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

m β (c 1 , c 2 , ρ) = m ′ ...(1.3.12)
the numerator approximating its limiting value m β (c 1 )/ρ and hence after a certain point
R(c 1 , c 2 , ρ, α, β) becomes an increasing function. [ Reference Figure 1.3.1 ]
5. But the rate of increase will decrease so that R(c 1 , c 2 , ρ, α, β) → m β (c 1 )/m 1−α (c 1 ) as
c 2 → ∞.
( The arguments provided above do not constitute a proof of the theorem. But, what is
being claimed is simple enough and the supporting arguments help in understanding the
main point.)
Thus, for a given c 1 there is a unique c 2 for which R(c 1 , c 2 , ρ, α, β) will be minimum. We
call this as c (c 1)
2 .
Note that c (c 1)
2 > c (c 1+1)
2
...(1.3.10)
and R(c 1 , c (c 1)
2 , ρ, α, β) > R(c 1 , c (c 1+1)
2 , ρ, α, β)
...(1.3.11)
Figure 1.3.1 presents the graph of R(c 1 , c (c 1)
2 , ρ, α, β) as a function of c 2 for α = 0.05, β =
0.10, ρ = 0.1 and c 1 = 1, 2, 3, 4.
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 , ρ, 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
the minimum sample size satisfying
m 1−α (c 1 , c 2 , ρ) = m
Construction Algorithm
Precisely we shall adopt the following steps.
Step 1. Choose c 1
Step 2. For c 2 < c (c 1)
2 , we check if R(c 1 , c 2 − 1, ρ, α, β) ≥ p ′ /p > R(c 1 , c 2 , ρ, α, β)
If it holds, then choose n = m β (c 1 , c 2 , ρ)/p ′ . Else,
Step 3. increase c 1 by one and go to step 1.
To facilitate the above tasks 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
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