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Variables Sampling Plans for Weibull Distributed Lifetimes Under Sudden Death Testing Chi-Hyuck Jun, Member, IEEE, S. Balamurali, and Sang-Ho Lee
lower specification limit regarding the lifetime of a part scale parameter of the Weibull distribution shape parameter of the Weibull distribution total number of specimens to be evaluated acceptable reliability level lot tolerance reliability level lot acceptance probability under a single sampling plan when the lot quality is lot acceptance probability at the first sample of a double sampling when the lot quality is lot acceptance probability at the second sample of a double sampling when the lot quality is number of specimens assigned in a group time to the first failure from the -th group
Abstract—Sudden death testing can be utilized for deciding upon the lot acceptance of manufactured parts. Variables single, and double sampling plans are proposed for the lot acceptance of parts whose life follows a Weibull distribution with known shape parameter. The proposed plans are different from the existing ones in that the lot acceptance criteria do not depend on the estimated scale parameter. Design parameters of both sampling plans are determined by using the usual two-point approach. The number of groups is determined independently of the group size, and even independently of the shape parameter. Also, the double sampling plan can reduce the average number of groups required. The effects of mis-specification of the shape parameter on the probability of accepting the lots under the single sampling plan are analyzed & discussed. Index Terms—Double sampling, lot acceptance, operating characteristic curve, single sampling.
AGN OC
, ,
ACRONYMS1 average group number operating characteristic
I. INTRODUCTION
NOTATION producer’s risk lot acceptance probability under a double sampling plan when the lot quality is average group number when the lot quality is consumer’s risk percentage point of the tail probability in a Chisquare distribution with degree of freedom number of groups of specimens under a single sampling plan number of groups at the first sample under a double sampling plan number of groups at the second sample under a double sampling plan cdf of a Chi-square distribution with degree of freedom parameter related to acceptance of the lot under a single sampling plan parameters related to acceptance of the lot under a double sampling plan
Manuscript received March 8, 2005; revised March 24, 2005 and September 6, 2005. This work was supported by MOST (KOSEF) through National Core Research Center for Systems Bio-Dynamics. Associate Editor: M. Xie. The authors are with the Division of Mechanical and Industrial Engineering, National Core Research Center for Systems Bio-Dynamics Pohang University of Science and Technology, Pohang, Kyungbuk 790-784, Korea (e-mail:
[email protected]). Digital Object Identifier 10.1109/TR.2005.863802 1The
singular and plural of an acronym are always spelled the same.
S
UDDEN death testing is frequently adopted by parts manufacturers to reduce testing time. The total number of specimens to be evaluated is divided into equal-sized groups according to the number of available experimental testers, for example [1], [2]. Thus, there are specimens in each group, and there are a total of groups so that equals times . The specimens in each group are tested identically & simultaneously on different testers. The first group of specimens is run until the first failure occurs. At this point, the surviving specimens are suspended & removed from testing. An equal set of new specimens numbering is next tested until the first failure. This process is repeated until one failure is generated from each of the groups. specimens are In the end, failures are observed while suspended. Balasooriya [3] first considers reliability sampling plans under the sudden death testing for the two parameter exponential distribution although the terminology ‘sudden death’ was not mentioned. Wu et al. [4] called this type of test a ‘limited failure-censored life test,’ and analyzed the expected test time. In most life testing, to reduce the test time of the experiment, a failure-censored (type-II) scheme, or time-censored (type-I) scheme is usually adopted. Acceptance sampling plans for the Weibull or some other distributions have been considered by many authors under various censoring schemes. Reliability sampling plans for the Weibull distribution under a type-II censoring scheme are proposed in [5], [6]. A different plan for the Weibull distribution having a known shape parameter is proposed in [7] using a cost model under some warranty policy. Lam [8] considers the sampling plan for the exponential distribution under a type-I censoring using a Bayesian approach.
