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

...(1.2.7)
To compare the relative changes in the OC function for changes in p i , we compute the
P C ′ i obtained by differentiating the OC function as given in equation (1.2.7) with respect to
m i . Thus,
Slope j = −P C ′ j = g(c j , m j )
r∏
i=1:i≠j
G(c i , m i ), j = 1, 2, ...r; c j ≥ 0, j = 1, 2, ...r
...(1.2.8)
Let m = m 1 + m 2 + .... + m r ; 0 ≤ m 1 ≤ m 2 ... ≤ m r ; ρ i = m i /m is fixed for i = 1, 2, ..., r
and only m varies. In this case, 0 ≤ c 1 ≤ c 2 ... ≤ c r .
Theorem 1.2.1 justifies to some extent the numerical results presented in Table 1.2.4.
Theorem 1.2.1
The function, H(m) = Slope i − Slope i+1 undergoes atmost one change of sign from positive
to negative for m ≥ 0.
Proof We need to show that the above statement is true for
F (m) = g(c i , mρ i )G(c i+1 , mρ i+1 ) − G(c i , mρ i )g(c i+1 , mρ i+1 ).
... (1.2.9)
Without loss of generality we take i = 1, so that F (m)/ [e −m m c 1
]=
c 1 ∑
i=0
m [ c 2−i
ρ c 2−i
2 ρ c 1
1 /((c 2 − i)!c 1 !)) − ρ c 1−i
1 ρ c 2
2 /((c 1 − i)!.c 2 !)) ] c 2 −c ∑ 1 −1
+ ((mρ 2 ) x /x!)(ρ c 1
1 /c 1 !)
0 ≤ c 1 ≤ c 2 < ∞
x=0
...(1.2.10)
We first note that all the coefficients of m x for x ≤ c 2 − c 1 − 1 are ≥ 0 in the second
series function of the right hand side in (1.2.10).
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