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

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x 1 + x 2 ≤ a 2 , ..., ..., x 1 + ... + x r ≤ a r ; and reject otherwise. In all the later discussions we use the notation x (j) for = x 1 + ... + x j , j = 1, 2, ..., r. Note that a i ≤ a j for j ≥ i. The OC function under Poisson conditions is given by, P A(a 1 , a 2 , .., a r ; m 1 , m 2 , ..., m r ) = a 1 ∑ x 1 =0 a r−x ∑(r−1) ... x r=0 r∏ g(x i , m i ) i=1 ...(1.2.17) To compare the relative changes in the OC function for changes in p i , we use the absolute value of the partial differential coefficient of the Poisson OC function as given in equation (1.2.17) with respect to m i ; m i = np i . This is used as a measure of discriminating power of the OC function w.r.t. j th attribute, treating it as function of m = m 1 + m 2 + ...m r and keeping m i /m fixed for i = 1, 2, ..., r Theorem 1.2.3 For i ≤ j the discriminating power of the plan (as defined) w.r.t. i th attribute is more than or equal to that for the j th attribute for every m = m 1 + m 2 ... + m r . Proof Denoting the partial differential coefficient of the Poisson OC function as given in equation (1.2.17) with respect to m i ; m i = np i by P A ′ i, we get −P A ′ i = P A(a 1 , a 2 , .., a i , ..., a r ; m 1 , m 2 , ..., m r )−P A(a 1 , a 2 , ..., a i −1, a i+1 −1, ..a r −1; m 1 , m 2 , ..., m r ) for a i > 0, and for a i = 0. −P A ′ i = P A(a 1 , a 2 , .., a i , , a r ; m 1 , m 2 , ..., m r ) ...(1.2.18) ...(1.2.19) Thus, the OC function is a decreasing function of m i or p i . Moreover, the OC function is 51
increasing in each a i . Note that a j = 0 implies a i = 0 for all i ≤ j. It therefore follows from (1.2.18) and (1.2.19) that ...(1.2.20) This proves the theorem. The above property of the plan A therefore allows us to order the attributes in order of relative discriminating power. If the attributes are ordered in the ascending order of AQL value, then it is possible to construct a sampling scheme ensuring an acceptable producer’s risk and also satisfy the condition Slope i ≥ Slope i+1 , for all m, i = 1, 2, ..., r − 1. For comparison m i /m is kept fixed, i = 1, 2, ..., r. 1.2.7 Construction of A kind MASSP’s using the sample size and AQL from MIL-STD-105D Using the set of n. AQL values chosen from MIL-STD-105D, we shall illustrate this for r = 3. There are 13 n. AQL values. We get 286 combinations such that n.AQL 1 < n.AQL 2 < n.AQL 3 . We further order the triplets lexicographically, considering the n.AQL’s in their natural order. We start with the first combination of triplet (0.1256, 0.1991, 0.3155) and choose a 1 = 1, a 2 = 1,and a 3 = 2. This gives producer’s risk of around 5.5 %. All other sets of acceptance numbers are worked out such that a plan positioned at a higher level will have a lesser producer’s risk, so that producer’s risk decreases up to a level of 5% and thereafter it is kept at less than 5%. The results of this exercise are presented in Table 1.2.4. Using the notation ρ i = AQL i /(AQL 1 + AQL 2 + AQL 3 ) we have defined p (3) 0.1 as the value of the p such that, P A(a 1 , a 2 , a 3 ; npρ 1 , npρ 2 , npρ 3 ) = 0.1. It is heartening to note that the ratio of the p (3) 0.1 to T otalAQL as defined is also decreasing from 2.3 to 1.4 with increase in n.T otalAQL, so that the OC appears to be quite steep. 1.2.8 The D kind plans Consider the MASSP where we take a sample of size n, observe the number of defectives (defects) for the i th attribute as x i for all i = 1, 2, ..., r and apply the acceptance rule: accept if x 1 + x 2 + ... + x r ≤ k ; reject otherwise The probability of acceptance at p = (p 1 , p 2 , ..., p r ) under Poisson conditions : P D(k; m 1 , m 2 , .., m r ) = k∑ x 1 =0 k−x ∑(r−1) ... x r=0 r∏ g(x i , m i ) i=1 52
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Multiattribute acceptance sampling
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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 and 34: curves i.e. the OC curve as a funct
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: Corollary For ρ 2 /ρ 1 > c 2 /c 1
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
Page 85 and 86: (j) Interaction of scrappable attri
Page 87 and 88: use of defective item is additive o
Page 89 and 90: 2.1.5 Approximation under Poisson c
Page 91 and 92: 2.2 The expected cost model for dis
Page 93 and 94: P (p (j) ) denotes the probability
Page 95 and 96: 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
Page 103 and 104: ( γ / ) 2 γ − ( m′− m e ) /
Page 105 and 106: 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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