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

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From (2.1.3) and (2.1.4), it can be easily shown that: ( K(N, n, p) = n S 0 + ∑ ) [( S i p i + (N − n) A 0 + ∑ ) ( A i p i P (p) + R 0 + ∑ i i i R i p i ) Q(p) ] , where P (p) denotes the average probability of <strong>acceptance</strong> at p. P (p) = ∑ ( ) r∏ n x p i i (1 − p i ) n−x i , x∈A i=1 x i and Q(p) = 1 − P (p). ...(2.1.5) B) When the defect occurrences are mutually exclusive. In this situation the expression for the probability of observing (x 1 , x 2 , ..., x r ) defective in a sample of size n from a lot of size N, containing (X 1 , X 2 , ..., X r ) defectives of types i = 1, 2, ..., r, will be multivariate hypergeometric as ( )( ) ( )( ) ( ) X1 X2 Xr N − X1 − X 2 ... − X r N P r(x 1 , x 2 , ..., x r | X 1 , ..., X r ) = ... / . x 1 x 2 x r n − x 1 − x 2 ... − x r n ...(2.1.6) At any process average the joint probability distribution of (X 1 , X 2 , ..., X r ) can be assumed to be multinomial such that ( )( ) ( ) N N − X(1) N − X(r−1) ( ) P r (X 1 , X 2 , ..., X r | p 1 , p 2 , ..., p r ) = ... p X (N−X(r) 1 1 ...p Xr ) r 1 − p(r) . X 1 X 2 X r . X (i) = X 1 + X 2 + ... + X i and p (i) = p 1 + p 2 + ... + p i ; i = 1, 2, ..., r. ...(2.1.7) From (2.1.6) and (2.1.7) it follows that average cost at p can be expressed as (2.1.5) replacing P (p) by P (p) = ∑ x 1 ,x 2 ,...,x r∈A x (i) = x 1 + x 2 + ... + x i for i = 1, 2, ..., r. ( )( ) ( ) n n − x(1) n − x(r−1) ( ) ... p x (n−x(r) 1 1 ...p xr r 1 − p(r) . x 1 x 2 x r ...(2.1.8) 83
2.1.5 Approximation under Poisson conditions As discussed in chapter 1.1 we use the phrase ‘Poisson conditions’ when Poisson probability can be used in the expressions of type B OC function in the two situations. 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 g(x i , np i ) = e −np i (np i ) x i /(x i )!. Under this condition the expression P (p) given in equation (2.1.5) can be modified as : P (p) = ∑ r∏ P r(x 1 , x 2 , ..., x r | p 1 , p 2 , ..., p r ) = g(x i , np i ). x 1 ,x 2 ,...,x r∈A i=1 ...(2.1.9) r∑ If we also make an additional assumption that p i → 0 then the expression P (p) given in i=1 equation (2.1.8) can also be modified as (2.1.9). (ii)Poisson as an exact distribution and the occurrences of defetc types independent. We assume the number of defects for r distinct characteristics in a unit 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 drawn from such a process will vary at random according to a Poisson law with parameter Np i for the i th characteristics 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 a Poisson variable with parameter np i . Independence of the different characteristics will be naturally maintained in the sample so that the type B probability of <strong>acceptance</strong> will be given by the expression of P (p) given as equation (2.1.9). In this situation we will be dealing with defects rather than defectives. 2.1.6 The genaralized cost model Thus under Poisson condition described as above the expression of K(N, n, p) is same as (2.1.5) with P (p) = ∑ s∈A ∏ ri=1 g(x i , np i ) and Q(p) = 1 − P (p) 84
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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
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ectification is defined by setting
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procurement of materials of the US
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average defective from such a proce
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0.4 Scope of the present inquiry 0.
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0.4.6 A generalized acceptance samp
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sembled units the general practice
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with the above purpose in mind. The
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equals to p i in the long run. We a
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m j , r∏ −P C j ′ = g(x j , m
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show that if we increase c 2 keepin
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independence and mutually exclusive
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different sampling schemes in terms
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curves i.e. the OC curve as a funct
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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
Page 85 and 86: (j) Interaction of scrappable attri
Page 87: use of defective item is additive o
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
Page 111 and 112: Hence, Q(m) = 1 − P (m) has the s
Page 113 and 114: where a 0 and n 0 are the parameter
Page 115 and 116: where, p = (p 1 , p 2 , ..., p r )
Page 117 and 118: situations as above. For the third
Page 119 and 120: 2.5.3 Optimal Bayesian plans in sit
Page 121 and 122: Table 2.5.2 : Optimal multi attribu
Page 123 and 124: Annexure Microsoft Visual Basic pro
Page 125 and 126: If Regret,OMREGRET(c3-1) Tjem Prevo
Page 127 and 128: 3.1 Bayesian single sampling multia
Page 129 and 130: f(p i , ¯p i , s i )dp i = e −v
Page 131 and 132: easonable to assume that the p i
Page 133 and 134: the minimum cost is obtained at a 1
Page 135 and 136: Table 3.1.1 (Contd.) : Results of 1
Page 137 and 138: 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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