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

Let M 2 < M 1 . Then G(c 1 , M 1 ρ) < G(c 1 , M 2 ρ) and G(c 2 , M 1 (1 − ρ)) < G(c 2 , M 2 (1 − ρ)) <
G(c 2 + 1, M 2 (1 − ρ)).
This makes G(c 1 , M 1 ρ).G(c 2 , M 1 (1−ρ)) < G(c 1 , M 2 ρ).G(c 2 +1, M 2 (1−ρ)) which is not true.
Hence M 2 ≥ M 1 .
If M 3 < M 1 then G(c 1 + 1, M 3 ρ)) > G(c 1 , M 1 ρ)) and G(c 2 , M 3 (1 − ρ)) > G(c 2 , M 1 (1 − ρ))
This makes
G(c 1 , M 1 ρ).G(c 2 , M 1 (1 − ρ)) < G(c 1 , M 3 ρ).G(c 2 + 1, M 3 (1 − ρ))
which is not true. Hence M 3 ≥ M 1 .
We now introduce the auxiliary function
R(c 1 , c 2 , ρ, α, β) = m β (c 1 , c 2 , ρ)/m 1−α (c 1 , c 2 , ρ)
R(c 1 , c 2 , ρ, α, β) is conveniently taken to ∞ when c 2 = −1
...(1.3.9)
Theorem 1.3.1
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 so that the function has a unique minimum in the range m > 0
We show this by the following partly demonstrative arguments which are supported by
extensive numerical computations (not included in the dissertation for the sake of brevity).
1. From the lemma preceeding the theorem, we observe that if we increase c 2 keeping c 1
as fixed i.e. when c 2 → ∞ the function m 1−α (c 1 , c 2 , ρ) increases. But G(c 1 , mρ) must be
greater than or equal to 1 − α, else G(c 1 , mρ).G(c 2 , m(1 − ρ)) becomes less than 1 − α.
2. It also follows that m 1−α (c 1 , c 2 , ρ) → m 1−α (c 1 )/ρ as c 2 → ∞. Similarly, m β (c 1 )/ρ[m 1−α (c)
and m β (c) are the Poisson fractiles of appropriate strengths as defined in section 1.3.1 ]
3. However, m β (c 1 , c 2 , ρ) > m 1−α (c 1 , c 2 , ρ) always. And for a small increase of c 2 both
m β (c 1 , c 2 , ρ) and m 1−α (c 1 , c 2 , ρ) will increase by a small amount relative to their original
values and therfore their ratio will decrease in general. Thus for small values of c 2 ,
R(c 1 , c 2 , ρ, α, β) is a decreasing function of c 2 .[Reference figure 1.3.1 ]
4. With the increase in c 2 beyond some point the increase in m 1−α (c 1 , c 2 , ρ) will be very
small but increase in m β (c 1 , c 2 , ρ) will still be finite. As c 2 increases the denominator
of R(c 1 , c 2 , ρ, α, β) approximates its limiting value m 1−α (c 1 )/ρ much earlier usually than
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