Multiattribute acceptance sampling plans - Library(ISI Kolkata ...
Multiattribute acceptance sampling plans - Library(ISI Kolkata ...
Multiattribute acceptance sampling plans - Library(ISI Kolkata ...
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B() denotes the cumulative binomial probability. From (2.4.3) we get<br />
Therefore,<br />
E(m 2 ) =<br />
c 1 ∑<br />
x 1 =0<br />
E(m 2 ) − E(m) =<br />
c 2 ∑+x 1<br />
x=x 1<br />
2(x + 1)b(x 1 , x, ρ).<br />
c 1 ∑<br />
x 1 =0<br />
c 2 ∑+x 1<br />
x=x 1<br />
(2x + 1)b(x 1 , x, ρ)<br />
∑c 1 x∑<br />
c 2 ∑+c 1 ∑c 1<br />
= (2x+1)b(x 1 , x, ρ)+ (2x+1)b(x 1 , x, ρ) = (c 1 +1) 2 ∑c 2<br />
+ [2(c 1 +i)+1]B(c 1 , c 1 +i, ρ).<br />
x=0 x 1 =0<br />
x=c 1 +1 x 1 =0<br />
i=1<br />
[<br />
]<br />
Further, E 2 ∑c 2 2<br />
(m) = (c 1 + 1) + B(c 1 , c 1 + i, ρ)<br />
i=1<br />
[<br />
= (c 1 + 1) 2 ∑c 2 ∑ c2 2<br />
+ 2(c 1 + 1) B(c 1 , c 1 + i, ρ) + B(c 1 , c 1 + i, ρ)]<br />
.<br />
i=1<br />
i=1<br />
[<br />
∑ c2 2<br />
∑c 2 ∑c 2<br />
Writing B(c 1 , c 1 + i, ρ)]<br />
= B(c 1 , c 1 + i, ρ) B(c 1 , c 1 + j, ρ)<br />
i=1<br />
i=1<br />
j=1<br />
and noting that B(c 1 , c 1 + j, ρ) ≥ B(c 1 , c 1 + i, ρ) for j ≤ i, we have<br />
[<br />
∑ c2 2<br />
∑c 2<br />
B(c 1 , c 1 + i, ρ)]<br />
≤ B(c 1 , c 1 +1, ρ)+3B(c 1 , c 1 +2, ρ)+...+(2c 2 −1)B(c 1 , c 1 +c 2 , ρ) = (2i−1)B(c 1 , c 1 +<br />
i=1<br />
i=1<br />
Thus,<br />
E 2 (m) ≤ (c 1 + 1) 2 +<br />
c 2 ∑<br />
i=1<br />
(2(c 1 + i) + 1)B(c 1 , c 1 + 2, ρ) ≤ E(m 2 ) − E(m).<br />
Hence, V (m) > E(m) and therefore n 0 < n.<br />
It therefore follows by mathematical induction that n 0 < n for all r ≥ 2.<br />
2.4.6 Obtaining a D kind plan with regret value lesser than that of the optimal<br />
C plan<br />
Using (2.2.9) the regret function of a C kind plan in the present context can be written as<br />
R(N, n) = n + (N − n)[γ 1 Q(p) + γ 2 P (p ′ )]<br />
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