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

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For a lot of size N, the quality is measured by the vector X = (X 1 , X 2 , ..., X r ), where X i = number of defectives in the lot in respect of the i th attribute, i = 1, 2, ..., r. A random sample of size n is drawn from a lot and the verification of non-conformity with respect to (w.r.t.) r specified attributes can be carried out in the sample in any order. The number of defectives vector for the sample is denoted as x = (x 1 , x 2 , ..., x r ). Two situations are considered viz., (i) Defective units w.r.t. the different attributes occur independently i.e. X 1 , X 2 , ..., X r are jointly independently distributed and x 1 , x 2 , ..., x r are consequently jointly independently distributed. (ii) Defective units w.r.t. the different attributes occur mutually exclusively i.e. the occurrence of one kind of defect precludes the occurrence of another kind of defect. In this case (x 1 , x 2 , ..., x r ) follows multinomial distribution with parameters n, p 1 , p 2 , ..., p r In both the situations mentioned, under Poisson conditions [ Hald (1981) ], which means r∑ p i → 0, np i finite, for all i and in addition p i → 0 in situation (ii) in the production setup i=1 assumed already, the type B probability for a realization of x 1 , x 2 , ..., x r in a sample of size n can be approximated as r∏ g(x i , np i ), i=1 where g(y, m) represents an individual term of Poisson distribution denoted by e −m m y /y!, y = 0, 1, 2... ...(0.5.1) Different sampling schemes and related acceptance criteria are studied. In all the schemes a single sample of size n is drawn from the lots. assuming the acceptance criterion to be as follows : if x ∈ A, accept the lot and otherwise reject it. Then, the type B probability of acceptance of the lot at p under Poisson conditions can be approximated as, P (p) = ∑ r∏ g(x i , np i ) x∈A i=1 ...(0.5.2) The inspection scenario may change from identification of a defective unit to the counting of defects in a unit. In case we count the number of defects per item for each characteristic we may construct a model for which the number of defects for a unit for i th the characteristic 17
equals to p i in the long run. We assume the number of defects for r distinct characteristics in a unit are indepedently distributed. The output of such a process is called a product of quality (p 1 , p 2 , .., p r ). In this case x i = number of defects w.r.t. attribute i in a random sample of size n, i = 1, 2, ..., r. The expression (0.5.1) holds here exactly for a realization of x = (x 1 , x 2 , ..., x r ) in a random sample size n and the probability of acceptance, P (p) given by (0.5.2) holds exactly too. Along with the two situations under appropriate assumptions mentioned before, where we count the number defective units w.r.t. different characteristics in the sample, we include under Poisson conditions the present situation of defect verifications also w.r.t. a number of characteristics under considerations. The type B OC function given in (0.5.2) for the appropriate acceptance criterion stated covers all these situations of defectives / defects under given assumptions discussed. 0.5.1.2 Chapter 2: Multiattribute sampling scheme based on AQL This chapter is devoted to the objective of establishing, for a multiattribute situation, a sampling scheme in line with available international standards tabulated on the basis of Acceptable Quality Level (AQL). The published plans most widely used is the US Military standard 105D (1963) [ abbreviated as MIL-STD-105D ], which has been adopted as an international standard (ISO 2859). The standard is highly suitable for the consumers. The sample size used in MIL-STD-105D is determined from the given lot size and by the choice of inspection level. The acceptance number (c) is arrived at, such that it remains more or less same for the same value of the product of sample size and the AQL to ensure that the Poisson probability of acceptance (rejection) remains same for the plans. This ensures a desired producer’s risk. There are 13 sets of n.AQL values and 11 acceptance numbers chosen in such a way that the producer’s risk (excepting for c = 0) varies from maximum of 9.02% to a minimum of 1.44%. With increase in the n.AQL values, c increases, so that producer’s risk decreases up to a level of 2% and thereafter it is kept at less than 2%. There is no theoretical foundation for the relation between sample size and lot size. As the lot size increases, the sample size increases but at a lesser rate such that n/N → 0 as N → ∞. The Standard prescribes that separate plans are to be chosen for the different classes of attributes. For example, the plan for a critical defect will have generally a lower AQL value than the plan for a major defect and the plan for a major defect will have an AQL value lower than the plan for a minor defect. To examine the consequences of constructing a sampling plan by this method for our multiattribute situation we first consider the effective producer’s risks. Secondly, we consider it reasonable to expect that the OC function should be more sensitive to the changes in the defect level of more important attributes, particularly, in a situ- 18
Page 1 and 2: Multiattribute acceptance sampling
Page 3 and 4: e) the cost functions are linear an
Page 5 and 6: Contents 0.1 Purpose of sampling in
Page 7 and 8: Part 0 : Introduction 0.1 Purpose o
Page 9 and 10: ectification is defined by setting
Page 11 and 12: procurement of materials of the US
Page 13 and 14: average defective from such a proce
Page 15 and 16: 0.4 Scope of the present inquiry 0.
Page 17 and 18: 0.4.6 A generalized acceptance samp
Page 19 and 20: sembled units the general practice
Page 21: with the above purpose in mind. The
Page 25 and 26: m j , r∏ −P C j ′ = g(x j , m
Page 27 and 28: show that if we increase c 2 keepin
Page 29 and 30: independence and mutually exclusive
Page 31 and 32: different sampling schemes in terms
Page 33 and 34: 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
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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
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Figure 1.3.1: The sketch of the fun
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2.1 General cost models 2.1.1 Scope
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(j) Interaction of scrappable attri
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use of defective item is additive o
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2.1.5 Approximation under Poisson c
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2.2 The expected cost model for dis
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P (p (j) ) denotes the probability
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250 200 Figure 2.2.1 : Lot Quality
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Table 2. 2. 1 Testing goodness of f
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2. 3 Cost of MASSP's of A kind 2.3.
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From (2.3.2) we observe that γ 2 1
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( γ / ) 2 γ − ( m′− m e ) /
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We now consider the following funct
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….(2.3.12) Here S A denotes the s
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∴There exists an optimal A plan f
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Hence, Q(m) = 1 − P (m) has the s
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where a 0 and n 0 are the parameter
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where, p = (p 1 , p 2 , ..., p r )
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situations as above. For the third
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2.5.3 Optimal Bayesian plans in sit
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Table 2.5.2 : Optimal multi attribu
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Annexure Microsoft Visual Basic pro
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If Regret,OMREGRET(c3-1) Tjem Prevo
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3.1 Bayesian single sampling multia
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f(p i , ¯p i , s i )dp i = e −v
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easonable to assume that the p i
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the minimum cost is obtained at a 1
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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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