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

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...(1.1.5) From (1.1.4) and (1.1.5) it follows that the unconditional probability is ( )( ) ( ) n n − x(1) n − x(r−1) P r(x 1 , x 2 , ..., x r | p 1 , p 2 , ..., p r ) = ... p x 1 1 ...p xr r (1 − p (r) ) (n−x(r)) , x 1 x 2 x r . where x (i) = x 1 +x 2 +...+x i , i = 1, 2, ..., r 0 ≤ x (1) ≤ x (2) ... ≤ x (r) ≤ n; 0 < p i < 1, i = 1, 2, ..., r; p (r) < 1. ...(1.1.6) Thus the average probability follows multinomial distribution with parameter (n, p 1 , p 2 , ..., p r ). 1.1.4 Poisson conditions Hald (1981) has used the phrase ‘under Poisson conditions’ when Poisson probability can be used as an approximation to the binomial in the expressions of type B OC function. We use the phrase to cover the situations as described below. 37
(i) Poisson as approximation to binomial and multinomial If p i → 0, n → ∞ and np i → m i then the binomial probability b(x i , n, p i ) tends to Poisson probability g(x i , np i ) where ( ) n x i and b(x i , n, p i ) = p x i i (1 − p i ) n−x i g(x i , np i ) = e −np i (np i ) x i /(x i )!. Under this conditions for i = 1, 2, ..., r, the expression given as equation (1.1.3) can be modified as r∏ P r(x 1 , x 2 , ..., x r | p 1 , p 2 , ..., p r ) = g(x i , np i ); x i ≥ 0, i = 1, 2, ..., r. i=1 ...(1.1.7) r∑ If we also make an additional assumption that p i → 0 then the equation (1.1.6) can also i=1 be modified as (1.1.7). (ii) Poisson as an exact distribution and occurrences of defect types independent In case we count the number of defects per item for each characteristic, we may construct a model for which the expected number of defects for a item for i th the characteristic equals to p i in the long run. We assume the number of defects for r distinct characteristics in a item are independently distributed. The output of such a process is called a product of quality (p 1 , p 2 , ..., p r ), the parameter vector representing the mean occurrence rates (of defects) per observational unit. The total number of defects for any characteristic in a lot of size N from such a process will vary at random according to a Poisson law with parameter Np i for the i th characteristic under usual circumstances. Similarly, the distribution of number of defects on attribute i, in a random sample of size n drawn from a typical lot will be be a Poisson variable with parameter np i . Independence of the different characteristics will be naturally maintained in the sample, so that the joint probability of occurrence can be expressed as (1.1.7). Here p i , i = 1, 2, ..., r denotes the average number of defects per item in respect of characteristics i instead of proportion defectives, since in this situation we are dealing with defects rather than defectives. 38
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 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: ...(1.1.2) If now X i is assumed to
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
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
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250 200 Figure 2.2.1 : Lot Quality
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Table 2. 2. 1 Testing goodness of f
Page 99 and 100:
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).
Page 149:
Taylor, E. F. (1957). Discovery sam
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