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

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i = j + 1, ..., r, then this plan will have a lesser regret value if, [p ′ (j−1) /p (j−1)] a j > [p ′ (j) /p (j)] a j+1 . Situation 4 : [p ′ j−1/p j−1 ] ≤ [p ′ j/p j ] In this case there exists a minimum regret A plan with ; a j−1 = a j . In the chapter 2.5 we verify these results by sample computations. 0.5.2.4 Chapter 4 : Cost of MASSP’s of C kind In this chapter we introduce the concept of the OC distribution and the OC moments of the C kind plans. The approach of moment equivalent plans was originally used by Hald (1981) for finding the approximate OC curve for a double sampling plan (in a single attribute situation ) from the OC of the equivalent single sampling plan. We use this approach in the case of MASSP of C kind to investigate some of its optimality properties. We first note that if we treat all ρ i = p i /p as fixed for i = 1, 2, ..., r; p = p 1 + p 2 + ... + p r , the Poisson OC of a given C kind plan with acceptance numbers c 1 , c 2 , ..., c r and sample size n can be considered as a function of m = np denoted by P (m). Since the P (m) is a monotonically decreasing function of m with the property: P (m) = 1 for m = 0 and P (m) = 0 for m → ∞, the function Q(m) = 1 − P (m) satisfies all the required properties of a distribution function, even though we do not regard m as a random variable in the present context. Thus, for a given (ρ 1 , ρ 2 , ..., ρ r ), the OC distribution for a C Plan is given by the distribution function 1 − P (m), 0 < m < ∞. For the OC distribution so defined we have obtained the expressions of moments of different orders. We may call them as OC moments. We may now construct a moment equivalent single sampling plan with parameters (a 0 , n 0 ) for a given value of (ρ 1 , ρ 2 , ..., ρ r ) such that the OC distribution of the single sampling plan has the same mean and variance as those of the OC distribution of the given C kind plan. For the moment equivalent single sampling plan we get a 0 = E 2 (m)/V (m); n 0 = nE(m)/V (m). It has been proved in this chapter, that the moment equivalent single sampling plan so obtained for any specified value of (ρ 1 , ρ 2 , ..., ρ r ) has sample size n 0 < n. We note that corresponding to the two process average points, p and p ′ of the two point prior distribution, we have two ρ vectors (ρ 1 , ρ 2 , ..., ρ r ) and (ρ ′ 1, ρ ′ 2, ..., ρ ′ r) and therefore we may construct two moment equivalent SSP’s (a 0 , n 0 ) and (a ′ 0, n ′ 0 ) corresponding to these two ρ vectors. We have proved that both n 0 and n ′ 0 are < n. Hald(1981) assumes near identity of the OC curves of the moment equivalent single sampling plan for estimating the probability of acceptance in case of a double sampling plan for different process average values. Hald’s assumptions has been corroborated by numerical computations. Following Hald’s argument we may also assume the near identity of the OC 27
curves i.e. the OC curve as a function of p for a given value of (ρ 1 , ρ 2 , ..., ρ r ) and that of the OC curve of the moment equivalent single sampling plan.We use the above logic (although mathematically non-rigorous) and numerically justify the near identity of the OC curve for a given plan of a MASSP of C kind and that of the equivalent SSP ( under the restriction of a specified ρ vector). Note that we require effectively this near identity only for the tail probabilities. We thereafter consider three possible scenario (a) a 0 > a ′ 0, n 0 < n ′ 0 (b) a 0 < a ′ 0, n 0 > n 0 ′ and (c) a 0 > a ′ 0, n 0 > n ′ 0. We show that corresponding to the MASSP of C kind with sample size n and acceptance numbers c 1 , c 2 , ..., c r the SSP (a 0 , n 0 ) and the SSP (a ′ 0, n ′ 0) will have lesser regret value in the situation (a) and the situation (b) respectively. In the situation (c) we notice ( using Hald’s results ) that the OC’s of two SSP’s (a 0 , n 0 ) and (a ′ 0, n ′ 0) intersect at some point p 0 (say). If p p ′ neither of the SSP (a 0 , n 0 ) and the SSP (a ′ 0, n ′ 0) may have lesser regret than the optimal C plan. In this case the plan (a ′′ 0, n ′′ 0); a ′′ > a 0 , n ′′ 0 > n 0 will have lesser regret if n > n ′′ 0. Further the Poisson OC of the MASSP D kind with (k, n) at any process average p is identical with the Poisson OC of the SSP (k, n) at p = p 1 + p 2 + ... + p r . It therefore follows that given an optimal MASSP of C kind it should be possible to construct a D kind plan with lesser regret values in first two situations as above. For the third situation we may have to satisfy the additional condition mentioned as above to obtain such a D kind plan. In the next chapter we undertake some numerical exercises to demonstrate these results. 0.5.2.5 Chapter 5: Results of numerical verification This chapter presents examples of comparison of the regret value of A kind, C kind and D kind plans for different situations using the results obtained in earlier chapters and provides a visual basic programme written as Excel Macro for obtaining optimal A kind and C kind plans. For the purpose of verification we consider the case where the number of characteristics r = 3. In chapter 2.2.2 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 cost parameters and the parameters of the two point prior distribution. Moreover, the P (p) denotes the type B probability of acceptance at p and Q(p) = 1 − P (p). We consider optimality properties for different acceptance criterion given the values of γ 1 , γ 2 , p and p ′ . 28
Page 1 and 2: Multiattribute acceptance sampling
Page 3 and 4: e) the cost functions are linear an
Page 5 and 6: Contents 0.1 Purpose of sampling in
Page 7 and 8: Part 0 : Introduction 0.1 Purpose o
Page 9 and 10: ectification is defined by setting
Page 11 and 12: procurement of materials of the US
Page 13 and 14: average defective from such a proce
Page 15 and 16: 0.4 Scope of the present inquiry 0.
