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

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a shirt with a button missing or a shirt with a wrong button), or when defects are classified in mutually exclusive classes e.g. critical, major or minor. It would therefore be necessary to take care of both the situations. As stated earlier that the primary focus of the present enquiry is to examine different alternative acceptance criteria rather than to locate an optimal plan in a given set up and to develop search alogorithm for the mentioned purpose. We make use of the cost models developed by Majumdar(1980), Majumdar(1990) and Majumdar (1997). The cost models mentioned along with some elementary results in the following chapter of the part have already been published as noted here. 2.1.4 A generalized cost model for nondestructive testing In our case we take a sample of size n and inspect each item for all the attributes. We observe x i ; i = 1, 2, .., r. as the number of defectives on the i th attribute. We denote the vector (x 1 , x 2 , ..., x r ) as x. We define set A as the set of x for which we declare the lot as acceptable. Let Ā be the complementary set for which we reject the lot. Let X i; denotes the number of defectives on i th characteristic in the lot, i = 1, 2, ..., r. Let the costs be when x ∈ A r∑ r∑ C(x) = nS 0 + x i S i + (N − n)A 0 + (X i − x i )A i i=1 i=1 ... (2.1.1) r∑ r∑ C(x) = nS 0 + x i S i + (N − n)R 0 + (X i − x i )R i i=1 i=1 when x ∈ Ā ...(2.1.2) The interpretations of cost parameters are as follows: S 0 is the common cost of inspection i.e. sampling and testing cost per item in the sample for all the characterstics put together; x i S i the cost proportional to the number of defectives of i th type in the sample which is the additional cost for an inspected item containing defects of i th type. The cost of acceptance is composed of two parts; (N − n)A 0 is cost proprtional to the r∑ items in the remainder of the lot, and another part (X i − x i )A i where A i is the cost of accepting an item containing defective for i th attribute. i=1 We assume the loss due to 81
use of defective item is additive over all the characterstics. This means if an item contains more than one defect say for i = 1 and 2 the loss will be the sum of the damages for both the characterstics put together. The assumption is reasonably valid under many situations. However, proportion of items containing more than one category of defects will usually be small. Costs of rejection consists of a part (N − n)R 0 proportional to the number of items in r∑ the remainder of the lot and another part, (X i − x i )R i proportional to the number of i=1 defective items rejected for all the attribute put together. If rejection means sorting, R 0 will give the sorting cost /item for all category of defects put together. The R i denotes the additional cost for items found with defective of i th category ( for example, cost of repair) and is additive over different category of defects. Note that there is only one sample of size n to be taken for inspection for all the characteristics and the cost model given here is only a multiattribute analogue of the cost model considered by Hald (1965) in the case of a single attribute. The average costs A) When defect occurrences are independent. In this case the probability of getting x i defectives in a sample of size n from a lot of size N containing X i defectives, for the i th attribute, i = 1, 2, ..., r,is given by hypergeometric probability as ( )( ) ( ) n N − n N P r(x i | X i ) = / x i X i − x i X i X i being the number of defectives of type i in a lot i = 1, 2, ..., r. ...(2.1.3) Further, r∏ P r (X 1 , X 2 , ..., X r ) = P r (X i ). i=1 If the lot quality X i ; i = 1, 2, ...r is distributed as binomial, then ( ) N X i P r(X i ) = p X i i (1 − p i ) N−X i . ...(2.1.4) The average cost per lot at a process average p = (p 1 , p 2 , ..., p r ) for the lots of size N is K(N, n, p) = ∑ r∏ C(x) P r(x i |X i ) + ∑ r∏ C(x) P r(x i |X i ) x∈A i=1 x∈Ā i=1 82
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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
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 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: (j) Interaction of scrappable attri
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 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
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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).
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Taylor, E. F. (1957). Discovery sam
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