Optimal design for step-stress accelerated test with random discrete stress elevating times based on gamma degradation process

Optimal design for step-stress accelerated test with random discrete stress elevating times based on
gamma degradation process

Morteza Amini,111Corresponding author, E-mail addresses: morteza.amini@ut.ac.ir, shemehsavar@khayam.ut.ac.ir, panzhq.yy@hotmail.com   Soudabeh Shemehsavar   and   Zhengqiang Pan
Department of Statistics, School of Mathematics, Statistics and Computer Sciences,
College of Science, University of Tehran, P.O. Box 14155-6455, Tehran,Iran
Department of Systems Engineering, College of Information Systems and Management,
National University of Defense Technology, Changsha, People’s Republic of China
Abstract

Recently, a step-stress accelerated degradation test (SSADT) plan, in which the stress level is elevated when the degradation value of a product crosses a pre-specified value, was proposed. The times of stress level elevating are random and vary from product to product. In this paper we extend this model to a more economic plan. The proposed extended model has two economical advantages compared with the previous one. The first is that the times of stress level elevating in the new model are identical for all products, which enable us to use only one chamber (oven) for testing all test units. The second is that, the new method does not require continuous inspection and to elevate the stress level, it is not necessary for the experimenter to inspect the value of the degradation continually. The new method decrease the cost of measurement and also there is no need to use electronic sensors to detect the first passage time of the degradation to the threshold value in the new method. We assume that the degradation path follows a gamma process. The stress level is elevated as soon as the measurement of the degradation of one of the test units, at one of the specified times, exceeds the threshold value. Under the constraint that the total experimental cost does not exceed a pre-specified budget, the optimal settings including the optimal threshold value, sample size, measurement frequency and termination time are obtained by minimizing the asymptotic variance of an estimated quantile of the lifetime distribution of the product. A case study is presented to illustrate the proposed method.

Keywords: Discrete times, Fisher information matrix, Gamma process, Inverse Gaussian distribution.

1 Introduction

Due to the rapid improvement of quality of products of today, it is difficult to assess the failure information of a product using traditional life testing procedures, which record only failure times. An alternative approach is to collect the degradation data at higher levels of stress, thus yielding more information about the lifetime distribution in a reasonable amount of time compared to traditional life tests. Such a life testing plan is called an accelerated degradation test (ADT). Mathematical models for analyzing ADT data have been proposed by Meeker & Escobar [7]. Meeker et al. [8] described accelerated degradation models that relate to physical failure mechanisms. Yu & Tseng [13] proposed a stopping rule for terminating an ADT. Chen & Zheng [2] proposed an approach for degradation analysis which makes inference directly on the lifetime distribution. Huang & Dietrich [3], used maximum likelihood estimation to estimate the model’s parameters. Joseph & Yu [4], developed a methodology for quality improvement when degradation data are available as the response in the experiments.

ADT is usually very costly, since it requires destroying a considerable number of products at each level of stress. Moreover, determining suitable levels of stress for the experiment is not straightforward. To handle these problems, step stress accelerated degradation tests (SSADT) were proposed by Tseng & Wen [12] in a case study of LEDs. In a SSADT framework, each product is first tested, subject to a pre-determined stress level for a specified duration, and the degradation data are collected. A product which survives until the end of the first step is again tested at a higher stress level and for a different time duration. The stress level is elevated step by step until an appropriate termination time is reached. The advantage of the SSADT is that the number of test units needed for conducting this test is relatively small.

To conduct a SSADT efficiently, special attention is paid to the number of tested units (sample size), measurement frequency and termination time. For ADT, Boulanger & Escobar [1] addressed the problem of optimal determination of the stress levels as well as the sample size for each stress level for a specified termination time. The problem of optimizing the test design have been extensively studied in recent years. Three commonly used optimization criteria are the minimum approximated variance (Avar) of the maximum likelihood estimators (MLE) of reliability, mean time to failure (MTTF) and the quantiles of the lifetime distribution. Tang et al. [11] discussed the optimal planning of a simple SSADT and minimized the test cost while achieving a requisite level of estimates, from which the optimal number of inspections of the whole test can be determined. Liao & Tseng [6] used a Wiener process to model a typical SSADT problem and then, under the additional constraint that the total experimental cost does not exceed a predetermined budget, obtained the optimal settings of the SSADT for minimizing the Avar of the estimated 100 percentile of the product’s lifetime distribution. They derived the optimal test plans for the sample size, measurement frequency and termination time. Tseng et al. [14] discussed the SSADT problem for a gamma process and gave the optimal settings for minimizing the Avar of the estimated MTTF of the lifetime distribution of the product.

