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

a
1
a
2
− x
(1)
a
j−2
− x
( j−3)
= ∑ ∑...
∑ ∏ g ( x , m ) g ( a
x
x1 = 0 x2
= 0 x
j−2
= 0
a
'
j+
1
∑
j+
1
= 0
a
'
j+
1
x
− x
∑
j+
2
j+
1
= 0
....
a
'
r
−(
x
j −2
i=
1
j+
1
x
r
i
+ ..... + x
∑
= 0
i
r−1
)
r
∏
i=
j + 1
j
− x , m ) g(0,
m ) .
g ( x
( j −2)
j −1
j
It becomes obvious by arguments similar to those used in theorems 2.3.2 and 2.3.3, from
equation (2.3.13) that
i
, m
i
) .
RA ( n;
a , a
2
,..., a
j 1
= a
j
, a
j
,..., ar
) − RA(
n;
a1
, a
2
,..., a
j − 1
= a
− 1, a
1 − j j r ≤ 0
if the given condition of the theorem is satisfied.
Thus, we see that if the condition of the theorem is satisfied, then
RA ( n ; a 1 …,a j-1 , a j , a j , a j+1 ,…, a r ) ≤ RA ( n ; a 1 ,…, a j-1 , a j-1 , a j , a j+1 ,…,a r ).
... a
)
…(2.3.14)
Let us consider the feasible interval for the (j-1) th acceptance number, denoted by c j-1 , viz.,
a j-1 ≤ c j-1 ≤ a j (recall a j-1 < a j given) and keep all other acceptance numbers fixed as in the
given optimal A plan.
Then, RA ( n ; a 1 …,a j-1 , c j-1 , a j , a j+1 , …, a r ) treated as a function of a single variable c j-1
keeping all other acceptance numbers fixed at their optimal values, has to exhibit one of the
following three features in the interval a j-1 ≤ c j-1 ≤ a j :
I. The function is monotonically decreasing in the interval.
II. The function is monotonically increasing in the interval.
III. The function first decreases monotonically and then increases monotonically.
[In the definition monotonically decreasing (increasing) includes the possibility of equality
also and does not mean strictly monotonically decreasing (increasing).]
The only possibility ruled out is first increasing and then decresing or a feature of multiple
waves. This is clearly not possible for a realistic regret function which can be supposed to
possess one of the three properties stated for any feasible segment of values of c j-1 .
Now because of (2.3.14), the only possibility is:
RA ( n ; a 1 …,a j-1 , a j-1 , a j ,..,a r )
= RA ( n ; a 1 …,a j-1 , a j-1 +1, a j , … ,a r )
= RA ( n ; a 1 , …,a j-1 , a j , a j ,…, a r )
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