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

2.5 Results of numerical investigations
2.5.1 Introduction
This chapter presents examples comparing the regret value of a) A, D and C type plans for
different situations using the results obtained in earlier chapters and b) proposes algorithm
for construction of A kind optimal plans. In chapter 2.2, we have noted that for a given two
point discrete prior distribution of p the regret function is
R(N, n) = n + (N − n)[γ 1 Q(p) + γ 2 P (p ′ )]
Where γ j ’s are functions of i) cost parameters (A 0 , A 1 , A 2 , ..., A r ), (S 0 , S 1 , S 2 , ..., S r ), (R 0 , R 1 , R 2 , ..., R r )
and ii) the parameters of prior distributions w 1 , w 2 , p and p ′ as defined in section 2.1.3.
Moreover, the P (p) denotes the type B probability of acceptance at p and Q(p) = 1 − P (p)
Further, if R 0 = S 0 and R i = S i , i = 1, 2, ..., r then γ 1 = 1.
We address the problem of obtaining the optimal plans under different acceptance criteria,
given the values of γ 1 , γ 2 , p and p ′ .
2.5.2 Optimal Bayesian plans in situation p ′ 1/p 1 = p ′ 2/p 2 = ... = p ′ r/p r
Table 2.5.1
In this case, let ρ i = p i /p and ρ ′ i = p ′ i/p ′ for i = 1, 2..., r. Here ρ i = ρ ′ i, ∀i. From the
results of chapter 2.4 it follows that for the optimal C plan with sample size n, acceptance
parameters c 1 , c 2 , ..., c r and the values of ρ i we can construct a moment equivalent D plan
with sample size n 0 and acceptance number k 0 , so that the two plans will have approximately
same probabilities of acceptance at p and at p ′ . Further, this D plan has smaller sample
size and lesser regret value than the optimal C plan.
Let r = 3. For a typical value of γ 1 = 1, γ 2 = 0.7, (p 1 , p 2 , p 3 ) = (0.002, 0.008, 0.020) and
p ′ 1/p 1 = p ′ 2/p 2 = p ′ 3/p 3 = 5, we compute the optimal parameters of the C plan and those of
the equivalent D type plan (which is effectively a single sampling plan for single attribute).
Parameters of these plans have been retained in real numbers. Integer appeoximation will
affect the equivalence to some extent.
We note from table 2.5.1 for lot sizes 1000 (1000) 10000 (5000) (50000) the regret value
of the moment equivalent D plan is less than that of optimal C plans. Moreover, the regret
value of the optimal D plans is still less than that of both these plans.
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