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c Heldermann Verlag Economic Quality Control ISSN 0940-5151 Vol 22 (2007), No. 1, 117 – 126 Sampling Risks for Chain Sampling Plans with Non-Constant Defective Probability A.R. Sudamani Ramaswamy and A.R. Manju Priya Abstract: This paper introduces modiﬁed producer ’s risk and consumer’s risk in the case of chain sampling plans obtained by modelling the defective probability p not as a constant, but as a random variable that follows a beta distribution. Keywords: Acceptable Quality Level, Chain Sampling, Lot Tolerance Proportion Defective. 1 Introduction The concept of chain sampling plans was introduced by Dodge [2] in 1955, to overcome the problem of lack of discrimination for single sampling plans with zero acceptance number c = 0. The procedure was developed to “chain” together the most recent inspections in a way that would build up the shoulder of the operating characteristic (OC)curve of c = 0 plans. This is especially desirable in a situation in which small samples are demanded because of economic or physical diﬃculties for obtaining a sample. The procedure of ChSP-1 is as follows: 1. From each lot, select a sample of n units. 2. Accept the lot, if (a) no defects are found in the sample, or (b) one defective is found in the sample, but no defective item was found in the previous i samples of n. Soundararajan [3] has presented procedures and tables for the construction of chain sampling plans and for selection of plans by speciﬁed properties under the conditions of the poisson model. The design of chain sampling plan is based on the producer’s risk α and the consumer’s risk β. Traditionally, α is deﬁned as the probability of rejecting a lot in which the defective probability, p, equals acceptable quality level (AQL) speciﬁed by the producer. α = P r(X > c | p = AQL) (1) Similarly, β is deﬁned as the probability of accepting a lot in which p is equal to the lot tolerance proportion defective (LTPD): β = P r(X ≤ c | p = LT P D) (2) 118 A.R. Sudamani Ramaswamy and A.R. Manju Priya The deﬁnitions of α and β are based on the assumption that incoming lots are formed from a production process that is stable with a constant defective probability p. Chun and Rinks [1] have assumed that the defective probability p is a random variable that follows beta distribution and derived modiﬁed producer’s risk and consumer’s risk α and β for single sampling inspec- tion plan. According to Chun and Rinks [1] the beta distribution represents the variations in the defective probability p better than any other distribution. The intent of this paper is to derive modiﬁed producer’s risk and consumer’s risk for chain sampling plans, assuming that the defective probability is a random variable that follows beta distribution. The producer’s risk and consumer’s risk displayed in Table 1 and Table 2 are derived on the assumption of perfect inspection even though errors are inevitable in any inspection process. 