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

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g(X i , N ¯p i , s i ) = Γ(s i + X i ) θ s i i .(1 − θ i ) X i s i ; θ i = X i !Γs i (s i + N ¯p i ) and X i non-negative integer. ...(3.1.6) [Note that we use g(x, θ) to denote the Poisson distribution term and g(x, θ 1 , θ 2 , ) to denote the gamma-Poisson term.] It follows that the distribution of the number of defectives or defects x i in the sample folows gamma-Poisson law with probability mass function given by g(x i , n ¯p i , s i ). Further there is stochastic independence of x 1 , x 2 , ...., x r . This is the framework for development of cost models in the present chapter. In the sections which follow, we use gamma prior distribution which work either approximately accurately or as exact distributions under different situations as explained. 3.1.4 The verification for the appropriate prior distribution Table 3.1.1 presents the inspection data for 86 lots containing about 25500 pieces of filled vials of an eye drop produced by an established pharmaceutical company based at <strong>Kolkata</strong>. Each vial is inspected for six attributes. From the criticality point of view, however, the defects can be grouped in two categories. The first category of defects is due to the presence of foreign matters viz. glass, fiber or impurities, which are critical from the user’s point of view. The other category of defects consists of breakage, defective sealing and leakage, reasonably obviously detectable and can be easily discarded by the user. Since the lot size ( 25500 ) is quite large, the observed lot quality variation approximates almost exactly the distribution of process average. We have therefore, instead of a gamma- Poisson distribution, fitted a gamma distribution with mean and the variance estimated from the observed data for both types of defects.The results are presented in Table 3.1.2 and table 3.1.3. Note that for both the attributes the ¯p i ’s are less than 0.1 ( ¯p 1 = 0.01708; ¯p 2 = 0.02302). The estimated s i = ¯p i 2 /V ar(p i ) are 3.4532 and 0.6229 respectively so that ¯p i /s i are much less than 0.2 justifying the use of gamma distribution as a substitute of beta distribution. The tables also present the usual χ 2 goodness of fit analysis. It can be seen that the computed χ 2 as a measure of the goodness of fit values are small (they are not statistically significant) enough for both types of attributes to justify the assumption that the prior distributions follow the assumed theoretical gamma distributions. Further, the scatter plot (see figure 3.1.1) of the observed numbers of defects of the second category against those of the first category exhibits no specific pattern. It would be therefore 125
easonable to assume that the p i ’s are independently distributed in the present context. It should be pointed out at this stage that the beta and gamma distribution are, however, not always appropriate. For example, we could not fit the gamma or beta distribution in case of quality variation of ceiling fans, garments and cigarettes the relevent data on which were collected. In such cases we will have to take recourse to direct computation of the cost function derived from the empirical distributions as observed. Numerically, the difficulty level in computation will not increase significantly. Nevertheless, since the gamma (or beta ) distribution is likely to be appropriate at least in some situations and neat theoretical expressions can be obtained in such cases, we will study the cost functions under such assumptions. 3.1.5 The expression of average costs under the assumption of independent gamma prior distributions of the process average vector Theorem 3.1.1 Let each p i be distributed with probability density function f(p i , ¯p i , s i ) for i = 1, 2, ..., r; p i ’s are jointly independent; the lot quality X i is distributed as g(X i , Np i ) ∀i and X i ’s are jointly independent. The optimal plan in this situation for a specified <strong>acceptance</strong> criteria x = (x 1 , x 2 , ..., x r ) ∈ A is obtained by minimizing the function : [ { ∑ r∑ } r∏ K(N, n)/(A 1 −R 1 ) = nk s+(N−n)k ′ r+(N−n) ′ d i ¯p i (s i + x i )/(s i + n ¯p i ) − d 0 g(x i , n ¯p i , s i ) x∈A i=1 i=1 where g(x i , n ¯p i , s i ) is a gamma-Poisson density given by, g(x i , n ¯p i , s i ) = Γ(s i + x i ) θ s i i .(1 − θ i ) x i ; θ i = x!Γs and x i non-negative integer. k ′ s = k s /(A 1 − R 1 ), k ′ r = k r /(A 1 − R 1 ). s i (s i + n ¯p i ) ...(3.1.7) Proof: Since, ∫ ∫ r∏ r∏ ... p i g(x j , np j )f(p j , ¯p j , s j )dp j = [ ¯p i (s i + x i )/(s i + n ¯p i ) g(x j , n ¯p j , s j ) ∫p 1 p 2 p r j=1 j=1 We get, ∫ ∫ ... d i p i P (p)dw(p 1 )dw(p 2 )..dw(p r ) = ∫p ∑ r∏ d i ¯p i (s i + x i )/(s i + n ¯p i ) g(x i , n ¯p i , s i ). 1 p 2 p r x∈A i=1 126 ]
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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
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Table 1.2.5: Three attribute A kind
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Table 1.2.5 (contd.): Three attribu
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Table 1.2.5 (contd.) : Three attrib
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Table 1.2.5 (contd.) : Three attrib
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1.3 Multiattribute sampling plans o
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Let M 2 < M 1 . Then G(c 1 , M 1 ρ
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α, β, and ρ. For α = 0.05, β =
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ma β (a 1 , a 2 , ρ) = m ′ ...(
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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 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: f(p i , ¯p i , s i )dp i = e −v
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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