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Journal of Applied Statistics Vol. 37, No. 3, March 2010, 453 460 A variables repetitive group sampling plan under failure-censored reliability tests for Weibull distribution Chi-Hyuck Jun*, Hyeseon Lee, Sang-Ho Lee and S. Balamurali Department of Industrial & Management Engineering, POSTECH, San 31 Hyoja-dong, Pohang 790-784, S. Korea (Received 11 March 2008; final version received 30 December 2008) We propose a variables repetitive group sampling plan under type-ii or failure-censored life testing when the lifetime of a part follows a Weibull distribution with a known shape parameter. The acceptance criteria do not involve unknown scale parameter differently from the existing plans. To determine the design parameters of the proposed plan, the usual approach of using two points on the operating characteristic curve is adopted and an optimization problem is formulated so as to minimize the average number of failures observed. Tables for design parameters are constructed when the quality of parts is represented by the unreliability or the ratio of the mean lifetime to the specified life. It is found that the proposed sampling plan can reduce the sample size significantly than do the single sampling plan. Keywords: acceptance sampling; consumer s risk; failure censoring; OC curve; producer s risk; progressive censoring 1. Introduction Suppose that the lifetime of a part follows a Weibull distribution with shape parameter m and scale parameter λ such that the cumulative distribution function is given by f(x)= 1 exp( (λx) m ), x 0. (1) Assume that the shape parameter m is known. Engineering experience with a particular type of application makes such an assumption quite reasonable because an estimate is readily available. If an estimate is not available, it should be estimated separately before the sampling plan is implemented. The quality of a part can be represented by the reliability, unreliability or its mean lifetime, etc. The unreliability of a part is defined by the probability that the lifetime (denoted by X) is shorter *Corresponding author. Email: chjun@postech.ac.kr ISSN 0266-4763 print/issn 1360-0532 online 2010 Taylor & Francis DOI: 10.1080/02664760802715914 http://www.informaworld.com
454 C.-H. Jun et al. than the specified time, μ 0 say. That is, p = P {X <μ 0 }=1 exp ( (λμ 0 ) m). (2) So, if we let w = (λμ 0 ) m, then the unreliability at time μ 0 is p = 1 exp( w). (3) Obviously, a part is considered as good if the unreliability is low. Often, it may be more convenient to represent the quality by the ratio of mean lifetime to a specified life. Since the mean life is derived by μ = Ɣ(1/m) mλ, (4) it follows that for the ratio b = μ/μ 0 ( ) Ɣ1/m m w = ln(1 p) =. (5) bm A part is considered as good if the ratio b is high. Note that the unreliability will be given when the mean ratio is specified and vice versa. Manufacturers would like to know from a life testing whether the quality level of their products meets the customer s requirements such as the minimum lifetime, reliability and so on. In most life testing a common restriction is the duration of the total time spent on testing. To reduce the test time of the experiment, many types of censoring schemes such as type-i (or time-censored), type- II (or failure-censored), a mix of type-i and type-ii [3], and progressive censoring [2] are usually adopted. Jun et al. [5] proposed and designed a variable sampling plan under sudden death testing. In this study, we consider a failure-censored sampling plan under Weibull distribution. Previous works in this area are available [4,8] for the unknown shape parameter in Weibull distribution, and the common approach is to utilize the extreme value distribution and the maximum likelihood estimation. As a result, designing a sampling plan is quite complicated for being used in practice. Although various sampling plans including single, double and sequential plans are available for normally distributed quality characteristics (see [7]), most of the plans for Weibull distributions are based on the single sampling plan. Sherman [9] proposed a new type of sampling