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Table 1.2.4 Summary table, giving information about slope comparisons for the 286 sampling plans with 3 classes of attributes Conditions Number of MASSP’s satisfying the given condition in the range: 0 plans with Slope 1 < 9 37 Slope 2 Number of plans with Slope 2 < 2 161 Slope 3 Number of plans with Slope 1 < 0 25 Slope 2 and Slope 2 < Slope 3 Number of plans with Slope 1 > Slope 2 > Slope 3 275 113 It is clear from the above table that, while using different AQL values as appropriate to the respective importance of the attributes and choosing the corresponding acceptance numbers there is no guarantee that we would get a good plan in terms of risks as well as the discriminating power in a multiattribute situation. As regards the ratio p (3) 0.1/ TotalAQL, it has ranged from 1.7 to 12. We now try to generalize the above results more formally for a class of plans where a sample of given size n is chosen and the acceptance numbers are different for different attributes. 1.2.5 The C kind plan and the properties of its OC Function We first note all these MASSP’s are based on the following acceptance criterion. We take a sample of size n, observe the number of defects for the i th attribute as x i and apply the following acceptance criterion, accept if: x i ≤ c i for all i = 1, 2, ..., r; reject otherwise. We will henceforth call these plans as C kind plans. The OC function under Poisson conditions is given by r∏ P C(c 1 , c 2 , ..., c r ; m 1 , m 2 , ..., m r ) = G(c i , m i ); i=1 m i = np i ∀i 47
...(1.2.7) To compare the relative changes in the OC function for changes in p i , we compute the P C ′ i obtained by differentiating the OC function as given in equation (1.2.7) with respect to m i . Thus, Slope j = −P C ′ j = g(c j , m j ) r∏ i=1:i≠j G(c i , m i ), j = 1, 2, ...r; c j ≥ 0, j = 1, 2, ...r ...(1.2.8) Let m = m 1 + m 2 + .... + m r ; 0 ≤ m 1 ≤ m 2 ... ≤ m r ; ρ i = m i /m is fixed for i = 1, 2, ..., r and only m varies. In this case, 0 ≤ c 1 ≤ c 2 ... ≤ c r . Theorem 1.2.1 justifies to some extent the numerical results presented in Table 1.2.4. Theorem 1.2.1 The function, H(m) = Slope i − Slope i+1 undergoes atmost one change of sign from positive to negative for m ≥ 0. Proof We need to show that the above statement is true for F (m) = g(c i , mρ i )G(c i+1 , mρ i+1 ) − G(c i , mρ i )g(c i+1 , mρ i+1 ). ... (1.2.9) Without loss of generality we take i = 1, so that F (m)/ [e −m m c 1 ]= c 1 ∑ i=0 m [ c 2−i ρ c 2−i 2 ρ c 1 1 /((c 2 − i)!c 1 !)) − ρ c 1−i 1 ρ c 2 2 /((c 1 − i)!.c 2 !)) ] c 2 −c ∑ 1 −1 + ((mρ 2 ) x /x!)(ρ c 1 1 /c 1 !) 0 ≤ c 1 ≤ c 2 < ∞ x=0 ...(1.2.10) We first note that all the coefficients of m x for x ≤ c 2 − c 1 − 1 are ≥ 0 in the second series function of the right hand side in (1.2.10). 48
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 and 22: with the above purpose in mind. The
Page 23 and 24: equals to p i in the long run. We a
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: ...(1.2.4) Note that, for single at
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 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 ) /
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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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