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

for j = 2.
...(2.2.7)
We assume that all the above functions are nonnegative and none are identical to 0. We also
assume that
k s (p (j) ) ≥ k m (p (j) ), j = 1, 2.
Further, let k s , k a , k r and k m denote the expected values of the corresponding cost functions
defined in (2.2.4), (2.2.5), (2.2.6) and (2.2.7) w.r.t. the prior. Note that these functions
are expressed as costs per unit. The average costs for the three cases without sampling inspection
i.e. the cases where,
(a) all lots are classified correctly,
(b) all lots are accepted, and
(c) all lots are rejected
then become k m , k a and k r respectively. Sampling inspection should only be taken recourse
to if
k Avg − k m < min[k a − k m , k r − k m ] where k Avg = K(N, n)/N.
Taking case (a) as the reference case we define the regret function R(N, n)
R(N, n) = [K(N, n) − K m (N, n)]/(k s − k m )
where K m (N, n) is the average minimum unavoidable cost per lot a multiattribute analogue
of what is given by Hald (1965).
...(2.2.8)
We may further simplify the above expression for R(N, n) as
R(N, n) = n + (N − n)[γ 1 Q(p (1) ) + γ 2 P (p (2) )]
as
γ j = w j |k a (p (j) ) − k r (p (j) )|/(k s − k m ); j = 1, 2.
Note that if R 0 = S 0 and R i = S i ; i = 1, 2, then k s = k r and γ 1 = 1.
...(2.2.9)
2.2.5 Using the model
In the above model γ j ’s are functions of the cost parameters and the prior distributions. The
change of acceptance criteria affects the cost function through the probability of acceptance.
This model therefore enables us to consider different kinds of MASSP’s and compare them in
terms of costs, which is the primary focus of our inquiry. We shall continue our discussions
to this end in the next chapters.
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