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

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Table 1.2.5 Explanatory notes relating to three attribute A kind single sampling plans for normal inspection 1. Construction setup This table is constructed using the set of n.AQL values chosen for MIL-STD-105D. There are 13 n.AQL values. We get 286 combinations each of size 3, such that n.AQL 1 < n.AQL 2 < n.AQL 3 . We order these combinations lexicographically, considering the AQL in their natural order and also keeping the TotalAQL in descending order. We have started with the first combination as the set (0.12, 0.20. 0.32) 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%. 2. Column identification Each plan is identified by its serial number given in column 1. The next three columns (column 2, column 3, column 4 ) contain the n.AQL i values for i = 1, 2, and 3 respectively. From column 5, column 6 and column 7, we obtain the three acceptance numbers of the A kind plan viz. a 1 , a 2 and a 3 . The next column i.e. the 8 th column gives the producer’s risk i.e. the probability of acceptance at the process average (AQL 1 , AQL 2 , AQL 3 ). The last column contains the ratio p (3) 0.10 to TotalAQL. 3. Obtaining an A kind sampling plan from the table We may use the table as follows: Suppose the lot size is 15000 and we use Inspection Level II. From the MIL-STD-105D we get the sample size as 315. Suppose there are three Attributes viz. Attribute 1, Attribute 2 and Attribute 3. We choose their respective AQL’s (say), as 0.15%, 1.0% and 4%. This gives n.AQL 1 = 0.4725, n.AQL 2 = 3.15, n.AQL 3 = 12.6. We may choose the plan number 145 and get the values of a 1 = 3, a 2 = 9, a 3 = 23. We would, therefore, take a sample of size of 315 from the lot of size 15000, inspect these 315 pieces for all the three attributes and observe the number of defects as x 1 , x 2 , and x 3 for the first, second and the third attribute respectively. If x 1 ≤ 3, x 1 +x 2 ≤ 9, x 1 +x 2 +x 3 ≤ 23, we shall accept the lot, else we shall reject the lot. For this plan, the producer’s risk at AQL = (0.001, 0.01, 0.04 ) is around 4.5%, as given in column 8. From column 9 we the p (3) 0.10/TotalAQL = 1.8. The probability of acceptance ( i.e. the consumer’s risk ) at this process average (0.0018, 0.018, 0.072) is around 10%. 55
Table 1.2.5: Three attribute A kind single sampling plans for normal inspection p ( ) /Total SL. n.(AQL 1 ) n.(AQL 2 ) n.(AQL 3 ) a 1 a 2 a 3 Over all Prod. 0. 10 No. risk AQL 1 0.1256 0.1991 0.3155 1 1 2 0.055 7.9 2 0.1256 0.1991 0.5 1 2 2 0.054 6.4 3 0.1256 0.3155 0.5 1 2 3 0.026 6.9 4 0.1991 0.3155 0.5 1 2 3 0.039 6.4 5 0.1256 0.1991 0.7924 1 2 3 0.034 5.9 6 0.1256 0.3155 0.7924 1 2 3 0.045 5.4 7 0.1991 0.3155 0.7924 2 2 3 0.050 5.1 8 0.1256 0.5 0.7924 1 2 4 0.038 5.4 9 0.1991 0.5 0.7924 2 2 4 0.043 5.1 10 0.1256 0.1991 1.256 1 2 4 0.030 5 11 0.3155 0.5 0.7924 2 3 4 0.030 4.9 12 0.1256 0.3155 1.256 1 2 4 0.040 4.7 13 0.1991 0.3155 1.256 2 2 4 0.044 4.5 14 0.1256 0.5 1.256 1 3 4 0.049 4.2 15 0.1991 0.5 1.256 2 4 4 0.049 4.1 16 0.3155 0.5 1.256 2 3 5 0.027 4.4 17 0.1256 0.7924 1.256 1 3 5 0.037 4.2 18 0.1991 0.7924 1.256 2 3 5 0.038 4.1 19 0.1256 0.1991 1.991 1 2 5 0.039 4 20 0.3155 0.7924 1.256 2 3 5 0.050 3.8 21 0.1256 0.3155 1.991 1 2 5 0.049 3.8 22 0.1991 0.3155 1.991 2 3 5 0.044 3.7 23 0.5 0.7924 1.256 3 4 5 0.048 3.6 24 0.1256 0.5 1.991 1 2 6 0.043 3.9 25 0.1991 0.5 1.991 2 2 6 0.048 3.8 26 0.3155 0.5 1.991 2 3 6 0.033 3.7 27 0.1256 0.7924 1.991 1 3 6 0.043 3.6 28 0.1991 0.7924 1.991 2 3 6 0.044 3.5 29 0.3155 0.7924 1.991 2 4 6 0.043 3.4 30 0.5 0.7924 1.991 4 6 6 0.050 3.2 31 0.1256 1.256 1.991 1 4 7 0.036 3.4 32 0.1991 1.256 1.991 1 4 7 0.049 3.3 33 0.1256 0.1991 3.155 1 2 7 0.035 3.4 34 0.3155 1.256 1.991 2 4 7 0.045 3.2 35 0.1256 0.3155 3.155 1 2 7 0.043 3.2 36 0.1991 0.3155 3.155 2 2 7 0.046 3.2 37 0.5 1.256 1.991 3 5 7 0.041 3.1 38 0.1256 0.5 3.155 1 3 7 0.047 3.1 39 0.1991 0.5 3.155 2 3 7 0.047 3 56
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Multiattribute acceptance sampling
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e) the cost functions are linear an
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Contents 0.1 Purpose of sampling in
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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 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: Figure 1.2.1 Absolute value of Slop
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
Page 107 and 108: ….(2.3.12) Here S A denotes the s
Page 109 and 110: ∴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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