Page 17 and 18: 0.4.6 A generalized acceptance samp
Page 19 and 20: sembled units the general practice
Page 21 and 22: with the above purpose in mind. The
Page 23 and 24: equals to p i in the long run. We a
Page 25 and 26: m j , r∏ −P C j ′ = g(x j , m
Page 27 and 28: show that if we increase c 2 keepin
Page 29 and 30: independence and mutually exclusive
Page 31: different sampling schemes in terms
Page 35 and 36: 0.5.3 Part3 Bayesian multiattribute
Page 37 and 38: K(N, n)/(A 1 − R 1 ) = nk ′ s +
Page 39 and 40: example, a sample of finished garme
Page 41 and 42: ...(1.1.2) If now X i is assumed to
Page 43 and 44: (i) Poisson as approximation to bin
Page 45 and 46: 1.2 Multiattribute sampling schemes
Page 47 and 48: We take this case and the case of t
Page 49 and 50: highest with respect to critical de
Page 51 and 52: ...(1.2.4) Note that, for single at
Page 53 and 54: ...(1.2.7) To compare the relative
Page 55 and 56: Corollary For ρ 2 /ρ 1 > c 2 /c 1
Page 57 and 58: increasing in each a i . Note that
Page 59 and 60: Figure 1.2.1 Absolute value of Slop
Page 61 and 62: Table 1.2.5: Three attribute A kind
Page 63 and 64: Table 1.2.5 (contd.): Three attribu
Page 65 and 66: Table 1.2.5 (contd.) : Three attrib
Page 67 and 68: Table 1.2.5 (contd.) : Three attrib
Page 69 and 70: 1.3 Multiattribute sampling plans o
Page 71 and 72: Let M 2 < M 1 . Then G(c 1 , M 1 ρ
Page 73 and 74: α, β, and ρ. For α = 0.05, β =
Page 75 and 76: ma β (a 1 , a 2 , ρ) = m ′ ...(
Page 77 and 78: Table:1.3.2: Construction parameter
Page 79 and 80: 1.3.5 D kind plans of given strengt
Page 81 and 82: Figure 1.3.1: The sketch of the fun
Page 83 and 84:
2.1 General cost models 2.1.1 Scope
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(j) Interaction of scrappable attri
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use of defective item is additive o
Page 89 and 90:
2.1.5 Approximation under Poisson c
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2.2 The expected cost model for dis
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P (p (j) ) denotes the probability
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250 200 Figure 2.2.1 : Lot Quality
Page 97 and 98:
Table 2. 2. 1 Testing goodness of f
Page 99 and 100:
2. 3 Cost of MASSP's of A kind 2.3.
Page 101 and 102:
From (2.3.2) we observe that γ 2 1
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( γ / ) 2 γ − ( m′− m e ) /
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We now consider the following funct
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….(2.3.12) Here S A denotes the s
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∴There exists an optimal A plan f
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Hence, Q(m) = 1 − P (m) has the s
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where a 0 and n 0 are the parameter
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where, p = (p 1 , p 2 , ..., p r )
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situations as above. For the third
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2.5.3 Optimal Bayesian plans in sit
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Table 2.5.2 : Optimal multi attribu
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Annexure Microsoft Visual Basic pro
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If Regret,OMREGRET(c3-1) Tjem Prevo
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3.1 Bayesian single sampling multia
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f(p i , ¯p i , s i )dp i = e −v
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easonable to assume that the p i
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the minimum cost is obtained at a 1
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Table 3.1.1 (Contd.) : Results of 1
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Table 3.1.2 : Testing goodness of f
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Figure 3.1.1 : Scatter plots of num
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Figure 3.1.2: Sketch of the cost fu
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Annexure Microsoft Visual Basic pro
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References Ailor, R. B., Schmidt, J
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Copenhagen. Hamaker, H. C. (1950).
Page 149:
Taylor, E. F. (1957). Discovery sam
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