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Type-I censoring can be generalized to random censoring, and some plans are developed for the exponential distribution [9], [10]. A sampling plan for the Weibull distribution is also available under a mixture of type-I & type-II censoring schemes [11]. Recently, various type-I & type-II progressive censoring plans are proposed in order to generalize test schemes for sufficiently large samples. Balasooriya & Low [12] propose a plan for the Weibull distribution under a type-I progressive censoring scheme by considering multiple failure modes. The sampling plans for the Weibull distribution under a type-II progressive censoring scheme are considered by many authors [13]–[15]. Similar plans under type-II progressive censoring for the two-parameter exponential distribution, and for the lognormal distribution, are available in [16], and [17], respectively. Sometimes, however, manufacturers do not have the freedom to choose among testing methods, but they are limited due to testing equipment & environment. Most decision rules regarding lot acceptance are based on the one in Lieberman & Resnikoff [18]. The common approach to deriving these rules for the Weibull distribution is to utilize the extreme value distribution, and the maximum likelihood estimation. So, the estimation of distribution parameters under various testing or censoring schemes is closely related to the research on the reliability acceptance sampling. See for example [19], [20]. Sudden death testing can be utilized for deciding upon the lot acceptance of manufactured parts. Particularly, when the number of test positions is limited, sudden death testing is preferred [1]. According to the characteristics of testers, a group size is usually specified, but the total number of groups should be determined. To determine the number of groups, a variables single sampling plan will be proposed in Section II, and a variables double sampling plan will be considered in Section III. It is assumed that the lifetime follows a Weibull distribution with known shape parameter, and that there is a lower specification limit regarding the lifetime. Engineering experience with a particular type of application might make such an assumption quite reasonable. The exponential assumption is nothing more than the assumption that the shape parameter is known to be equal to one. It is also assumed that a desirable, and an undesirable levels of lot quality can be specified at the producer’s, and the consumer’s risk, respectively. The proposed plans are different from those in [3] in that the lot acceptance criteria in [3] depend on the estimated distribution parameters, while ours do not. II. SINGLE SAMPLING PLAN Suppose that the lifetime of a part follows a Weibull distribution with shape parameter , and scale parameter , such that the cumulative distribution function is given by (1)
where denotes the lifetime of a part. Hence, if is given, then is obtained from (2) through the corresponding (3) Let us propose the following single sampling variables plan under sudden death testing. , and allocate 1) Draw a random sample of size parts to each of groups. 2) Perform sudden death testing, and observe , the time . to the first failure from the -th group 3) Calculate the quantity (4) 4) Accept the lot if , and reject the lot otherwise. It should be noted that the above acceptance criterion does not involve the scale parameter of the Weibull distribution. In the following, we will describe how to determine the parameters , and of the above single sampling variables plan. As mentioned earlier, is not a design parameter, but it is assumed to be specified by the characteristics of testers. Note first that the should be i.i.d., and its distribution is obtained as follows. (5) are lifetimes of parts in a group. Therefore, where follows i.i.d. Weibull with shape parameter , and scale parameter . Because follow i.i.d. exponential dis, the quantity in (4) follows the tributions with parameter . Hence, the lot Gamma distribution with parameters acceptance probability, when the quality is , is given by (6) in (6) follows the Chi-square distribution The quantity with degree of freedom , so (6) reduces to (7) where is given by (3), and is the distribution function of a Chi-square random variable with degree of freedom . of the above sampling To determine the parameters plan, the usual approach can be adopted, which is to use two points on an operating characteristic (OC) curve. As in Fertig & ( is Mann [5], the probability of acceptance should be called producer’s risk) at the acceptable reliability level , and the probability of acceptance should be (this is called consumer’s risk) at the lot tolerance reliability level . We need to solve the following two inequalities: (8) (9)
Assume that the shape parameter is known. Also assume that there is a lower specification limit regarding the lifetime. So, the fraction nonconforming, or unreliability, is expressed by
are values given in (3) corresponding to , where , and denotes the percentage point of and , respectively. If tail probability in the Chi-square distribution with degree of freedom , then (8), and (9) can be rewritten as
(2)
(10)