2 Modiﬁed Producer’s and Consumer’s Risk If p is considered constant, then for a chain sampling plan given by (n, i) the probability of accepting a lot P a(p) is obtained by means of the binomial distribution: P a(p) = (1 − p)n + (np)(1 − p)n−1 (1 − p)ni (3) The producer’s risk α is given by α = 1 − P a(AQL) = 1 − E[P a(AQL)] = E[1 − P a(AQL)] = E[1 − P a(p) | p = AQL] (4) Following the representation (4) Chun and Rinks [1] deﬁned the modiﬁed producer’s risk α when p is modelled by a random variable as the conditional expectation as follows: α = E[1 − P a(p) | 0 ≤ p ≤ AQL] ˜ (5) ˜ The modiﬁed consumer’s risk β is deﬁned analogously as follows: ˜ β = E[P a(p) | LT P D ≤ p ≤ 1] (6) When p is modelled as a constant α and β for a chain sampling plan with parameters n and i are calculated using the binomial distribution: α = 1 − (1 − AQL)n + nAQL(1 − AQL)n−1 (1 − AQL)ni (7) n n−1 ni β = (1 − LT P D) + nLT P D(1 − LT P D) (1 − LT P D) (8) Let f (p | a, b) be the density function of the beta distribution with parameters a and b. Then the following relation holds: pa−1 (1 − p)b−1 f (p | a, b) = for 0 < p < 1 (9) B(a, b) Γ[a]Γ[b] where B(a, b) = Γ[a + b] Moreover, the incomplete beta function is given by: Sampling Risks for Chain Sampling Plans with Non-Constant Defective Probability 119 y 1 Iy (a, b) = pa−1 (1 − p)b−1 dp (10) B(a, b) 0 Proposition 1: ˜ Let A = AQL, then the modiﬁed producer’s risk α for a chain sampling plan with parameters n and i is given by: B(a, n+b) IA (a, n+b) B(a+1, n+ni+b−1) IA (a+1, n+ni+b−1) α=1− ˜ +n (11) B(a, b) IA (a, b) B(a, b) IA (a, b) ˜ and the modiﬁed consumer’s risk β is given by: ˜ B(a, n+b) 1−IL (a, n+b) + n B(a+1, n+ni+b−1) 1−IL (a+1, n+ni+b−1) β= (12) B(a, b) 1−IL (a, b) B(a, b) 1 − IL (a, b) Proof: ˜ The producer’s risk α is deﬁned as: α = 1 − E[P a(p) | p ≤ A] ˜ (13) For a chain sampling plan the conditional expectation based on the binomial sampling model is given as: E[P a(p) | p ≤ A] = E q n + npq n−1 q ni | p ≤ A = E [q n | p ≤ A] + nE pq n−1 q ni | p ≤ A (14) If p has beta distribution with parameters a and b then the conditional expectation becomes: A a−1 b−1 A a−1 b−1 q n p B(a,b) dp q pq n+ni−1 p B(a,b) dp q 0 E[P a(p) | p ≤ A] = A + n0 A pa−1 q b−1 pa−1 q b−1 B(a,b) dp B(a,b) dp 0 0 A A pa−1 q n+b−1 pa q n+ni+b−2 B(a,b) dp B(a,b) dp 0 = A + n0 A pa−1 q b−1 pa−1 q b−1 B(a,b) dp B(a,b) dp 0 0 B(a, n+b) IA (a, n+b) = + B(a, b) IA (a, b) B(a+1, n+ni+b−1) IA (a+1, n+ni+b−1) n (15) B(a, b) IA (a, b) Thus from equations (14) and (15) the modiﬁed producer’s risk (11) is obtained. ˜ The modiﬁed consumer’s risk β as given by (12) is obtained analogously. Corollary 1: 1−A For a chain sampling plan with parameters n and i, let k = A for a given acceptable quality 120 A.R. Sudamani Ramaswamy and A.R. Manju Priya ˜ level A = AQL. then the modiﬁed producer’s risks α converges to the classical producer’s risk α as a → ∞ and b = ka: ˜ lim α = α a→∞ (16) b=ka where a and b are the parameters of the considered beta distribution of p. Proof: 1−A Consider the beta distribution with parameters a and b = ka for a ﬁxed value k = A . With (7) we will show that B(a, n + b) lim a→∞ = (1 − A)n (17) b=ka B(a, b) and B(a + 1, n + ni + b − 1) lim a→∞ = A(1 − A)n−1 (1 − A)ni (18) b=ka B(a, b) Consider B(a, n + b) Γ[a]Γ[n + b] Γ[a + b] lim a→∞ lim = a→∞ (19) b=ka B(a, b) b=ka Γ[a + n + b] Γ[a]Γ[b] and