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

More documents

Recommendations

Info

m β (c 1 , c 2 , ρ) = m ′ ...(1.3.7) Note that the smallest value of n (as a real number) is obtained as n = m β (c)/p ′ as the plan giving the smallest sample size. 1.3.3 Construction of C kind <strong>plans</strong> of given strength The Problem We consider the case r = 2 and we want to find a C kind plan satisfying the equations: G(c 1 , np 1 ).G(c 2 , np 2 ) = 1 − α G(c 1 , np ′ 1).G(c 2 , np ′ 2) = β ...(1.3.5) Restricting to the situation where p 1 /(p 1 + p 2 ) = p ′ 1/(p ′ 1 + p ′ 2) = ρ we write p = p 1 + p 2 , p ′ = p ′ 1 + p ′ 2, m = np, m ′ = np ′ . For a given ρ, we define m P (c 1 , c 2 , ρ) as the value of m satisfying the equation G(c 1 , mρ)G(c 2 , m(1 − ρ)) = P ...(1.3.6) Note: The function m P (c 1 , c 2 , ρ) used for the C kind MASSP is similar to the function m P (c) defined in the context to the single <strong>sampling</strong> plan for single attribute. We must have n, c 1 , c 2 such that m 1−α (c 1 , c 2 , ρ) = m Lemma Let G(c 1 , M 1 ρ)G(c 2 , M 1 (1−ρ)) = G(c 1 , M 2 ρ)G(c 2 +1, M 2 (1−ρ)) = G(c 1 +1, M 3 ρ).G(c 2 , M 3 (1−ρ)) = P. Then M 2 ≥ M 1 and M 3 ≥ M 1 ...(1.3.8) Proof 65
Let M 2 < M 1 . Then G(c 1 , M 1 ρ) < G(c 1 , M 2 ρ) and G(c 2 , M 1 (1 − ρ)) < G(c 2 , M 2 (1 − ρ)) < G(c 2 + 1, M 2 (1 − ρ)). This makes G(c 1 , M 1 ρ).G(c 2 , M 1 (1−ρ)) < G(c 1 , M 2 ρ).G(c 2 +1, M 2 (1−ρ)) which is not true. Hence M 2 ≥ M 1 . If M 3 < M 1 then G(c 1 + 1, M 3 ρ)) > G(c 1 , M 1 ρ)) and G(c 2 , M 3 (1 − ρ)) > G(c 2 , M 1 (1 − ρ)) This makes G(c 1 , M 1 ρ).G(c 2 , M 1 (1 − ρ)) < G(c 1 , M 3 ρ).G(c 2 + 1, M 3 (1 − ρ)) which is not true. Hence M 3 ≥ M 1 . We now introduce the auxiliary function R(c 1 , c 2 , ρ, α, β) = m β (c 1 , c 2 , ρ)/m 1−α (c 1 , c 2 , ρ) R(c 1 , c 2 , ρ, α, β) is conveniently taken to ∞ when c 2 = −1 ...(1.3.9) Theorem 1.3.1 If we increase c 2 keeping c 1 as fixed then as c 2 → ∞, R(c 1 , c 2 , ρ, α, β) first decreases to a minimum and then increases so that the function has a unique minimum in the range m > 0 We show this by the following partly demonstrative arguments which are supported by extensive numerical computations (not included in the dissertation for the sake of brevity). 1. From the lemma preceeding the theorem, we observe that if we increase c 2 keeping c 1 as fixed i.e. when c 2 → ∞ the function m 1−α (c 1 , c 2 , ρ) increases. But G(c 1 , mρ) must be greater than or equal to 1 − α, else G(c 1 , mρ).G(c 2 , m(1 − ρ)) becomes less than 1 − α. 2. It also follows that m 1−α (c 1 , c 2 , ρ) → m 1−α (c 1 )/ρ as c 2 → ∞. Similarly, m β (c 1 )/ρ[m 1−α (c) and m β (c) are the Poisson fractiles of appropriate strengths as defined in section 1.3.1 ] 3. However, m β (c 1 , c 2 , ρ) > m 1−α (c 1 , c 2 , ρ) always. And for a small increase of c 2 both m β (c 1 , c 2 , ρ) and m 1−α (c 1 , c 2 , ρ) will increase by a small amount relative to their original values and therfore their ratio will decrease in general. Thus for small values of c 2 , R(c 1 , c 2 , ρ, α, β) is a decreasing function of c 2 .[Reference figure 1.3.1 ] 4. With the increase in c 2 beyond some point the increase in m 1−α (c 1 , c 2 , ρ) will be very small but increase in m β (c 1 , c 2 , ρ) will still be finite. As c 2 increases the denominator of R(c 1 , c 2 , ρ, α, β) approximates its limiting value m 1−α (c 1 )/ρ much earlier usually than 66
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 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: 1.3 Multiattribute sampling plans o
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 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
show all

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

Create successful ePaper yourself

Delete template?

Save as template?