plan for the inspection of attribute quality characteristics called the repetitive group sampling plan. The operation of the plan is similar to that of the sequential sampling plan. According to Sherman [9], the repetitive group sampling plans are not nearly as efficient as the sequential sampling plans but they are usually more efficient than the single sampling plan in terms of the average sample number. The variables repetitive group sampling plan for a normal distribution has been studied by Balamurali and Jun [1]. In this work, we propose the repetitive group sampling plans under failure censored testing for a Weibull distributed quality characteristic under the assumption as mentioned earlier. As a result, it turns out that designing a sampling plan is quite simple when it is assumed that the shape parameter in Weibull distribution is known. We also consider a sampling plan under progressively censoring as a special case. 2. Proposed sampling plan In a variables single sampling plan under type-ii censoring scheme, n units are placed on a test at time zero and test continues until r( n) failures to be observed and the remaining (n r) units are removed. If the degree of censoring q is given as in the work of Schneider [8], then the number r should be a design parameter and the sample size n is determined by n = r/(1 q). If the cost of
Journal of Applied Statistics 455 a test unit is high, the degree of censoring should be minimal to make the test units removed as small as possible. The proposed variables repetitive group sampling (VRGS) plan for the Weibull distribution is as follows: (1) Draw a random sample of size n from a lot. (2) Perform testing until r failures are observed and record X (i), the ith failure time (i = 1,..., r). (3) Calculate the quantity ν = r i=1 ( X(i) μ 0 ) m + (n r) ( X(r) μ 0 ) m. (6) (4) Accept the lot if ν k a and reject the lot if ν<k r.ifk r ν<k a, then repeat above steps (1) (4) through resampling. First note that design parameters are just (r, k a, k r ). It is seen that this reduces to a single sampling plan if k a = k r. Since (X (1), X (2),..., X (r) ) is the order statistic from a Weibull distribution, (X m (1),Xm (2),...,Xm (r) ) is the order statistic from an exponential distribution with parameter λm. So, it can be shown that the quantity ν in Equation (6) follows a Gamma distribution with parameters (r, w), where w is given by Equation (5). Further, 2νw follows a Chi-square distribution with degree of freedom 2r. So, the lot acceptance probability based on a single sample (group) when the lot quality is p will be P a (p) = P {ν k a p} =1 G 2r (2k a w), (7) where G φ is the distribution function of a Chi-square random variable with degree of freedom φ. On the other hand, the lot rejection probability is given by P r (p) = P {ν <k r p} =G 2r (2k r w). (8) Then, the lot of quality p will be finally accepted with probability of P a (p) A(p) = P a (p) + P r (p) = 1 G 2r (2k a w) 1 G 2r (2k a w) + G 2r (2k r w). (9) Note that the following holds for an integer r and a positive k: r 1 1 e kw (kw) j G 2r (2kw) =. j! In order to determine the design parameters (r, k a, k r ) of the VRGS plan, we use the two points on the operating characteristic (OC) curve. As in the work by Fertig and Mann [4], the probability of acceptance should be 1 α (α is called producer s risk) at the acceptable reliability level (ARL), p 2 say, and the probability of acceptance should be at β (this is called consumer s risk) at the lot tolerance reliability level (LTRL), p 1 say. Then these parameters can be chosen so as to satisfy the following two inequalities: j=0 A(p 1 ) β, A(p 2 ) 1 α. (10a) (10b) There may exist multiple solutions, we alternatively determine these parameters to minimize the average number of failures to be observed at b 2, which is analogous to minimizing the average
