of our interest, d < 1 for γ 2 < γ 1 . The regret function value
for the SSP (a 0 , n 0 ) is
RE = n 0 + (N − n 0 )γ 1 d 1 ≤ n 0 + (N − n 0 )γ 1 d ≤ n + (N − n)γ 1 d.
Hence, RE ≤ RC.
Thus the SSP with a 0 and n 0 as acceptance number and sample size respectively will have
lesser regret value than the optimal C kind plan.
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