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

show that if we increase c 2 keeping c 1 as fixed then as c 2 → ∞, R(c 1 , c 2 , ρ, α, β) first decreases
to a minimum and then increases and go on increasing so that the function has a unique
minimum in the range m > 0. We call this as c (c 1)
2 . We have further noted that c (c 1)
2 > c (c 1+1)
2 .
The task is now to find uniquely the value of c 1 , and c 2 such that: R(c 1 , c 2 − 1, ρ, α, β) >
p ′ /p ≥ R(c 1 , c 2 , ρ, α, β) for the smallest value of c 1 . Since c 1 , c 2 are all integers we must
consider the set of c 1 , c 2 values for which R(c 1 , c 2 , α, β) is less than p ′ /p and choose the
c 1 and c 2 for which m β (c 1 , c 2 , ρ) is minimum. This will ensure us the minimum sample
size obtained as n = m β (c 1 , c 2 , ρ)/p ′ , where p ′ = p ′ 1 + p ′ 2, satisfying the stipulations of the
producer’s and consumer’s risks at specified points.
We could, therefore develop a step by step procedure to obtain the desired sample plan. To
facilitate the above task we may construct a table containing c 1 , c 2 , m β (c 1 , c 2 , ρ), m 1−α (c 1 , c 2 , ρ)
arranged in descending order of R(c 1 , c 2 , ρ, α, β) for a given α, β, and ρ. We present the table
for α = 0.05, β = 0.10 and ρ = 0.1.
For the A kind plans of given strength we want to satisfy the equations : P A(a 1 , a 2 ; np 1 , np 2 ) =
1 − α; P A(a 1 , a 2 ; np ′ 1, np ′ 2) = β; P A(a 1 , a 2 ; np 1 , np 2 ) denotes the probability of acceptance
at (p 1 , p 2 ). We restrict to the situation where p 1 /(p 1 + p 2 ) = p ′ 1/(p ′ 1 + p ′ 2) = ρ. For a given
x 1 ∑=a 1
ρ we define ma P (a 1 , a 2 , ρ) as the value of m satisfying the equation : g(x 1 , mρ)G(a 2 −
x 1 , m(1−ρ)) = P. We have to obtain n, a 1 , a 2 for a given ρ, α and β such that ma 1−α (a 1 , a 2 .ρ) =
m and ma β (a 1 , a 2 , ρ) = m ′ .
As before we introduce the auxiliary function Ra(a 1 , a 2 , ρ, α, β) = ma β (a 1 , a 2 , ρ)/ma (1−α) (a 1 , a 2 , ρ).
We find that ma P (a 1 , a 2 , ρ) is an increasing function of a 1 , a 2 ; Ra(a 1 , a 2 , ρ, α, β) is a decreasing
function of a 1 and a 2 , keeping the other parameters fixed.
We obtain the smallest a 2 = a ∗ 2 for which Ra(0, a ∗ 2, ρ, α, β) > p ′ /p ≥ Ra(a ∗ 2, a ∗ 2, ρ, α, β) and
then find a ∗ 1 such that, Ra(a ∗ 1 − 1, a ∗ 2, ρ, α, β) > p ′ /p ≥ Ra(a ∗ 1, a ∗ 2, ρ, α, β);. and the desired
value of n is obtained by dividing the value of ma β (a ∗ 1, a ∗ 2, ρ) by p ′ .
To facilitate this task we construct a table containing a 1 , a 2 , ma β (a 1 , a 2 , ρ), ma 1−α (a 1 , a 2 , ρ)
arranged in descending order of Ra(a 1 , a 2 , α, β, ρ). For α = 0.05, β = 0.10 and ρ = 0.1 the
result has been presented in this chapter.
We now introduce a MASSP of D kind as the one with the following rule : from each lot
of size N, take a sample of size n, accept if total number of defects of all types put together
is less than or equal to k, otherwise reject the lot.
For obtaining the D type MASSP of given strength, we use the fact that under Poisson
conditions the OC at (p 1 , p 2 , ..., p r ) is identical with that of single sampling plan for single
attribute with sample size n and with acceptance number k, at p = p 1 +p 2 +..+p r . Thus, the
fixed risk D type plan satisfies Rd(k − 1) > p ′ /p ≥ Rd(k) where Rd(k) = md β (k)/md 1−α (k);
G(k, md P ) = P. We obtain n from n = md β (k)/p ′ ; p ′ = p ′ 1+p ′ 2+...+p ′ r. p = p 1 +p 2 +...+p r .
These plans have been extensively tabulated by Hald(1981) for different values α, β, and p ′ /p
x 1 =0
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