Recently, Pan & Balakrishnan [9] proposed a step-stress accelerated degradation test (SSADT) plan in which the stress level is elevated when the degradation value of a product crosses a pre-specified value. In such a plan the times of elevating the stress levels are random and vary from product to product. However, such a plan can be extended to a more economic plan in two ways. First one can extend the model such that the times of stress level elevating in the new model be identical for all products, which enable us to use only one chamber (oven) for testing all test units. Second, one can conduct the plan such that it does not require continuous inspection of the degradation by the experimenter (or may be electronic sensors). In the later we don’t need to spend any money on electronic sensors and also we have decreased the cost of measurement.

To extend the model, we assume that the measurements are made at the end of consecutive intervals of fixed duration. The experimenter elevates the stress level as soon as a new measurement of at least one of the test units exceeds the threshold value.

In this paper, we propose a SSADT model when the degradation path follows a gamma process. The measurements are made at times , for , where is the measurement frequency in unit of time and is the number of measurements. The stress level is elevated as soon as a new measurement of the degradation of one at least of the test units exceeds the threshold value. Next, under the constraint that the total experimental cost does not exceed a pre-specified budget, the optimal settings including the optimal threshold value, sample size, measurement frequency and termination time are obtained by minimizing the Avar of an estimated quantile of the lifetime distribution of the product.

The paper is organized as follows. The model is described in Section II. An optimization algorithm is presented in Section III to obtain the optimal settings under budget constraints. Section IV presents a real data example for illustrating the proposed algorithm and some concluding remarks are made in Section V.

2 Model description

Suppose that test units are subject to a degradation test. Let denote the degradation path of the product under a level of stress at time . Denote the use stress level (under the normal conditions) by . The lifetime of the unit under the normal conditions, , is defined the first time that crosses the critical value , that is

(1)

in which , denotes the degradation value of the unit under the stress at time .

Pan & Balakrishnan [9], proposed a SSADT model in which the stress level of the unit, , , is elevated as soon as the degradation of the unit passes the threshold value (). Hence, the testing stress of the unit, , under such a SSADT model can be expressed for the stress values as follows

(2)

where is the termination time of the test and

(3)

in which , denotes the degradation value of the unit under the stress levels at time . We should keep in mind that the degradation values do not depend on the random variables and the random variables depend on the degradation values. In other words, we have

(4)

Also, it is worth noting the difference of notations in this paper with the paper of Pan and Balakrishnan [9]. In this paper, denotes the cumulative degradation value of item during the test, under the stress level, while in Pan and Balakrishnan [9], it denotes the cumulative degradation value of item during the stress level.

They considered both Wiener and Gamma processes for the degradation path of the products. In the following, we assume that the independent increments of the degradation path of the products follows a gamma process (Lawless & Crowder [5]); that is, for fixed and ,

(5)

where stands for the gamma distribution with shape and scale parameter and , respectively. Thus, we have, for

Park and Padgett [10] provided the approximate distribution of iid random differences , for as Birnbaum-–Saunders distribution with cdf

(6)

Under the above settings and using (6), Pan and Balakrishnan [9] derived the approximate joint pdf of as

(7)

The idea of using random time for elevating the stress level of SSADT is interesting. However, implementing the above SSADT model with stress model (2) is not economic, since the main results for conducting a SSADT is that, we can economically use only one chamber (oven), which allows us to elevate gradually the stress levels at specified (or random) times for all testing units. From equation (2), it is clear that chambers should be used simultaneously, since with current metrology technology it is not possible to conduct such an SSADT plan within a single chamber.

A more economic model with random stress-elevating times can be conducted with the common stress

(8)

for all test units, in which

(9)

Under the SSADT plan with stress model (8), the stress level is elevated for all units as soon as the degradation of at least one of the units exceeds the threshold value . The test terminated at time .

Using (7), one can obtain the joint distribution of , which is a rather complicated formula.

For the case , since are independent and identically distributed, using (6) has the approximate cdf

(10)

Also, for the case , using (6), the approximate joint survival function of and is obtained as follows

where

(11)

It is easy but a bit tedious to extend the above discussion for . For instanse, for , we have

The second step for conducting a more economic plan is to extend the model with test stress model (8) such that it does not require continuous inspection of the degradation level of test units by the experimenter or electronic sensors. In the following, we describe the extended model with discrete inspection times.