456 C.-H. Jun et al. sample number in a usual double sampling plan [10]. The average failure number (AFN) for the lot of quality p is obtained by AFN(p) = r P a (p) + P r (p) = r 1 G 2r (2k a w) + G 2r (2k r w). (11) Therefore, we consider the following optimization problem to determine parameters (r, k a, k r ). Minimize AFN(p 1 ) = Subject to A(p 1 ) = r 1 G 2r (2k a w 1 ) + G 2r (2k r w 1 ). (12a) 1 G 2r (2k a w 1 ) β, (12b) 1 G 2r (2k a w 1 ) + G 2r (2k r w 1 ) 1 G 2r (2k a w 2 ) A(p 2 ) = 1 α, (12c) 1 G 2r (2k a w 2 ) + G 2r (2k r w 2 ) r 1, k a k r 0. (12d) In the above, w 1 and w 2 are the values of w in Equation (5) at p = p 1 and p = p 2, respectively. Note that the objective function of Equation (12a) is evaluated at p = p 1 because it is larger than that at p = p 2, so it may be reasonable to minimize the larger value. To solve the above nonlinear optimization problems given in Equation (12), the sequential quadratic programming (SQP) proposed by Nocedal and Wright [6] can be used. The SQP has been implemented in Matlab Software using the routine fmincon. As we need an integral value of r, we first solved Equation (12) for k a and k r by various fixed r values (in increasing order) and then selected r to minimize the AFN. It should be noted that the solution may not be globally optimal because the problem is nonlinear. The issue of the existence and uniqueness of the solution was addressed in Nocedal and Wright [6]. The above problem can be solved to obtain the design parameters by a simple search using an Excel sheet when the related algorithm or software is not available. As mentioned earlier, the proposed plan reduces to the single sampling plan if k r = k a = k. In such a case, the problem (12) reduces to the one finding the minimum r and k satisfying: 2kw 1 χ 2 β,2r, 2kw 2 χ 2 1 α,2r, (13a) (13b) where χq,φ 2 denotes the percentage point of tail probability q in the Chi-square distribution with degree of freedom φ. It can be shown that the proposed plan approaches the single sampling plan with r = 1 and k = ln β/ln(1 p 1 ) if p 2 becomes very small. This follows because A(p 2 ) = 1 for a very small p 2 and the objective function will be minimal at 1. Table 1 shows the design parameters of the VRGS plan according to various two points of (p 1, β) and (p 2,1 α). Here, we chose α = 0.05, β = 0.1. Note that the design parameters can be determined independently of the shape parameter of the Weibull distribution. These tables also show the AFN at p 1 and the failure number from the single sampling plan, which indicates that the number of failures to be observed can be significantly reduced by using the VRGS plan as compared with the single sampling plan. It can be also seen that the AFN of the VRGS is smaller than two times r, which means that the decision can be made just from the first sample in most cases. From this table it can be observed that the proposed plan reduces to a single sampling plan when p 2 is chosen relatively low. It is also observed that for a fixed p 2 the number of failures to be observed increases as p 1 decreases. When the quality of products is represented by the ratio of the true mean to the specified life through b = μ/μ 0, two points of (b 1, β) and (b 2,1 α) should be specified to determine the design parameters. In this case, we need to solve Equation (12) by using the relation of Equation (5). If
Journal of Applied Statistics 457 Table 1. Design parameters of VRGS plans according to (p 1, p 2 ). p 1 p 2 r k a k r AFN r(single) 0.5 0.2 5 12.125 8.585 6.353 8 0.1 2 6.201 3.190 2.778 3 0.05 2 5.953 5.953 2 2 0.01 1 4.567 4.567 1 1 0.005 1 6.593 6.593 1 1 0.3 0.1 4 20.201 12.411 5.580 7 0.05 2 11.872 6.615 2.637 3 0.03 2 11.667 11.667 2 2 0.01 1 6.672 5.022 1.080 2 0.005 1 9.243 9.243 1 1 0.1 0.05 10 142.097 102.478 14.127 18 0.02 3 52.327 39.741 3.426 4 0.01 2 37.116 35.264 2.033 3 0.005 1 25.217 9.475 1.425 2 0.001 1 43.144 43.144 1 1 0.05 0.02 6 194.748 123.625 8.890 11 0.01 3 107.971 79.685 3.487 4 0.005 2 76.485 70.593 2.054 3 0.001 1 99.479 48.794 1.082 1 0.0005 1 97.354 97.354 1 1 a table is prepared similar to Table 1 having various pairs of (p 1, p 2 ), the design parameters can be read from this table. But, when preparing the tables by specifying the mean ratios, the shape parameter should be specified as well. When the lot is accepted under the two points of (b 1, β) and (b 2,1 α), it can be said that the true mean of products will be greater than or equal to