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

The individual Poisson term is denoted as g( x,
m)
= Exp ( −m)
m / x!
and the cumulative
c
x
x
Poisson as G( c,
m)
= ∑ Exp ( −m)
m / x!
. We also introduce h ( x,
m)
= m / x!
. Further the
x=
0
Poisson probability of acceptance at ( p , p ,..., p ) , for the Plan A with sample size n and
1 2 r
acceptance number a , a ,..., a is denoted as: PA , ,..., ; , ,..., )
1 2 r
( a1 a2
ar
m1
m
2 mr
and the
corresponding probability of rejection at ( p
1,
p2,...,
pr
) as QA(
a , a ,..., a ; m , m ,..., m )
1 2 r 1 2 r
= 1 − PA(
a , a ,..., a ; m , m ,..., m ) .
1 2 r 1 2 r
x
For the D kind plan with sample size n and acceptance number k the probability of
acceptance at ( p , p ,..., p ) is denoted as PD ( k,
m)
. The corresponding probability of
1 2 r
rejection at ( p , p ,..., p ) is denoted by QD ( k,
m)
.
1 2 r
For given values of γ 1 , γ 2 the regret function defined in equation ( 2.2.9 ) for lots of size N
using an A kind, with sample size n and with acceptance numbers ( a , a ,..., a ) is written
as :
1
2
r
RA(
n;
a
, a
,..., a
1 2 r
) =
n + ( N − n)[
γ QA(
a , a ,..., a ; m , m , a , a ,..., a m ) + γ PA(
a
, a
,..., a
; m'
, m'
,..., m'
1 1 2 r 1 2 1 2 r r 2 1 2 r 1 2 r
Similarly the regret function of the plan D with sample size n and acceptance number k :
RD ( n,
k ) = n + N − n)[
γ QD(
k,
m)
+ γ PD(
k,
')]
(
1 2
m
)]
2.3.2 Situation 1
Theorem (2. 3.1)
Assume γ
1
, γ
2
, N , p, p'
to be given. Given an optimal D plan with parameters (k, n), the
regret function value of the plan A with acceptance criterion:
x() i
≤ k −1;
for i = 1, 2,…,r-1; is less than the regret function value of the plan D (k, n) if
k
k + 1
[ p '( r − 1) / p(
r −1)
] > ( p'
/ p)
…. (2.3.1)
Proof:
Since D (n, k) is an optimal plan RD (n, k+1) - RD (n, k) ≥ 0
⎛ γ
k + 1
⎞
⎜ 2 ⎟
− ( m′−m)
⎛ m′
⎞
e ⎜ ⎟ > 1 …. (2.3.2)
⎝ γ
1 ⎠ ⎝ m ⎠
Now RD (n; k) - RA (n; a 1 = k-1, a 2 = k-1 ,..., a r-1 = k-1, a r = k) / (N-n) =
γ2 [ g ( k,m′
( r 1)
) g (0,m′
r
)] − γ1
[ g ( k,m(
r 1 )
) g (0,mr
)]
−
−
95