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1、612 IEEE TRANSACTIONS ON RELIABILITY, VOL. 58, NO. 4, DECEMBER 2009 censoring times of products subject to elevated stress levels. However, this approach may offer little help for highly reliable products which are not likely to fail in a short period of time. If there exists a quality characteristi

2、c (QC of the product whose degradation over time can be related to reliability, then collecting degradation data can provide useful information about the reliability of the product. Early work on degradation models has been reviewed by Nelson 14. Meeker & Escobar 12 presented an updated survey o

3、n various approaches used to assess reliability information from degradation data. For some recent developments on degradation models and associated inferential issues, one may refer to Chen & Zheng 4, Huang & Dietrich 8, Joseph & Yu 9, Giorgio et al. 6, Hsieh & Jeng 7, Bae et al. 1,

4、 Ng 15, and Gebraeel et al. 5. Bagdonavicius & Nikulin 2 have given models and methods for statistical analysis of accelerated degradation tests (ADT, along with a number of pertinent references. Meeker et al. 13 described accelerated degradation reliability models that correspond to physical-fa

5、ilure mechanisms. This approach also introduced a connection between degradation models, and failure-time lifetime models. They also discussed Monte Carlo, and bootstrap sampling methods for inference about the lifetime distribution of the product. Yu & Tseng 20 proposed a stopping rule for term

6、inating an ADT. To conduct a degradation test more efciently, Yu & Tseng 21 proposed a quasilinear model to describe the degradation path, and the optimal determination of design variables such as sample size, measurement frequency, and termination time by minimizing the variance of the estimate

7、d -th percentile of the lifetime distribution of the product under the constraint that the total experimental cost does not exceed a pre-specied budget. Earlier, Boulanger & Escobar 3 addressed for ADT the problem of optimal determination of both the stress levels, and sample size for each stres

8、s level under a pre-xed termination time. Although ADT is an efcient life-test method, it is usually very expensive to conduct. For example, if we adopt three levels of stress with 25 test units at each level, then three ovens, and 75 test units are needed. For a newly developed product, or an expen

9、sive product, it may not be possible to have so many test units on hand. Moreover, the selection of suitable levels of stress is not straightforward. In this situation, a constant-stress ADT is therefore not suitable. Recently, Tseng & Wen 19 proposed a SSADT to handle this problem. In a SSADT e

10、xperiment, an item is rst tested, subject to a pre-determined stress level for a specied length of time. If it does not fail, it is tested again at a higher stress level for another specied length of time. The stress on the specimen is thus increased step by step until an appropriate termination tim

11、e is reached. Obviously, the advantage of the SSADT is that only a few test units are needed for conducting this life test. To conduct a SSADT efciently, special attention needs to be paid to the above mentioned design variables (sample size, measurement frequency, and termination time. Liao & T

12、seng 11 formulated a SSADT model with a Wiener process, and discussed such an optimal design problem. From the denition of a Wiener process, it is known that the degradation path is not a strictly increasing function. Generally, a gamma process (with monotone increasing pattern is more suitable for

13、describing the degradation path of some spe- cic products, and especially in the case of crack tests. Some well-known references on the gamma degradation process are Singpurwalla 18, Lawless & Crowder 10, and Park & Padgett 16, 17. The design of an efcient SSADT plan for a gamma degradation

14、path is therefore of great interest. In this paper, under the constraint that the total experimental cost does not exceed a pre-specied budget, we deal with the optimal SSADT plan (including the optimal settings for the sample size, measurement frequency, and termination time for a gamma degradation

15、 process by minimizing the approximate variance of the estimated MTTF of the lifetime distribution of the product. The rest of this paper is organized as follows. In Section II, the problem of a SSADT experiment is formulated. In Section III, we introduce the SSADT model with a gamma degradation pro

16、cess. In Section IV, the optimal test plan is presented. In Section V, an example is presented to illustrate the method proposed in the preceding sections. Finally, some concluding remarks are made in Section VI. II. PROBLEM DESCRIPTION, AND FORMULATION denote the degradation path of the product und

17、er Let , and its lifetime can be suitably dea typical-use stress ned as the rst passage time when crosses a critical value . Hence, we have (1 In the following, we assume that the independent increments of the degradation path of the product follows a gamma process (Lawless & Crowder, 10. For xe

18、d , and , (2 where ; , and are shape, and scale parameters of the gamma distribution, respectively. Recently, Park & Padgett 16 proposed a simple approximate formula for the MTTF of the product under typical-use as stress (3 For a typical highly-reliable product within a short life-testing time,

