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

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Further, ∫ ∫ ... P (p)dw(p 1 )dw(p 2 )...dw(p r ) = ∫p ∑ r∏ x ∈ A g(x i , n ¯p i , s i ). 1 p 2 p r i=1 Using the above we get the result. 3.1.6 Bayesian Plans Using the expression (3.1.7), we may now construct optimal A kind, C kind and D kind plans for a given lot size N, cost parameters and the parameters of the prior distributions and compare their relative merit in a given situation. We demonstrate this with one real life example obtained in respect of plastic containers used for cosmetics. In this case the defects which can be categorized as major type are defects like colour variation, improper neck finishing, prominent marks, weak body. The cost of testing this is approximately around Rs. 0.06 per unit. The minor type of defects are black spot, shrink marks, less visible parting lines which can be verified at a cost of around Rs 0.14 per unit. ( Note that verification for critical defect is less costly. ) An item containing the major defects will be discarded at a cost of Rs.1.50 and an item containing minor defects can be resold to another consumer at a reduced price such that the cost of rejection in this case is around Rs.0.50. An item containing defects of major category, if found during filling at the consumer’s end, costs the producer around Rs.5.50 ( price of the product ) and a product containing minor defects will cost the producer around Rs.3.20 ( manufacturing cost + transportation costs ). Using the notation of cost model we find that the cost parameters are S 0 = R 0 = 0.20, S 1 = R 1 = 1.50, R 2 = S 2 = 0.50, A 1 = 5.50 and A 2 = 3.20. From the past data we compute the ¯p 1 = 0.0105, ¯p 2 = 0.035 and Standard deviations as 0.0091 and 0.0055 respectively, so that the parameters of the gamma priors s 1 and s 2 work out to be around 1.2 and 40 respectively. This gives k s ′ = 0.05, d 0 = 0.05, d 1 = 1, d 2 = 0.675. We now construct optimal A kind, D kind and C kind plans minimizing the cost function [K(N, n)/(A 1 − R 1 )]. The table 3.1.4 gives the optimal A, C and D plans for lot sizes 20000 (10000) 50000. We observe that optimal A kind plans are cheaper than the optimal C kind plans and the optimal D kind plans. The optimal C kind plans are cheaper than the optimal D kind plans. For a lot of size 30000 the behaviour of cost function near the neighbourhood of the optimum k and n for the Plan D is presented in figure 3.1.2. For the optimum plan, k = 25 and n = 319. For the same lot size, 30000, the optimum A kind plan has a 2 = 25 and n = 250. If we vary a 1 from 0 to 25, we may see from figure 3.1.3 that the minimum cost is obtained at a 1 = 8. If we choose n = 319 (the sample size of optimal D) plan and a 2 = 25 and vary a 1 , 127
the minimum cost is obtained at a 1 = 10. The Visual Basic program as Excel Macro “Costcalculation” for obtaining the cost as relevant to Figure 3.1.3 has been included at the end of this chapter. The output of the programme is the cost of an A kind plan with a 2 = 25, n = 250, N = 30000, and for a 1 = 5(1)20 for the cost parameters and parameters of the prior gamma distribution as given in this section. For N = 30000, the optimal C kind plan has sample size n = 250 and acceptance numbers c 1 = 8, c 2 = 30. We have therefore, demonstrated that it should be possible to obtain an optimal MASSP by the above methods. While doing so, it should be noted that the Bayesian solutions rest on the assumption that the prior distribution is stable and that no outliers occur, so that for small and medium lot sizes the Bayesian plans will often reduce to ‘accept without inspection’ [Hald (1981)]. It is, however, clear that the choice of acceptance criterion does affect the costs of optimal MASSP’s. However, in the present chapter we do not attempt at obtaining general theoretical results for choosing the acceptance criteria as in the case of discrete prior distributions. 128
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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
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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 and 130: f(p i , ¯p i , s i )dp i = e −v
Page 131: easonable to assume that the p i
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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