JUN et al.: VARIABLES SAMPLING PLANS FOR WEIBULL DISTRIBUTED LIFETIMES UNDER SUDDEN DEATH TESTING
(11)
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TABLE I DESIGN PARAMETERS OF SINGLE SAMPLING PLANS WHEN = 0:05, = 0:1
Dividing (10) by (11) yields (12) Therefore, can be obtained by the smallest integer satisfying (12), and can be calculated from the equal case in either (10) or (11). If there is no restriction on the integer solution for , then it can be determined by the equality case in (12). Note that the number of groups can be determined independently of the group size , and even independently of the shape paramour proposed plan reduces eter . Note also that when to the single sampling plan under complete testing. In fact, the above sampling plan under sudden death testing can be reduced to a single sampling plan under complete testing as in Hisada & Arizino [21] when the shape parameter is known. For example, , , , , we have when ; and so is obtained by 5.1, and is calcu, should be 202.56. lated by 2,025.6. So, for the case of corresponding to Table I shows the values of when , and . It can be observed that, for small values of & , the . It folnumber of groups only depends on the ratio of lows because, for small values of , in (12),
TABLE II SAMPLE SIZE REQUIRED FOR TWO PLANS WHEN = 0:05, = 0:1
(13) The following example shows how to use the single sampling plan when deciding upon the acceptance of the lot. Example 1: Suppose that the quality assurance in a bearing manufacturer states that at , and at . The number of test positions is limited to . Because from Table I, & are selected, we may , and allocate 10 bearings draw a random sample of size to each of 5 groups. Now suppose that the first failure times , from each -th group under sudden death testing is , , , and . Bearings , are known to have Weibull distributed lifetimes with and the lower specification of 100. Then is calculated by , which is less than . So, the lot should be rejected. As mentioned earlier, the designed parameters can be determined independently of the Weibull shape parameter. So we may use the results in [3] for the exponential distribution. Table II compares two plans (Balasooriya’s, and our proposed) in terms of the number of groups required (values are rounded to integers). It is seen that the sample size required for the proposed plan is smaller than that for Balasooriya’s plan.
2)
Perform sudden death testing, and observe to the first failure from the -th group Then, calculate the quantity
, the time .
(14) 3) 4)
5)
, and reject the lot if Accept the lot if . , then draw a second sample If , and allocate parts to each of of size groups. Perform sudden death testing, and calculate
III. DOUBLE SAMPLING PLAN To reduce the total number of groups, the following double sampling variables plan can be considered: , and 1) Draw a first random sample of size allocate parts to each of groups.
(15) 6)
Accept the lot if otherwise.
, and reject the lot
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The probability of lot acceptance based on the first sample is given by
DESIGN PARAMETERS
TABLE III DOUBLE SAMPLING PLANS WHEN = 0:05, = 0:1 OF
(16) The probability of lot acceptance based on the second sample can be expressed by
(17) Because , and are independent, and they follow , and , Chi-square distributions with degrees of freedom respectively, then (17) reduces to
(18)
Then, the final lot acceptance probability is obtained by (19) The parameters associated with the above plan, ( , , , ), can be obtained by solving two equations as before.
,
(20) (21) Alternatively, we determine these parameters to minimize the average number of groups required at , which is analogous to minimizing the average sample number in a usual double sampling plan (See [22]). The average group number (AGN) for lot quality is obtained by
(22) Therefore, we may consider the following optimization problem to determine those parameters. (23) (24) (25) We see again for this double sampling plan that the num& , can be determined independently of bers of groups, the shape parameter because the lot acceptance probability is obtained regardless of . Table III shows the design parameters of the double sampling when , plan corresponding to various pairs of . It is assumed that to reduce the number of parameters. Because the objective function is nonlinear, the
above problem may result in a different solution according to a different initial value. In this case, we take the solution at which the AGN is minimized from several trials. Table III also shows the AGN at . because the AGN for the single sampling plan is just , it is seen that the AGN for the double sampling plan is smaller than that for the single sampling plan. Example 2: We have a similar situation as in Example 1, but we decided to use a double sampling plan. From Table III, , , , and . Draw a first random , and allocate 10 bearings to each of 3 sample of size , groups. Now suppose that the first failure times are , . Then, is calculated by , which is between , and . So, we , and allocate 10 bearings draw a second sample of size to each of 3 groups. Suppose now that the failure times are , , . We have . Hence the lot is rejected because .
