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

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a 1 a 2 − x (1) a j−2 − x ( j−3) = ∑ ∑... ∑ ∏ g ( x , m ) g ( a x x1 = 0 x2 = 0 x j−2 = 0 a ' j+ 1 ∑ j+ 1 = 0 a ' j+ 1 x − x ∑ j+ 2 j+ 1 = 0 .... a ' r −( x j −2 i= 1 j+ 1 x r i + ..... + x ∑ = 0 i r−1 ) r ∏ i= j + 1 j − x , m ) g(0, m ) . g ( x ( j −2) j −1 j It becomes obvious by arguments similar to those used in theorems 2.3.2 and 2.3.3, from equation (2.3.13) that i , m i ) . RA ( n; a , a 2 ,..., a j 1 = a j , a j ,..., ar ) − RA( n; a1 , a 2 ,..., a j − 1 = a − 1, a 1 − j j r ≤ 0 if the given condition of the theorem is satisfied. Thus, we see that if the condition of the theorem is satisfied, then RA ( n ; a 1 …,a j-1 , a j , a j , a j+1 ,…, a r ) ≤ RA ( n ; a 1 ,…, a j-1 , a j-1 , a j , a j+1 ,…,a r ). ... a ) …(2.3.14) Let us consider the feasible interval for the (j-1) th acceptance number, denoted by c j-1 , viz., a j-1 ≤ c j-1 ≤ a j (recall a j-1 < a j given) and keep all other acceptance numbers fixed as in the given optimal A plan. Then, RA ( n ; a 1 …,a j-1 , c j-1 , a j , a j+1 , …, a r ) treated as a function of a single variable c j-1 keeping all other acceptance numbers fixed at their optimal values, has to exhibit one of the following three features in the interval a j-1 ≤ c j-1 ≤ a j : I. The function is monotonically decreasing in the interval. II. The function is monotonically increasing in the interval. III. The function first decreases monotonically and then increases monotonically. [In the definition monotonically decreasing (increasing) includes the possibility of equality also and does not mean strictly monotonically decreasing (increasing).] The only possibility ruled out is first increasing and then decresing or a feature of multiple waves. This is clearly not possible for a realistic regret function which can be supposed to possess one of the three properties stated for any feasible segment of values of c j-1 . Now because of (2.3.14), the only possibility is: RA ( n ; a 1 …,a j-1 , a j-1 , a j ,..,a r ) = RA ( n ; a 1 …,a j-1 , a j-1 +1, a j , … ,a r ) = RA ( n ; a 1 , …,a j-1 , a j , a j ,…, a r ) 103
∴There exists an optimal A plan for which c j-1 = a j . Thus, the original supposition is contradicted. ∴ The theorem is proved. 2.3.6 Conclusion From the above results, we now know how the ordering of attributes and the process average values at the two states effect the cast of an A kind plan . In the chapter 2.5 we attempt to illustrate numerically the results obtained in this chapter. 104
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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
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 ) /
Page 105 and 106: We now consider the following funct
Page 107: ….(2.3.12) Here S A denotes the s
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
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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