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

S 0 denotes the summation with respect to x i ’ s over the domain indicated in the line preceding
line and, a′ j + l
= a
j + l
− a
j
for l = 1 , 2,...,
r − j and
D 1 is obtained from N 1 by replacing
m′
i
by m i .
…. (2.3.8)
Let,
i=
r
S ∏ h(
x , m′
)
0 i i
i=
j+
1
W =
.
i=
r
S ∏ h(
x , m )
0 i i
i=
j+
1
The ratio of the first term of to the second term of (2.3.8) can be written as
a j
= ( γ
2 / γ ) Exp(
-(
m'-m))
.(
m ′
1
(j- 1) / m(j-
1) ) .W . The theorem is proved if we can show
that this quantity is greater than 1 under the given condition.
Now,
( j )
a j
∆ PA = PA ( a1 = ... a
j − 1
, a
j
+ 1, a
j + 1
, a
j + 1
,...., ar
; m
1 ,..., mr
)
j ) (a j
- PA ( a1 = ... = a
j
; a
j + 1
,..., ar
; m1
,..., mr
)
a′
−1
j + 1
a′
r
−1−
( x
j+ 1
+ ... + xr−
1
) r
= g ( a
j
+ 1, m ( j )
) ∑
∑ ∏ g ( x i , m i ) …(2.3.9)
x
x = 0 i=
j + 1
( j )
j + 1=
0
∆ PA and ∆ QA,
when used in the regret function lead to the regret difference
( j )
a j
denoted as ∆ R )
a j
(
1
r
( j )
∆ ( R1 ) ≥ 0 ⇒
a j
j + 1
i=
r
i=
r
( γ
′
∏ ′
2 / γ
1 ). Exp(-(
m'-m)).(
m(
j ) / m(
j ) ) S1
h(
xi
, mi
) / S1
∏ h(
xi
, mi
)
i=
j + 1
i=
j + 1
≥ 1.
…(2.3.9a)
S denotes the summation with respect to
1
Note that the set of tuples ,..., x )
(
j 1 r
x i
' s over the domain indicated in (2.3.9).
x +
coming under
of x ’s coming under S
i
0
(call this set as T 0 ).
Let T 2 = T 0 – T 1 . Let S 2 denote the summation on all
S (call this set as T 1 ) is a subset
1
x combinations appearing in T 2 .
i
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