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

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2.5 Results of numerical investigations 2.5.1 Introduction This chapter presents examples comparing the regret value of a) A, D and C type plans for different situations using the results obtained in earlier chapters and b) proposes algorithm for construction of A kind optimal plans. In chapter 2.2, we have noted that for a given two point discrete prior distribution of p the regret function is R(N, n) = n + (N − n)[γ 1 Q(p) + γ 2 P (p ′ )] Where γ j ’s are functions of i) cost parameters (A 0 , A 1 , A 2 , ..., A r ), (S 0 , S 1 , S 2 , ..., S r ), (R 0 , R 1 , R 2 , ..., R r ) and ii) the parameters of prior distributions w 1 , w 2 , p and p ′ as defined in section 2.1.3. Moreover, the P (p) denotes the type B probability of acceptance at p and Q(p) = 1 − P (p) Further, if R 0 = S 0 and R i = S i , i = 1, 2, ..., r then γ 1 = 1. We address the problem of obtaining the optimal plans under different acceptance criteria, given the values of γ 1 , γ 2 , p and p ′ . 2.5.2 Optimal Bayesian plans in situation p ′ 1/p 1 = p ′ 2/p 2 = ... = p ′ r/p r Table 2.5.1 In this case, let ρ i = p i /p and ρ ′ i = p ′ i/p ′ for i = 1, 2..., r. Here ρ i = ρ ′ i, ∀i. From the results of chapter 2.4 it follows that for the optimal C plan with sample size n, acceptance parameters c 1 , c 2 , ..., c r and the values of ρ i we can construct a moment equivalent D plan with sample size n 0 and acceptance number k 0 , so that the two plans will have approximately same probabilities of acceptance at p and at p ′ . Further, this D plan has smaller sample size and lesser regret value than the optimal C plan. Let r = 3. For a typical value of γ 1 = 1, γ 2 = 0.7, (p 1 , p 2 , p 3 ) = (0.002, 0.008, 0.020) and p ′ 1/p 1 = p ′ 2/p 2 = p ′ 3/p 3 = 5, we compute the optimal parameters of the C plan and those of the equivalent D type plan (which is effectively a single sampling plan for single attribute). Parameters of these plans have been retained in real numbers. Integer appeoximation will affect the equivalence to some extent. We note from table 2.5.1 for lot sizes 1000 (1000) 10000 (5000) (50000) the regret value of the moment equivalent D plan is less than that of optimal C plans. Moreover, the regret value of the optimal D plans is still less than that of both these plans. 113
2.5.3 Optimal Bayesian plans in situation where p ′ i/p i ’s are not same for all i Table 2.5.2 For r = 3, let p ′ 1 = 0.01, p ′ 2 = 0.04, p ′ 3 = 0.10, p ′ 1/p 1 = 5, p ′ 2/p 2 = 5, p ′ 3/p 3 = 3 and we take γ 1 = 1, γ 2 = 0.7. Since p ′ 1/p 1 = p ′ 2/p 2 = 5 from the results of Theorem 2.3.4, we must have a 1 = a 2 for the optimal A plan. Note that in this case p ′ (2) /p (2) = 5 and p ′ /p = 3.462 . Thus, for k ≥ 4 we find the inequality (p ′ (2) /p (2)) k > (p ′ /p) (k+1) satisfied. This means there exists an A plan cheaper than the optimal D plan for which the k value is greater than or equal to 4. (Theorem 2.3.1). Table 2.5.2 presents the parameters of the optimal D plan, optimal A plan and the corresponding regret values for lot sizes 1000 (1000) 10000. Note that for all the optimal A plans, a 1 = a 2 and a 3 > a 2 . This table also gives the parameters of the optimal C plan and the corresponding regret value, which is higher than that of the optimal D plan and the optimal A plan. Table 2.5.3: For r = 3 let p ′ 1 = 0.01, p ′ 2 = 0.04, p ′ 3 = 0.10, p ′ 1/p 1 = 8, p ′ 2/p 2 = 5, and p ′ 3/p 3 = 3. Let, γ 1 = 1, and γ 2 = 0.7 In this case p ′ (1) /p (1) = 8 and p ′ (2) /p (2) = 5.405 and p ′ /p = 3.523. For k > 2 the inequality (p ′ (2) /p (2)) k > (p ′ /p) (k+1) is satisfied. This means there exists an A plan cheaper than the optimal D plan for which the k value is greater than 2.( Theorem 2.3.1 ) Also by Theorem 2.3.3, a 3 > a 2 for a 3 > 2. Further, (p ′ (1) /p (1)) a 2 > (p ′ (2) /p (2)) a 2+1 for a 2 > 4 and hence a 2 > a 1 for a 2 > 4. Table 2.5.3 presents the parameters of the optimal D plan, optimal A plan and the corresponding regret values for lot sizes 1000 (1000) 10000. Note that for all the optimal A plans a 2 > a 1 and a 3 > a 2 . This table also gives the parameters of the optimal C plan and the corresponding regret value, which is higher than that of the optimal D plan and the optimal A plan. The annexures contains Microsoft Visual Basic Programmes used to construct the optimal plans. 114
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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
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 ) /
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: situations as above. For the third
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
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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