b 1 times the specified life at the confidence level 1 β and that the producer s risk will be less than α when the true mean is b 2 times the specified life. Usually, b 1 = 1 is used in practice, in which case the true mean is directly compared with the specified life. Tables 2 4 show the design parameters of the proposed sampling plan according to (b 1, b 2 ) when m = 1 (exponential case), m = 2 and m = 3, respectively. Here, it is assumed that α = 0.05 and β = 0.1. Table 2. Design parameters of VRGS plans according to (b 1, b 2 ) when m = 1. b 1 b 2 r k a k r AFN r(single) 1 2 10 15.116 10.439 15.108 19 3 5 8.4872 5.7108 6.6729 8 4 3 5.9165 3.0695 4.5593 6 5 3 5.563 3.9942 3.5477 4 10 2 3.9358 3.5328 2.0748 3 1.5 3 10 22.674 15.659 15.108 19 4 6 14.744 10.105 8.1278 10 5 4 10.86 6.5158 5.7081 7 10 2 6.4238 3.3686 2.742 3 2 3 26 68.397 53.113 41.128 54 4 10 30.231 20.879 15.108 19 5 6 20.09 12.436 9.1729 11 10 3 11.126 7.9884 3.5477 4 3 4 49 180.65 148.91 79.475 106 5 17 71.002 52.369 26.584 34 6 10 45.347 31.318 15.108 19 10 4 21.72 13.032 5.7081 7
458 C.-H. Jun et al. Table 3. Design parameters of VRGS plans according to (b 1, b 2 ) when m = 2. b 1 b 2 r k a k r AFN r(single) 1 2 3 7.533 3.908 4.559 6 3 2 5.096 4.014 2.188 3 4 1 3.647 0.917 1.754 2 5 1 3.244 1.550 1.278 2 10 1 2.937 2.937 1 1 1.5 3 3 16.950 8.793 4.559 6 4 2 12.061 6.937 2.585 3 5 2 11.312 11.312 2 2 10 1 6.604 6.527 1.003 2 2 3 8 63.591 44.027 11.377 14 4 3 30.133 15.633 4.559 6 5 2 22.220 10.589 2.927 4 10 1 12.978 6.200 1.278 2 3 4 14 229.338 166.654 21.289 27 5 5 99.715 59.534 7.587 9 6 3 67.798 35.173 4.559 6 10 2 54.341 44.736 2.103 2 Table 4. Design parameters of VRGS plans according to (b 1, b 2 ) when m = 3. b 1 b 2 r k a k r AFN r(single) 1 2 2 5.7515 3.8874 2.3577 3 3 1 3.5185 1.8639 1.2248 2 4 1 4.5614 4.5614 1 1 5 1 3.264 3.264 1 1 10 1 3.2336 3.2336 1 1 1.5 3 2 19.411 13.12 2.3577 3 4 1 12.938 4.1925 1.533 2 5 1 11.203 8.8893 1.0629 2 10 1 23.914 23.914 1 1 2 3 4 81.023 49.548 5.6046 7 4 2 46.012 31.099 2.3577 3 5 1 32.406 7.8549 1.7893 2 10 1 63.134 63.134 1 1 3 4 7 426.67 283.67 10.173 13 5 3 249.95 135.62 4.0909 5 6 2 155.29 104.96 2.3577 3 10 1 89.622 71.114 1.0629 2 When comparing the above three tables, it is seen that the number of failures to be observed (r) in the proposed sampling plan tends to decrease as the shape parameter (m) of the Weibull distribution increases. If we compare the case of m = 2 with the case of m = 3, the number of failures to be observed does not differ much particularly when b 1 is lower and b 2 is higher. So, the proposed sampling plan may not be seriously sensitive to the misspecified value of the shape parameter of the Weibull distribution in those cases. Example 1 Suppose a manufacturer of ball bearings wants to know whether his products have the mean lifetime greater than the specified life 10,000 (cycles) at 10% of consumer s risk if it is accepted. It is known that the lifetime of a ball bearing follows a Weibull distribution with shape parameter of m = 2. They select a repetitive group sample plan and would like to determine the plan parameters satisfying the above requirement and yielding the producer s risk of the acceptance smaller than 5% when the true mean is 30,000 cycles. In this example, it is set that
Journal of Applied Statistics 459 b 1 = 1 and b 2 = 3, so from Table 3 the plan parameters are r = 2, k a = 5.096 and k r = 4.014. So, the manufacturer is required to test ball bearings until two failures are observed. Suppose now that the first failure was observed at 12,010 and the second at 16,800 when five bearings were put to test. Then, the quantity in Equation (6) is obtained by ν = 12.732 > k a = 5.096. So, the manufacture can accept the lot from the first sample. 