Γ[a + b] Γ[n + b] Γ[a + b] (b + n − 1) · · · bΓ[b] lim = lim a→∞ b=ka Γ[a + b + n] Γ[b] a→∞ b=ka (a + b + n − 1) · · · (a + b)Γ[a + b] Γ[b] (b + n − 1) · · · b = lim a→∞ b=ka (a + b + n − 1) · · · (a + b) (ka + n − 1) · · · ka = lim a→∞ (a + ka + n − 1) · · · (a + ka) k + n−1 · · · k a = lim a→∞ k + 1 + n−1 · · · (k + 1) a n n kn k 1 = = = 1− (20) (k + 1)n k+1 k+1 Analogously we obtain: n−1 ni B(a + 1, n + ni + b − 1) 1 1 1 lim a→∞ = 1− 1− (21) b=ka B(a, b) k+1 k+1 k+1 For b = ka = 1−A a we obtain from the beta distribution: A a E[p] = =A (22) a+b 1−A Thus, for k = A in (20) and (21) the limiting value is given by: B(a, n + b) lim a→∞ = (1 − A)n (23) b=ka B(a, b) B(a + 1, n + ni + b − 1) lim a→∞ = A(1 − A)n−1 (1 − A)ni (24) b=ka B(a, b) Sampling Risks for Chain Sampling Plans with Non-Constant Defective Probability 121 ˜ Thus, the limiting value of α is obtained: B(a, n + b) B(a + 1, n + ni + b − 1) lim α = 1 − a→∞ a→∞ ˜ lim − n a→∞ lim b=ka b=ka B(a, b) b=ka B(a, b) = 1 − (1 − A)n + nA(1 − A)n−1 (1 − A)ni = α (25) 1−L In the same way, we obtain for k = L with L = LTPD the limit: ˜ lim β = β (26) a→∞ b=ka 3 Numerical Examples for the Modiﬁed Producer’s Risk ˜ In Table 1 the values of α for various chain sampling plans are displayed . The defective probability p is modelled by a beta distribution with parameters ((a, b). For comparison the case of a constant defective probability with p = AQL is also included. Four diﬀerent AQL ˜ values are considered. Comparing the α values in case of variable p with that of constant the case, one can see that the risk is maximum when p is constant. Moreover, it is revealed that the ˜ modiﬁed producer’s risk α has its lowest value when p has a beta distribution with parameter (1, 1). Below the considered cases for the sampling plan, the beta distribution and the AQL are listed. sampling plan n 10 parameters i 1,2,3,4,5,6 beta distribution (a, b) (1,1), (2,18), (4,36) parameters AQL: A 0.005, 0.025, 0.050, 0.150 4 Numerical examples for the Modiﬁed Consumer’s Risk ˜ In Table 2 values for the modiﬁed consumer’s risk β for chain sampling plans with n = 10 and i = 1, 2, . . . , 6 are displayed for the case that the defective probability is modelled by various beta distributions. The considered LTPD values are set equal to 0.10, 0.15 and 0.20. Comparing the ˜ values β of the modiﬁed risks for the diﬀerent beta distributions shows that the risk is maximum for the case of a constant defective probability p. 122 A.R. Sudamani Ramaswamy and A.R. Manju Priya ˜ Table 1: Modiﬁed Producer’s Risk α for Chain Sampling Plans with n = 10. Average Quality Level(AQL) i Betaparameters 0.005 0.025 0.050 0.150 (a,b) Modiﬁed Producer’s Risk, α 1 (1,1) 0.001160 0.024679 0.081225 0.361239 (2,18) 0.001715 0.034487 0.105681 0.356257 (4,36) 0.002283 0.045504 0.137535 0.423776 constant 0.003432 0.069135 0.212586 0.734726 2 (1,1) 0.001923 0.038141 0.116344 0.426683 (2,18) 0.002841 0.053044 0.150438 0.427417 (4,36) 0.003775 0.069521 0.193477 0.502790 constant 0.005654 0.103700 0.288295 0.789660 3 (1,1) 0.002658 0.049362 0.141066 0.455137 (2,18) 0.003922 0.068337 0.181371 0.459815 (4,36) 0.005205 0.089000 0.230819 