Suppose we want to make measurements for each unit with a measurement frequency per unit of time. Note that for the stress model (8), since the stress elevating time is random, we can only determine the total number of measurements and the number of measurements under each stress level is random. Under the stress level, , we elevate the stress level, as soon as a new measurement of the degradation of at least one of the test units exceeds the threshold value . The testing stress of the unit, , under such a model can be expressed as follows

(12)

where

(13)

in which stands for the smallest integer greater than and is defined in (9).

The random variables are discrete with the obvious property that , with probability 1. Also, it is straightforward that, if for some , , we have , for all and . Therefore, if for some , , the testing stress takes only the values during the total time interval, .

For the case , we have

(14)

and

(15)

where is given in (10).

Also, for the case , the joint pmf of and is

(16)

and simillar expressions are derived easily for , and , which are all approximated using the joint survival function given in (11). Similar formulas can also be obtained for the joint pmf of , for .

Suppose that the relation between and the stress level is modeled by the Arrhenius reaction rate model as

for two unknown parameters and .

For and , let denote the degradation increment of the unit at time relative to time , that is

where is the level of degradation of the unit at time . We have for all for with , , and .

Using (5), the conditional distribution of given , for , is , , in which we assume and . Therefore, given and , the likelihood function of is given by

(17)

where is given in (14) to (16) for and similar formulas can also be obtained for the joint pmf of , for .

3 The Optimal Design

We develope the optimization algorithm for the case . The generalization of the proposed algorithm for larger values of is straightforward. For the case , the optimization of the SSADT model in (12) consists of finding the optimal values of , , , and the optimal value of . We concern the problem of minimizing the Avar of an estimate of a quantile of the product’s lifetime distribution as an optimization criterion. The approximated cdf of the lifetime of the product under the use stress level, , which is defined as in (1), is (see Park and Padgett, [10])

(18)

where

Using (18), one can obtain the 100 percentile of as . The MLE of , , is obtained by substituting the MLEs of and into . These values can be obtain by maximizing the likelihood function of in (17).

The Avar of can be obtained as a function of , based on Avar of the MLE (The inverse of the Fisher information matrix). Using the delta method we have

where is the corresponding pdf of the cdf in (18), the transpose of vector is

and is the Fisher information matrix of the likelihood in (17) which is calculated and given in Appendix and is the MLE of for a fixed .

We have

and

where and stands for the pdf of the standard normal distribution.

The total cost of the experiment is given by

where is the unit cost of operation per unit of time, is the unit cost of measurement, and is the unit cost of items. The optimization criterion is to find the values , which minimize , subject to , where is the total budget for conducting the degradation experiment.

Since the parameter spaces and are independent we have

(19)

Thus, although there is no analytic expression for the solution of the optimization problem of finding , due to the integer restriction on the optimality parameters , and , the global and unique minimum of can be found by searching the minimum through all possible values of , and (see for example Tseng et al. [14]), for a fixed value of and then applying the common minimization algorithms for minimization of the continuous function of .

Summing up, the optimal solution of the above optimization problem can be determined by the Algorithm I below.

Algorithm I:

Step 1) Define the function with domain as follows:

Step 1-1) Compute the largest possible number for , when and (one measurement for each stress level), which is equal to

Step 1-2) Set .

Step 1-3) Compute the largest possible number for , for the fixed , and , which is

Step 1-4) Set .

Step 1-5) Let .

Step 1-6) Compute .

Step 1-7) Set , and repeat steps 1-5 and 1-6 until .

Step 1-8) Set and repeat steps 1-3 through 1-7 until .

Step 1-9) Let .

Step 1-10) Return

Step 2) Let and .

It is worth noting that the above optimization algorithm can easily be extended for larger values of . For example generalization to the case , can be easily made by replacing and in Step 1 of Algorithm I with and , respectively.

4 Numerical illustration

minimum C.V.
0.1 292795.6 0.1389 0.0507 0.9999950
0.2 307152.4 0.1358 0.0505 0.9999951
0.3 317950.3 0.1335 0.0504 0.9999952
0.4 327481.6 0.1318 0.0503 0.9999953
0.5 336650.3 0.1300 0.0502 0.9999954
0.6 346075.7 0.1283 0.0501 0.9999955
0.7 356450.0 0.1266 0.0500 0.9999956
0.8 368981.1 0.1245 0.0499 0.9999957
0.9 387073.5 0.1218 0.0497 0.9999959
Table 1: Optimal SSADT plan for minimizing for different values of , under the budget constraint.

