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

….(2.3.12)
Here S A denotes the summation with respect to x 1 , x 2 ,…, x j-1 over the specified domain in
(2.3.11) and S B denotes the summation with respect to x j+1 ,x j+2 , …, x r over the specified
domain on the same expression. The respective domains are indicated in (2.3.11)
For any set of x 1 , x 2 ,…,x j-1 included in LHS of (2.3.12) we find that the following fraction:
a − x
j−1
( j−2)
∑
x = 0
j−1
a − x
j−1
( j−2)
∑
x = 0
j−1
⎛ m'
⎜
≥
⎜ m
⎝
j −1
j −1
a
⎞
⎟
⎟
⎠
h(
x
h(
x
j −1
j −1
, m'
− x
j ( j − 2)
, m
.
j −1
j −1
) h(
a
) h(
a
j
j
− x
− x
( j −1)
( j −1)
, m'
, m
j
j
)
)
⎛ m'
⎜
≥
⎜ m
⎝
j−1
j−1
⎞
⎟
⎟
⎠
a − x
j−1
( j−2)
⎛ m'
⎜
⎜ m
⎝
j
j
⎞
⎟
⎟
⎠
a −a
j j−1
∴ LHS of (2.3.12 ) ≥
( γ
/ γ ). Exp(-(
m′
-m)).(
m'
/m
2 1
j-1
j−1
)
a
j
− x
( j−2 )
S
S
A
A
i=
j −2
∏ h(
x , m′
) S
i=
1
i=
j −2
∏ h(
x
i=
1
i
i
, m
i
i
) S
B
i=
i=
r
∏ h(
x , ′
i
mi
)
j + 1
.
= r
∏ h(
x , m )
i
B
i=
j + 1
i
i
Thus the value of the above expression is less than or equal to 1.
… (2.3.13)
We now compare the regret function of two A kind plans, one with (j-1) th acceptance
number raised to a j and the other with (j-1) th acceptance number as a j -1, whereas the other
acceptance number are retained unchanged in both the plans. Since,
PA , a ,..., a , a ,..., a m , m ,..., m , m ,..., m )
( a
1 2 j j r , 1 2 j j + 1 r
- PA , a ,..., a − 1, a ,..., a , m , m ,..., m , m ,..., m ) =
( a
1 2 j j r 1 2 j j + 1 r
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