0.536124 constant 0.007768 0.130534 0.333625 0.800474 4 (1,1) 0.003367 0.058740 0.158691 0.469281 (2,18) 0.004961 0.080967 0.202975 0.476258 (4,36) 0.006575 0.104817 0.255919 0.551552 constant 0.009779 0.151366 0.360766 0.802604 5 (1,1) 0.004049 0.066601 0.171425 0.477155 (2,18) 0.005960 0.091420 0.218231 0.485392 (4,36) 0.007888 0.117679 0.272918 0.559294 constant 0.011691 0.167538 0.377016 0.803023 6 (1,1) 0.004707 0.073210 0.180754 0.481947 (2,18) 0.006920 0.100091 0.229130 0.490849 (4,36) 0.009146 0.128153 0.284522 0.563455 constant 0.013509 0.180093 0.386745 0.803105 5 Numerical Examples for Sample Size and Producer’s Risk For designing a chain sampling plan, the sample size n and the number i are speciﬁed for attaining a certain level of the two risks. Table 3 shows the sample size n required to obtain ˜ the producer’s risk α = 0.05 for given i = 1, 2, . . . , 6. As in Chun and Rinks [1], we observe that the required sample size increases with decreasing AQL. Again, the defective probability is modelled by beta distributions with parameters (1, 1), (2, 18) or (4, 36) or that p is a constant equal to the AQL value. Sampling Risks for Chain Sampling Plans with Non-Constant Defective Probability 123 Table 2: Modiﬁed Consumer’s Risk for Chain Sampling Plans with n = 10. i Beta parameters LTPD (a,b) 0.10 0.15 0.20 1 (1,1) 0.041347 0.022240 0.011477 (2,18) 0.269898 0.150922 0.079159 (4,36) 0.316003 0.181848 0.096393 constant 0.483764 0.265274 0.136197 2 (1,1) 0.033724 0.018429 0.009878 (2,18) 0.216980 0.121555 0.066042 (4,36) 0.253804 0.145553 0.079714 constant 0.395780 0.210340 0.110469 3 (1,1) 0.032199 0.017973 0.009770 (2,18) 0.204084 0.117319 0.064986 (4,36) 0.237794 0.139962 0.078275 constant 0.365102 0.199526 0.107706 4 (1,1) 0.031833 0.017909 0.009762 (2,18) 0.200662 0.116664 0.064896 (4,36) 0.233416 0.139060 0.078146 constant 0.354405 0.197396 0.107410 5 (1,1) 0.031736 0.017900 0.009761 (2,18) 0.199704 0.116558 0.064888 (4,36) 0.232167 0.138910 0.078134 constant 0.350675 0.196977 0.107378 6 (1,1) 0.031709 0.017898 0.009761 (2,18) 0.199426 0.116540 0.064887 (4,36) 0.231800 0.138850 0.078133 constant 0.349375 0.196895 0.107375 Example: Consider the case of AQL=0.005 and i = 2. For a constant p, the chain sampling plan with the sample size n = 32 gives a producer’s risk of α = 0.05. If p is assumed to follow a beta distribution with parameters (4, 36), then n should be 40 to achieve the same value of the producer’s risk. If p is assumed to be a beta random variable with parameters (2, 18), then the sample size value must be increased to n = 47 to achieve the same value of the producer’s risk.If p is assumed to follow beta distribution with parameters (1, 1) the sample size should be even larger, namely n = 58. Thus we can see that when the variance of p increases with the mean ﬁxed at a certain value, the sample size n for a ﬁxed value of i has to be increased in order to guarantee the same level of the producer’s risk. 124 A.R. Sudamani Ramaswamy and A.R. Manju Priya ˜ Table 3: Sample Size Requirements for chain sampling plans and a producer’s risk of α = 0.05. Average Quality Level (AQL) i Betaparameters 0.005 0.025 