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

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1. p ′ 1/p 1 = p ′ 2/p 2 = p ′ 3/p 3 From the results of chapter 2.2.4 it follows that for the optimal C plan with sample size n, acceptance parameters c 1 , c 2 , ..., c r and given values of p i /p; p ′ i/p ′ , we can construct a moment equivalent D plan with sample size n 0 and acceptance number k 0 which will have approximately same probability of acceptance as the plan C at p and at p ′ . Further, this plan has lesser regret than the optimal C plan. For a typical value of γ 1 = 1, γ 2 = 0.7, (p 1 , p 2 , p 3 ) = (0.002, 0.008, 0.02) 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. We find that regret value of the moment equivalent SSP is less than the optimal C plan. Moreover the regret value of the optimal D plan is still lesser than that of each of these two plans. 2. p ′ i /p i are not same for all i a) For r = 3, let p ′ 1 = 0.01, p ′ 2 = 0.04, p ′ 3 = 0.10, p ′ 1/p 1 = 5, p ′ 2/p 2 = 5, p ′ 3/p 3 = 3 and we take γ 1 = 1, γ 2 = 0.7. Since p ′ 1/p 1 = p ′ 2/p 2 = 5 we notice from the results of the chapter 2.3 (situation 4 ) that a 1 = a 2 for the optimal A plan. Also in this case p ′ (2) /p (2) = 5 and p ′ /p = 3.46. Thus for k ≥ 4 we satisfy the inequality (p ′ (2) /p (2)) k > (p ′ /p) (k+1) . This means there exists an A plan cheaper than the optimal D plan for which the k value is greater than or equal to 4. (Situation 1 in chapter 2.4). We have computed the parameters of the optimal D plan, optimal A plan and the corresponding regret value for lot sizes 1000 (1000) 10000. We find that for all the optimal A plans a 1 = a 2 and a 3 > a 2 . Further we compute the corresponding regret value of the optimal C kind which is higher than that of the optimal D plan and the optimal A plan. b) For r = 3, let p ′ 1 = 0.01, p ′ 2 = 0.04, p ′ 3 = 0.10, p ′ 1/p 1 = 8, p ′ 2/p 2 = 5, p ′ 3/p 3 = 3, γ 1 = 1, γ 2 = 0.7. For k > 2 we satisfy the inequality (p ′ (2) /p (2)) k > (p ′ /p) (k+1) . This means there exists an A plan cheaper than the optimal D plan for which the k value is greater than 2. (Situation 1 of chapter 2.3). Also by results of the chapter (situation 3 ) a 3 > a 2 for a 3 > 2. Further (p ′ (1) /p (1)) a 2 > (p ′ (2) /p (2)) a 2+1 for a 2 > 4 and hence a 2 > a 1 for a 2 > 4 (situation 2). We present the parameters of the optimal D plan, optimal A plan and the corresponding regret values for lot sizes 1000 (1000) 10000.For all the optimal A plan a 2 > a 1 and a 3 > a 2 . We have also presented the parameters of the optimal C plan and the corresponding regret value which is higher than that of the optimal D plan and the optimal A plan. 29
0.5.3 Part3 Bayesian multiattribute sampling inspection plans for continuous prior distribution 0.5.3.1 Chapter 1: Bayesian sampling inspection plans for continuous prior distribution In this chapter it will be assumed that the process average defective (defects) for each attribute has a continuous prior distribution. We examine in particular, the problems of choice of a theoretical distribution as relevant to a multiattribute situation. The most widely used continuous prior distribution for the process average quality p i is the beta distribution. β(p i , s i , t i ) = p i s i −1 (1 − p i ) t i−1 /β(s i , t i ), s i > 0, t i > 0. We express the above using the parameters ; ¯p i and s i where ¯p i = s i /(s i + t i ). When ¯p i and ¯p i /s i are small ; more precisely, if ¯p i < 0.1 and ¯p i /s i < 0.2, this distribution can be approximated by a gamma distribution: f(p i , ¯p i , s i )dp i = e (−v i) (v i ) s−1 dv i /Γ(s i ); v i = s i p i / ¯p i with mean E(p i ) = ¯p i and the shape parameter, s i . Hald (1981) used this approximation to tabulate the optimal sampling single sampling plans. Most of his results are based on assuming gamma as the right distribution in the effective range. Corresponding to a beta distribution of the single attribute process average of quality p i , the distribution of the lot quality denoted by X i as well as sample quality x i become a beta-binomial distribution which can similarly be approximated as a gamma-Poisson distribution. The gamma Poisson distribution assumed appears to be the right distribution when we are counting number of defects instead of defectives. Thus, in all the situations where we are counting number of defectives / defects we write the joint prior distribution of p = (p 1 , p 2 , ..., p r ) under the assumption of independence. The joint distribution term of x = (x 1 , x 2 , ..., x r ) is thus assumed to be the product of r gamma-Poisson distribution terms. The expression for the average costs connected with a sampling scheme is to be worked out from the general cost model expression, obtained under these assumptions. To verify how far the assumed probability distributional forms fits into a real life situation, we collected the inspection data for 86 lots containing about 25500 pieces each of filled vials of an eye drop produced by an established pharmaceutical company based at Kolkata. Each vial is inspected for six characteristics. From the criticality point of view, however, the defects can be grouped in two categories. We have been able to justify, by using the χ 2 goodness of fit analysis, the assumption that the distributions of lot quality follow the assumed theoretical gamma-Poisson distributions. Further, the scatter plot of the observed numbers of defects of the second category against 30
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 and 32: different sampling schemes in terms
Page 33: curves i.e. the OC curve as a funct
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
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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
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Table 2. 2. 1 Testing goodness of f
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2. 3 Cost of MASSP's of A kind 2.3.
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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).
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Taylor, E. F. (1957). Discovery sam
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