IV. EFFECTS OF MIS-SPECIFICATION OF SHAPE PARAMETER Because the shape parameter is assumed to be known, one may be concerned about the mis-specification of the shape parameter. The effect of mis-specification of the Weibull shape parameter was studied for the case of Weibull-to-exponential transformation by several authors including Keats et al. [23], and Xie et al. [24]. We would like to analyze the effect of misspecification of the shape parameter on the probability of accepting the lots under the proposed single sampling plan. be the true Let be the specified shape parameter, and shape parameter. Then, the lot acceptance probability under the
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TABLE IV LOT ACCEPTANCE PROBABILITY AT p WHEN = 0:05, = 0:1
TABLE V LOT ACCEPTANCE PROBABILITY AT p WHEN = 0:05, = 0:1
proposed single sampling plan when the quality of lot is be
will
(26)
As long as is not equal to , does not follow an follows an i.i.d. i.i.d. exponential distribution. Instead, exponential distribution with scale parameter . To derive the , let us first consider the probability distribution of
However, the convolution of two or more Weibull distributions does not have a closed-form solution. In this study, we used a Monte Carlo simulation to calculate the probability in (26) by generating Weibull random variates with repetitions of 1, 000, 000 times. Tables IV, and V show the lot acceptance probabilities at , and , respectively, for several combination of according to values of when , . Here, the value of was chosen as the closest integer to the exact value , the probabilities shown in Table I. So, even when , and those in Table IV are not exactly equal to 0.95 in Table V are not exactly equal to 0.1 . There may not be a serious mis-specification problem when is relatively small, while there may be when is large. It is also observed that the mis-specification problem can be alleviated by taking larger (number of specimen in a group). V. SUMMARY AND CONCLUSIONS
(27) follows a Weibull distribution with shape paTherefore, , and scale parameter . To obtain the above rameter lot acceptance probability in (26), we need to know the -fold convolution of this Weibull distribution. Note that this proba, not on the individual bility only depends on the ratio of & . When , (26) reduces to values of (28)
This paper presents single, and double sampling variables plans for deciding upon the lot acceptance of manufactured parts under sudden death testing. The sudden death testing is frequently adopted by parts manufactures to reduce testing time. The number of groups to be tested will be the key parameter for designing the sudden death testing, and for the proposed sampling plans. The procedure of determining design parameters will be described. The usual approach of using two points on an operating characteristic curve is employed. Particularly for designing the double sampling plan, a nonlinear optimization problem of minimizing the average group number is considered with constraints of acceptance probabilities at the above two points. 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 known. This assumption has a theoretical
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advantage of resulting in a simpler analysis, which may give us an insight for interpreting the results. By adopting this assumption, many other sampling plans under various censoring schemes can be modified relatively easily. This assumption may not be restrictive, particularly when manufacturers have accumulated knowledge from the past history. One interesting result is that the number of groups in a sudden death testing can be determined independently of the group size and even independently of shape parameter. To reduce the number of groups, a double sampling plan is preferred to the single sampling plan. ACKNOWLEDGMENT We would like to thank the associate editor, and anonymous referees for their helpful comments. REFERENCES [1] F. G. Pascual and W. Q. Meeker, “The modified sudden death test: planning life tests with a limited number of test positions,” Journal of Testing and Evaluation, vol. 26, no. 5, pp. 434–443, 1998. [2] B. L. Vlcek, R. C. Hendricks, and E. V. Zaretsky, Monte Carlo Simulation of Sudden Death Bearing Testing, NASA, Hanover, MD, USA, 2003. [3] U. Balasooriya, “Failure-censored reliability sampling plans for the exponential distribution,” Journal of Statistical Computation and Simulation, vol. 52, no. 4, pp. 337–349, 1995. [4] J.-W. Wu, T.-R. Tsai, and L.-Y. Ouyang, “Limited failure-censored life test for the Weibull distribution,” IEEE Transactions on Reliability, vol. 50, no. 1, pp. 107–111, 2001. [5] K. W. Fertig and N. R. Mann, “Life-test sampling plans for two-parameter Weibull populations,” Technometrics, vol. 22, no. 2, pp. 165–177, 1980. [6] H. Schneider, “Failure-censored variables-sampling plans for lognormal and Weibull distributions,” Technometrics, vol. 31, no. 2, pp. 199–206, 1989. [7] Y. I. Kwon, “A Bayesian life test sampling plan for products with Weibull lifetime distribution sold under warranty,” Reliability Engineering and System Safety, vol. 53, no. 1, pp. 61–66, 1996. [8] Y. Lam, “Bayesian variable sampling plans for the exponential distribution with type I censoring,” The Annals of Statistics, vol. 22, no. 2, pp. 696–711, 1994. [9] Y. Lam and S. T. B. Choy, “Bayesian variable sampling plans for the exponential distribution with uniformly distributed random censoring,” Journal of Statistical Planning and Inference, vol. 47, no. 3, pp. 277–293, 1995. [10] J. Chen, S. T. B. Choy, and K.-H. Li, “Optimal Bayesian sampling acceptance plan with random censoring,” European Journal of Operational Research, vol. 155, no. 3, pp. 683–694, 2004. [11] J. Chen, W. Chou, H. Wu, and H. Zhou, “Designing acceptance sampling schemes for life testing with mixed censoring,” Naval Research Logistics, vol. 51, no. 4, pp. 597–612, 2004. [12] U. Balasooriya and C.-K. Low, “Competing causes of failure and reliability tests for Weibull lifetimes under type I progressive censoring,” IEEE Transactions on Reliability, vol. 53, no. 1, pp. 29–36, 2004.