3. Special cases Note that the VRGS plan reduces to a single sampling plan if k a = k r whose procedure can be proceeded up to Step 4 mentioned in Section 2. It should also be noted that the single sampling plan (case of k a = k r ) can be utilized to design a progressive censoring scheme, where a certain proportion (called the censoring fraction) of functioning items will be removed from the test whenever a failure is observed. The following sampling plan is a new progressively censored reliability sampling plan similar to the one proposed by Balasooriya et al. [2]. If q i denotes the censoring fraction at the ith failure, then the sample size required can be determined by ν = Let us consider the following progressive censored sampling plan: r ( 1 r i=1 q i). (14) (1) Draw a random sample of size n from a lot. (2) Perform testing until r failures are observed. When the ith failure occurs, record its time X (i) and remove R i ( = nq i ) functioning items (i = 1,..., r). (3) Calculate the quantity r ( ) m X(i) ν = (R i + 1). (15) i=1 (4) Accept the lot if ν k and reject the lot, otherwise. It turns out that the quantity ν in Equation (15) follows Gamma distribution with parameters (r, w). So, the design parameters (r, k) can be determined by using the inequalities (13) or the optimization problem in Equation (12). Note in the above plan that the sample size is determined by r i=1 q i, not by the individual q i. For example, when p 0 = 0.01 at α = 0.05 and p 1 = 0.05 at β = 0.1, Balasooriya et al. [2] estimate the sample size n = 77 for q 1 = 0.42, q 2 = 0, q r = 0.28 while n = 74 for q 1 = 0.24, q 2 = 0.23, q r = 0.23. However, in ours determined the sample size as n = 3.82/(1 0.7) = 12.73 for both cases. 4. Concluding remarks We proposed variables repetitive group sampling plans under failure-censored reliability tests for Weibull distribution with known shape parameter, where the unknown scale parameter is not included in the acceptance criteria. It can be seen that a variables sampling plan for the Weibull distribution will be designed relatively in a compact way when the shape parameter is assumed known. This assumption may not be restrictive particularly when manufacturers have accumulated knowledge from the past history. The simpler analysis may also give us an insight for interpreting the results. It turned out that the design parameters of the proposed plan can be determined independently of the shape parameter when the quality, if represented by the unreliability, and that the existing plans for exponential distribution, if any, can be utilized. It was observed that the proposed plan can significantly reduce the number of failures to be observed in a life test as compared with the single sampling plan. μ 0
460 C.-H. Jun et al. To determine the number of units put on test initially and the number of failures to be observed simultaneously, a certain cost model may be needed for the economic design of the proposed plan. This approach can also be used when considering the amount of time required to reach a decision on the lot acceptability. Acknowledgements The authors thank anonymous reviewers for their valuable comments. This work was supported by KOSEF through the National Core Research Center for System Bio-Dynamics at POSTECH. References [1] S. Balamurali and C.-H. Jun, Repetitive group sampling procedure for variables inspection, J. Appl. Stat. 33 (2006), pp. 327 338. [2] U. Balasooriya, S.L.C. Saw, and V.G. Gadag, Progressively censored reliability sampling plans for the Weibull distribution, Technometrics 42 (2000), pp. 160 168. [3] J. Chen, W. Chou, and H. Zhou, Designing acceptance sampling schemes for life testing with mixed censoring, Naval Res. Logist. 51 (2004), pp. 597 612. [4] K.W. Fertig and N.R. Mann, Life-test sampling plans for two-parameter Weibull populations, Technometrics 22 (1980), pp. 165 177. [5] C-H. Jun, S. Balamurali, and S.-H. Lee, Variables sampling plans for Weibull distributed lifetimes under sudden death testing, IEEE Trans. Reliab 55 (1980), pp. 53 58. [6] J. Nocedal and S.J. Wright, Numerical Optimization, Springer, New York, 1999. [7] E.G. Schilling, Acceptance Sampling in Quality Control, Marcel Dekker, New York, NY, 1982. [8] H. Schneider, Failure-censored variables-sampling plans for lognormal and Weibull distributions, Technometrics 31 (1989), pp. 199 206. [9] R.E. Sherman, Design and evaluation of repetitive group sampling plan, Technometrics 7 (1965), pp. 11 21. [10] D.J. Sommers, Two-point double variables sampling plan, J. Qual. Technol. 13 (1981), pp. 25 30.