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

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has the same mean and variance of the OC distribution for a given C kind plan. For the single sampling plan, let m 0 = n 0 p denote the parameter and let b = m 0 /m. Then E(m 0 ) = bE(m); V ar(m 0 ) = b 2 V (m). Equating the first two moments we get a 0 + 1 = E 2 (m)/V (m); n 0 = nE(m)/V (m). ....(2.4.5) It may be noted here that, for a fixed ρ i ; i = 1, 2, ..., r the OC curve for a multiattribute C kind plan is looked upon as a function of np since in this case (p 1 , p 2 , ..., p r ) = (ρ 1 p, ρ 2 p, ..., ρ r p). In the following discussions we treat ρ i , i = 1, 2, ..., r as fixed. 2.4.5 Sample size of the moment equivalent plan Hald (1981) assumes near identity of the OC curves of the moment equivalent single sampling plan for estimating the probability of acceptance in case of a double sampling plan for different process average values. Hald’s assumption has been corroborated by numerical computations. The expression used by Hald may be considered as the first term of series expansion of Khamis (1960) and has the same accuracy as the Edgeworth and Cornish- Fisher approximation. Following Hald’s argument we may also assume the near identity of the OC curves i.e. the OC curve as of the given C kind plan as a function of p for a given vector (ρ 1 , ρ 2 , ..., ρ r ) and that of the OC curve of the moment equivalent single sampling plan(SSP). We use the above logic (although mathematically non-rigorous) and numerically justify the near identity of the OC curves of a MASSP of C kind and that of the moment equivalent SSP (under the restriction of a specified set of values for the ρ i ’s). As we shall see later that we require effectively this near identity only for the tail probabilities. Theorem 2.4.2 Let p [r] = (p 1 , p 2 , ..., p r ) and P (p [r] ) denote the OC function value at p [r] where there are r attributes. Further, P (p [r] ) = G(c 1 , np 1 )G(c 2 , np 2 )...G(c r , np r ) ≃ G(a 0 , n 0 (p 1 + p 2 + ... + p r )) 107
where a 0 and n 0 are the parameters of the moment equivalent plans obtained from (2.4.5) for ρ i = p i /(p 1 + p 2 + ... + p r ) for i = 1, 2, ..., r, then n 0 < n. ....(2.4.6) Proof For a fixed s ≥ 2 let (2.4.6) hold for all r, 2 ≤ r ≤ s and we take r = s. Consider a C kind plan with sample size n and acceptance numbers c 1 , c 2 , ..., c s . Now, P (p [s] ) = G(c 1 , np 1 )G(c 2 , np 2 )...G(c s , np s ) = G(a 0 , n 0 p (s) ), where p (s) = p 1 +p 2 +...+p s and a 0 and n 0 have been obtained from (2.4.5) for ρ i = p i /p (s) i = 1, 2, ...s. Adding one more attribute to the study system, P (p [s+1] ) = G(c 1 , np 1 )G(c 2 , np 2 )....G(c s , np s )G(c s+1 , np s+1 ) = G(a 0 , n 0 p (s) )G(c s+1 , np s+1 ) = G(a 0 , n(n 0 /n)p (s) )G(c s+1 , np s+1 ) = G(a 0 , n.p ′ (s) )G(c s+1, np s+1 ), where p ′ (s) = (n 0/n)p (s) < 1. Note that n 0 /n < 1 by supposition and therefore p ′ (s) < p (s) < 1. We may now construct a SSP with parameter (a 1 , n 1 ) for the C kind plan with number of attributes, r =2 using (2.4.5) such that, G(a 1 , n 1 (p s+1 + p ′ (s)) = G(a 0 , n.p ′ (s))G(c s+1 , np s+1 ). Further let this be equal to G(a 1 , n 2 .(p s+1 + p (s) )). But p ′ (s) < p (s) and, therefore, n 2 < n 1 < n. This implies that if (2.4.6) holds for r = 2 then it holds for all r ≥ 2, by mathmatical induction. We now prove that (2.4.6) holds for r = 2. To prove that n 0 < n we must show that E(m 2 ) − E 2 (m) > E(m) or E(m 2 ) − E(m) > E 2 (m). Defining ρ = m 1 /m = p 1 /p, E(m) = c 1 ∑ x 1 =0 c 2 ∑+x 1 x=x 1 b(x 1 , x, ρ) = and b(x 1 , x, ρ) = ( x x 1 ) ρ x 1 (1 − ρ) x−x 1 ,we get c 1 ∑ x∑ x=0 x 1 =0 b(x 1 , x, ρ)+ c 2 ∑+c 1 c 1 ∑ x=c 1 +1 x 1 =0 b(x 1 , x, ρ) = (c 1 +1)+ c 2 ∑ i=1 B(c 1 , c 1 +i, ρ). 108
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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
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K(N, n)/(A 1 − R 1 ) = nk ′ s +
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example, a sample of finished garme
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...(1.1.2) If now X i is assumed to
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(i) Poisson as approximation to bin
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1.2 Multiattribute sampling schemes
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We take this case and the case of t
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highest with respect to critical de
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...(1.2.4) Note that, for single at
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...(1.2.7) To compare the relative
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Corollary For ρ 2 /ρ 1 > c 2 /c 1
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increasing in each a i . Note that
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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
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: Hence, Q(m) = 1 − P (m) has the s
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
Page 139 and 140: Figure 3.1.1 : Scatter plots of num
Page 141 and 142: Figure 3.1.2: Sketch of the cost fu
Page 143 and 144: Annexure Microsoft Visual Basic pro
Page 145 and 146: References Ailor, R. B., Schmidt, J
Page 147 and 148: Copenhagen. Hamaker, H. C. (1950).
Page 149: Taylor, E. F. (1957). Discovery sam
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