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

( γ / )
2 γ − ( m′−
m
e
) /
⎤
1 ⎢⎣
⎡ m
( ′ m
r −1) ( r −1)
⎥⎦
a
r−1
+ 1
. F
> 1
( r −1)
a r
implies ∆ R1
> 0.
−1
Here,
⎡ ar
−ar− 1
−1
⎤
= ⎢ ∑ ′
x
F m r r
/ xr
! ⎥ /
⎢⎣
xr
= 0
⎥⎦
We note that F > F1 by (2.3.6).
⎡
⎢
⎢⎣
a
r
−a
x
r
r
∑
−1
−1
= 0
x
m r r
/ x
r
⎤
! ⎥
⎥⎦
Now R 1 - R 2 > 0 if
a
⎡
m ′ / m
⎤ r−1
>
⎡
m ′ / m
⎤ −1
⎢⎣ ( r − 2) ( r − 2) ⎥⎦ ⎢⎣ ( r−1)
( r−1)
⎥⎦
and therefore we get (2.3.4) and therefore the theorem 2.3.2 is proved.
a
r
+ 1
2.3.4 Situation 3
Consider a set of A plans such that for some j (1 ≤ j < r), a i = a j < a j+1 i = 1,2,…, j-1 i.e.
to say that all acceptance numbers are all equal for i ≤ j but a j+1 > a j .
Theorem (2.3.3)
Let the optimal plan from the set specified at the beginning of the section 2.3.5 has the
parameter n, a
1
= a
2
= .. = a
j
, a
j + 1
,..., ar
for a given N, γ
1, γ 2,
p and p' . If we now
construct a plan with acceptance numbers b i such that b
1
= b2
= ... b j − 1 = a
j
− 1, (assuming
a j ≥ 1), and
b = for i = j, j + 1,...,
r then the latter plan will have a lesser regret value if:
i
a i
Proof:
[ ′ a
j
p ] > [ ′ p ]
p ( j −1)
/ ( j −1
)
( j ) /
( j )
a j
+ 1
p … (2.3.7)
We denote the regret function value of the optimal plan in the specified set up as R 1 and that
of the constructed second plan as R 2 .
(R 1 – R 2 ) / (N-n)
= γ
2
g ( a j
, m′
( j −1
) ) g ( 0 , m′ j
) N1 − γ
1
g ( a
j
, m(
j −1
) ) g ( 0 , m
j
)
where
a'
j+
a '
r
−(
x
j+ 1
+ ... + xr−
1
) r
N = ∑ ... ∑ ∏ g ( x
1
i , m'
i )
x 0 x = 0 i=
j + 1
1
j+ 1 =
= S 0 ∏
r
g ( x i , m'
i ) .
i=
j + 1
r
D
1
98