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

1.3 Multiattribute sampling plans of given strength
1.3.1 Scope
We considered in the last chapter two schemes: one, consisting of plans of A kind and,
another consisting of plans of C kind, which differed in terms of acceptance criteria. In this
chapter we try to construct MASSP’s defined by two equations Q(p) = α and P (p ′ ) = β,
where p = (p 1 , p 2 , ..., p r ) is the process average vector for the satisfactory quality level and
p ′ = (p ′ 1, p ′ 2, ..., p ′ r) is the process average vector for an unsatisfactory quality level; Q(p)is
the probability of rejection at p and P (p ′ ) is the probability of acceptance at p ′ . Further, α
and β are the stipulated producer’s and consumer’s risks, respectively.
1.3.2 Single attribute situation
Before we start, we look at the method of obtaining the Poisson solution for sampling plan
of a given strength for the case of single attribute as outlined by Hald (1981). For the case
of single attribute, we define a sampling plan of strength (p, α, p ′ , β) as the plan satisfying
two equations Q(p) = α, and P (p ′ ) = β, p and p ′ denote the process average values for
the satisfactory and unsatisfactory quality levels, respectively. Allowing c and n to be real
numbers, these two equations determine (c, n) uniquely. However, in practice c and n are
integers. We therefore find a sampling plan such that the two conditions are satisfied as
nearly as possible and actual risks are less than or equal to the stipulated risks.
For this we define m P (c) as the value of m such that G(c, m P (c)) = P . Under Poisson
conditions the problem boils down to solving the two equations: G(c, np) = 1 − α and
G(c, np ′ ) = β. Let G(c, m P (c)) = P then we solve the equations
and
np = m 1−α (c)
np ′ = m β (c).
...(1.3.1)
The ratio R(c, α, β) = m β (c)/m 1−α (c) is a decreasing function of c for given (α, β).
We may therefore obtain c uniquely from the equation:
R(c, α, β) = p ′ /p
since c is an integer we obtain c from : R(c − 1, α, β) > p ′ /p ≥ R(c, α, β)
and obtain an integer n from m β (c)/p ′ ≤ n ≤ m 1−α (c)/p
...(1.3.2)
...(1.3.3)
...(1.3.4)
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