19、 we are interested in designing an efcient SSADT experiment in such a way that the MTTF of the product can be predicted as precisely as possible. A SSADT experiment can be denote higher stress expressed as follows. Let levels such that To begin with, suppose there are test units subject to a degrada

20、tion test (with a measurement frequency per units time under stress , and the duration time of degradation test under the is ; next, we increase the stress level to , and the stress is up to . Continue the process until duration time under , and the experiment is terminated at . the stress is up to

21、Thus, the testing stress, , of an SSADT experiment can be expressed as if , . . . . . . if . TSENG et al.: OPTIMAL STEP-STRESS ADT PLAN FOR GAMMA DEGRADATION PROCESSES 613 Let denote the quality characteristic of the product at time , . For , under the stress let denote the total number of measureme

22、nts under stress . Then, we have , and , , , denote a set of observed degradation paths under a SSADT experiment. denote the estimated , and Let denote the total cost of conducting an SSADT experiment. The choice of , , and will affect the cost of the experiment, as well as the estimated precision o

23、f the MTTF of the product. Hence, the optimal settings of , , and can be obtained by solving the following optimization problem. Fig. 1. The relationship between L (t, and fL(tjS g lying degradation path follows a linear pattern. when the under- by using the -independent increment property of the ga

24、mma has a gamma distribution, i.e., process, we know that (4 denotes the approximate variance of , and denotes the total budget for conducting the degradation experiment. We will discuss this optimization problem in the following section. where III. SSADT WITH GAMMA DEGRADATION PROCESS , assume that

25、 , and that there exists a relationship between and stress , i.e., . Note that the reason for changing the value of the shape parameter rather than the scale parameter with the degradation level is due to the fact that we can get the reproductive property of the gamma distribution which is only “add

26、itive” with respect to the shape parameter , and be the degradation path not the scale parameter . Let of a SSADT with a gamma degradation model. Then, the relaand can be expressed as tionship between follows. Under the stress , and for , we have Given (6 In the following, we assume that temperature

27、 is an accelerating variable, and that the following Arrhenius reaction rate model can be used to model the relationship between , and the temperature stress . (7 , Now, given a set of observed SSADT paths, , , the likelihood function of the SSADT model for a gamma process is given by (8 is the cumu

28、lative number of measurewhere , and . Then, the maxments up to , of can imum likelihood estimator (MLE be obtained by a numerical optimization method. The estimated , can be obtained by substituting , , MTTF under , , and into (3, and (7 directly. In the following section, we will derive the optimal

29、 test plan. IV. THE OPTIMAL TEST PLAN When we increase the stress up to at time , by the additive property with respect to shape parameter of the gamma distribecomes bution, the degradation path for Similarly, for general obtain , , and , we (5 and , Fig. 1 shows the relationship between , when the

30、underlying degradation path is linear. with Let denote the SSADT degradation path of the -th , , test sample at time . For . Then, from (5, and let To conduct a SSADT efciently, the framework for solving the optimization problem in (4 consists of two parts: , 1 the computation of the approximate var

31、iance of and 2 the total cost of experimentation A. Computation of By the best asymptotically normal (BAN property of the follows a multivariate normal distribution, i.e., MLE, 614 IEEE TRANSACTIONS ON RELIABILITY, VOL. 58, NO. 4, DECEMBER 2009 where (9 shown at the bottom of the page.The matrix in

32、(9, found at the bottom of the page is the Fisher information , and expressions for all the entries of are matrix of given in the Appendix. Then, by the -method, the approximate variance of is found to be (10 where and denotes the transpose of . Fig. 2. Simulated SSADT sample of carbon-lm resistors.

33、 TABLE I OPTIMAL SSADT PLAN WHICH IS BASED ON A PILOT STUDY B. Cost Function The total experimental cost consists of three parts: , 1 the cost of conducting an experiment is denotes the unit cost of operation; where , where 2 the cost of measurement is denotes the unit cost of measurement; and , whe

34、re denotes 3 the cost of testing the product is the unit cost of device. Therefore, the total cost of conducting a SSADT experiment is given by (11 C. Optimization Model From the expressions given above, the optimization problem can be expressed as Minimize (12 subject to (13 . where Due to the comp

35、lexity of the objective function, an analytic expression does not exist for the solution of the optimization problem in (12, and (13. However, with the simplicity in the structure of the constraint, and the integer restriction on the decision variables, the optimal solution can be determined through the following algorithm. , where Step 1 Set is the oor of (the largest integer that is less than ; is the largest possib