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

From (2.3.2) we observe that
γ
2
1
g ( k , m'
γ g ( k , m
( r −1)
( r −1)
) g (0, m'
)
) g
( 0, m)
)
>
γ
γ
2
1
− ( m ′− m )
e
k + 1 ≥
[ m′
/ m]
1
This implies , RD (n; k) > RA (n; a 1 = k-1,..., a r-1 = k-1, a r = k). (Proved)
Remarks:
A. The condition (2.3.1) is a sufficient condition and holds if
( p ′ / p′
) < ( p / p )
r
( r−1)
r
( r −1)
B. For r = 2 the condition (2.3.1) becomes
(
p
k
k + 1
′
1
/ p
1
) > ( p ′ / p )
For r = 2 this implies
p
′
p
1
>
1
.
p′
p
We now try to obtain more general results.
2.3.3 Situation 2
Let us consider the set of A plans with acceptance criteria
x ≤ a for i = 1, 2,..., r-1
x
( i ) r − 1
( r )
≤ a r
.
Note that the probability of acceptance at a process average p = ( p1 ,..., pr
) for this plan is
PA ( a1 , a
2
,..., a
r
; m1
, m
2 ,...,mr
) =
a a −x
r
= ∑ − 1 r ( r
.... ∑ − 1) r
∏ g(
x i
, m i
)
x = x = 0 i=
1
1
0
r
a
r
∑ − 1
= g ( x( r − ),
m
x( r )
− 1
= 0
ar
− x
1 ( r −1)
) ∑
x = 0
r
( r−1)
( x r
, m r
)
g ... (2.3.3)
Theorem (2.3.2)
Let n, a r-1 , a r be the optimal parameters with acceptance criterion specified at the beginning
of the section 2.3.4, for a given lot size N, γ 1 , γ 2 , p, p' etc. and assumption a r-1 ≥ 1. If we
now construct a plan with the same sample size but with acceptance criterion changed to:
x ( r − 2 )
≤ a
r − 1
− 1 , x(
r − 1)
≤ a
r −1
, x(
r )
≤ a
r then the latter plan will have lesser
regret value than the former plan, optimal in the specified set up, if
′ r −1
[ ( r −2
)
p( r −2
) / p
]
a
> [ p ′
( r −1)
/ p
( r −1)
]
a r −1
+ 1
... (2.3.4)
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