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

2.3.5 Situation 4
Theorem (2.3.4) :
Assume N γ 1 , γ , p, p' are given.
, 2
There exists a minimum regret A plan with a = a
p p '/
p
' j −1 / p
j − 1 <
j
j
j −1
j
, if
….(2.3.10)
Proof:
Suppose there does not exist an optimal A plan with a j-1 = a j , but the given condition is
satisfied.
Let us suppose (a 1 ,…,a j-1 , a j ,..,a r ) is an optimal A plan with a j-1 < a j .
Let us use the notation
( j )
a
[ 1 2 j −1
j r 1 2 j −
j−1
1
∆ PA ( a , a ,..., a , a ,..., a ; m , m ,..., m , m ,..., m )] =
= PA ( a1 ,...., a
j −1
, a
j
,..., a
r
; m1
,..., mr
) − PA(
a1
,..., a
j −1
, a
j −1
, a
j + 1
,..., a
r
; m
1 ,..., mr
)
a1
= ∑ ∑ ... ∑ ∏ g ( xi
, mi
) ∑ g ( x j −1
, m j −1
) g ( a j − x(
j −1
) , m j ) .
x1
= 0
a2
− x(1)
x2
= 0
a j−2
− x(
j−3)
j −2
x j−2
= 0
i=
1
a j−1
− x(
j−2 )
x j−1
= 0
j
r
a ' j+
1 a ' j+
2 − x j+
1 a ' r −(
x j+
1 + x j+
2 + ,... + xr−
1 ) r
∑ ∑ .....
∑ ∏ g ( xi
, mi
)
x = 0 x = 0
x = 0 i=
j + 1
j+
1
Where ,
a ' = a − a , ∀i
≥ j + 1.
i
i
j+
2
j
Since (a 1 ,…,a j ,…,a r ) must satisfy the optimal criteria,
r
( j )
a j −1
…(2.3.11)
∆ RA, ≤ 0 where the notation
represents the difference of the regret function value for the A plan as explained in the proofs
of theorems 2.3.2 and 2.3.3. The condition is equivalent to,
( γ
2
≤ 1
/ γ Exp m′
1
). (-( -m)).
S
A
S
A
i = j − 2
∏ h(
x m′
i
,
i
)
i = 1
a
j
i = j − 2
∏ h(
x
i
, m
i
)
i = 1
A
j
−1
− x
( j − 2)
i = r
∑ h(
x
∏ ′
j −1
, m'
j −1
) h(
a
j
− x
( j −1)
, m'
j
) S
B
h(
x
i
, m
i
)
x
j −1
= 0
i = j + 1
−1
− x
( j − 2)
i = r
∑ h(
x
j −1101
, m
j −1
) h(
A
j
− x
( j −1)
, m
j
) S
B ∏ h(
x
i
, m
i
)
x
j −1
= 0
i = j + 1