[13] U. Balasooriya, S. L. C. Saw, and V. Gadag, “Progressively censored reliability sampling plans for the Weibull distribution,” Technometrics, vol. 42, no. 2, pp. 160–167, 2000. [14] S.-K. Tse and C. Yang, “Reliability sampling plans for the Weibull distribution under type II progressive censoring with binomial removals,” Journal of Applied Statistics, vol. 30, no. 6, pp. 709–718, 2003. [15] H. K. T. Ng, P. S. Chan, and N. Balakrishnan, “Optimal progressive censoring plans for the Weibull distribution,” Technometrics, vol. 46, no. 4, pp. 470–481, 2004. [16] U. Balasooriya and S. L. C. Saw, “Reliability sampling plans for the two-parameter exponential distribution under progressive censoring,” Journal of Applied Statistics, vol. 25, no. 5, pp. 707–714, 1998. [17] U. Balasooriya and N. Balakrishnan, “Reliability sampling plans for lognormal distribution, based on progressively-censored samples,” IEEE Transactions on Reliability, vol. 49, no. 2, pp. 199–203, 2000. [18] G. J. Lieberman and G. J. Resnikoff, “Sampling plans for inspection by variables,” Journal of the American Statistical Association, vol. 50, pp. 457–516, 1955. [19] N. Balakrishnan, N. Kannan, C. T. Lin, and H. K. T. Ng, “Point and interval estimation for Gaussian distribution, based on progressively type-II censored samples,” IEEE Transactions on Reliability, vol. 52, no. 1, pp. 90–95, 2003. [20] A. A. Soliman, “Estimation of parameters of life from progressively censored data using Burr-XII model,” IEEE Transactions on Reliability, vol. 54, no. 1, pp. 34–42, 2005. [21] K. Hisada and I. Arizino, “Reliability tests for Weibull distribution with varying shape-parameter, based on complete data,” IEEE Transactions on Reliability, vol. 51, no. 3, pp. 331–336, 2002. [22] D. J. Sommers, “Two-point double variables sampling plan,” Journal of Quality Technology, vol. 13, no. 1, pp. 25–30, 1981. [23] J. B. Keats, P. C. Nahar, and K. M. Korbell, “A study of the effect of mis-specification of the Weibull shape parameter on confidence bounds based on the Weibull-exponential transformation,” Quality and Reliability Engineering International, vol. 16, pp. 27–31, 2000. [24] M. Xie, Z. Yang, and O. Gaudoin, “More on the mis-specification of the shape parameter with Weibull-to-exponential transformation,” Quality and Reliability Engineering International, vol. 16, pp. 281–290, 2000.
Chi-Hyuck Jun was born in Seoul, Korea in 1954. He received a B.S. (1977) in mineral and petroleum engineering from Seoul National University, an M.S. (1979) in industrial engineering from KAIST, and a Ph.D. (1986) in operations research from University of California, Berkeley. Since 1987, he has been with the department of industrial and management engineering, POSTECH; and he is now a professor, and the department head. He is interested in reliability and quality analysis, and data mining techniques. He is a member of IEEE, INFORMS, and ASQ.
S. Balamurali received his M.Sc., and Ph.D. degrees in Statistics from Bharathiar University, India. He is presently working as a Research Professor in the Division of Mechanical and Industrial Engineering, POSTECH. His research interests include statistical process control, and acceptance sampling. His recent publications have appeared in the engineering, and applied statistics journals.
Sang-Ho Lee was born in Samcheok, Korea in 1980. He Received a B.S. (2004) in industrial engineering from POSTECH. He is now a Ph.D. student in the Department of Industrial and Management Engineering at POSTECH. His interests are in data mining, and applied statistics.