0.050 0.100 (a,b) Required sample size n 1 (1,1) 74 14 7 3 (2,18) 59 12 6 3 (4,36) 51 10 5 3 constant 41 8 4 2 2 (1,1) 58 11 5 2 (2,18) 47 9 5 2 (4,36) 40 8 4 2 constant 32 6 3 1 3 (1,1) 50 10 5 2 (2,18) 40 8 4 2 (4,36) 34 7 3 2 constant 27 5 2 1 4 (1,1) 45 9 4 2 (2,18) 36 7 3 2 (4,36) 30 6 3 1 constant 24 4 2 1 5 (1,1) 41 8 4 2 (2,18) 33 6 3 1 (4,36) 28 5 2 1 constant 22 4 2 1 6 (1,1) 38 7 3 1 (2,18) 30 6 3 1 (4,36) 26 5 2 1 constant 21 4 2 1 6 Numerical Examples for Sample Size and Consumer’s Risk ˜ Table 4 shows the sample size n required to achieve a consumer’s risk β ≈ 0.10 without exceeding it. A The LTPD values are taken as 0.10,0.15 and 0.20, which are the same as in Chun and Rinks [1]. From the table it is seen that required sample size decreases with increasing LTPD value for i = 1, 2, . . . , 6. Example: Consider a LTPD=0.15 and i = 2, then n = 15 is necessary for achieving β ≈ 0.10 when p is a constant. When p is a random variable following a beta distribution, the required sample size is smaller than in the case of a constant defective probability. For illustration let LTPD=0.15 and i = 2, then a sample size n = 12 is necessary for the beta random variable with parameters (4, 36) and ˜ n = 11 for (2, 18) to achieve β ≈ 0.10. Sampling Risks for Chain Sampling Plans with Non-Constant Defective Probability 125 From the numerical results we conclude that, as the variance of p increases, the required sample size n gets smaller for achieving a ﬁxed consumer’s risk. Table 4: Sample Size Requirements for Chain Sampling Plans and a Consumer’s Risk of ˜ β ≈ 0.10. i Beta parameters LTPD (a,b) 0.10 0.15 0.20 Required sample size n 1 (1,1) 7 6 5 (2,18) 17 12 10 (4,36) 18 13 10 constant 24 16 12 2 (1,1) 6 5 4 (2,18) 15 11 9 (4,36) 17 12 10 constant 23 15 11 3 (1,1) 6 5 4 (2,18) 15 11 9 (4,36) 17 12 10 constant 22 15 11 4 (1,1) 6 5 4 (2,18) 15 11 9 (4,36) 17 12 10 constant 22 15 11 5 (1,1) 6 5 4 (2,18) 15 11 9 (4,36) 17 12 10 constant 22 15 11 6 (1,1) 6 5 4 (2,18) 15 11 9 (4,36) 17 12 10 constant 22 15 11 7 Conclusion In the design of acceptance sampling plan , random variations in the defective probability p must be taken appropriately accounted for. If this is done, it is shown by means of a chain sampling plan that the resulting modiﬁed producer’s risk and consumer’s diﬀer considerably from the classical producer’s risk and consumer’s risk. The related problem can be solved by reducing the variations in p. 126 A.R. Sudamani Ramaswamy and A.R. Manju Priya References [1] Chun, Y.H. and Rinks, D.B. (1998): Three Types Of Producer’s and Consumer’s Risks in the Single Sampling Plan. Journal of Quality Technology 30, 254-268. [2] Dodge, H.F.(1955): Chain Sampling Inspection Plans. Industrial Quality Control 11, 10- 13. [3] Soundararajan, V.(1978): Producer’s and Tables for the Construction and Selection Of Chain Sampling Plans (ChSP-1), Part-1. Journal of Quality Technology 10, 56-60. A.R. Sudamani Ramaswamy A.R. Manju Priya Department of Mathematics PSG College of Technology Avinashilingam Deemed University Coimbatore-641004 Coimbatore-641043 Tamilnadu Tamilnadu India India

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