Figure 1: Plot of versus .

Figure 2: Plot of minimum C.V. as a function of .

We consider the data from the carbon-film-resistor problem described by Meeker & Escobar [7], for the purpose of illustration of the proposed procedure. Increasing the resistance value of the carbon-films resistors over time reduces the performance quality of the products and leads to failure. The failure occurs as soon as the percent increase in resistance hits a threshold value , under the use operating temperature C. Suppose that the cost of operation, is $1.9 per unit of time, the cost of measurement, is $1.3 for each measurement, the cost of each item, is $53, the unit time is 4 hours and the total budget is $1500.

For obtaining the optimal design we need some initial values for the parameters. These initial values are usually obtained from a pilot study. The pilot study need not to be even a step stress test. Tseng et al. [14] used this data set and obtained an initial estimate of the parameters as follows

We use the above estimates as the true parameter vector of the SSADT model with gamma degradation process and C and C. Algorithm I is used with software R.2.14 for the optimization process. Interestingly, the optimal sampling setting is obtained as

for all values of . Hence, the optimal total time of experiment is equal to . The optimal threshold values , are given in Table 1 which also includes the estimated , minimum C.V. (the minimum coefficient of variation of equal to , where is defined in Algorithm I) and the probability of stress level elevation before the end of the experiment, that is for . For example, for estimating the median time to failure of the product (), the optimal setting is to elevate the stress level as soon as a new measurement of the degradation of at least one of the test units exceeds the threshold value , which occurs with probability 0.9999954.

Figure 1 shows the plot of versus . The optimal value of decreases as increases. The reason is that the degradation values under stress level provide more information about upper quantiles ( for greater values of ) than those under .

Also the minimum C.V. is plotted as a function of in Figure 2. As we can see from Figure 2, the estimation precision increases for upper quantiles. This means that the SSADT plan is more suitable for estimation of the upper quantiles of the product’s lifetime.

5 Sensitivity and stability analysis

Bias() MSE() Bias() MSE() Bias() MSE()
(13, 52, 7, 0.0504) 0.125667 0.503172 0.460590 0.212146 -0.001530 9.228e-05
(13, 52, 7, 0.0502) 0.204601 0.912977 0.460766 0.212310 -0.003645 9.973e-05
(13, 62, 6, 0.0818) 0.153269 0.647985 0.460649 0.212201 -0.002000 0.000117
(13, 62, 6, 0.0912) 0.133570 0.535452 0.460604 0.212159 -0.004489 0.000113
(13, 204, 2, 0.6181) 0.188061 0.595449 0.460837 0.212375 -0.005635 0.000303
Table 2: Stability analysis of parameter estimation for different values of .
min.C.V.
0 0 0 0.1335 0.0504 13 52 7
48.5% 0 -3.5% 0.0519 0.1517 13 62 6
-7.9% 0 4.9% 0.2959 0.6181 13 204 2
0 1.9% -1.9% 0.2797 0.6181 13 204 2
0 -6.6% 4.3% 0.0992 0.0818 13 62 6
-1.8% 1.8% 0 0.2887 0.6181 13 204 2
6.5% -6.5% 0 0.0877 0.0893 13 62 6
2.8% 2.8% -1.9% 0.2778 0.6181 13 204 2
-3.9% -1.8% 5.8% 0.1224 0.0744 13 62 6
min.C.V.
0 0 0 0.1300 0.0502 13 52 7
41.5% 0 -2.5% 0.0519 0.1517 13 62 6
-7.8% 0 4.7% 0.2868 0.6181 13 204 2
0 1.8% -1.8% 0.2704 0.6181 13 204 2
0 -8.0% 8.0% 0.0903 0.0912 13 62 6
-1.6% 1.6% 0 0.2763 0.6181 13 204 2
7.2% -7.0% 0 0.08219 0.0923 13 62 6
2.7% 2.7% -2.2% 0.2690 0.6181 13 204 2
-9.3% -9.4% 9.3% 0.1012 0.0832 13 62 6
min.C.V.
0 0 0 0.1266 0.0500 13 52 7
75% 0 -67% 0.0289 0.1090 13 62 6
-31.5% 0 31.5% 0.4516 0.9999 13 204 2
0 0.9% -7% 0.2500 0.6181 13 204 2
0 -9.9% 9.0% 0.0802 0.1020 13 62 6
-1.5% 1.5% 0 0.2669 0.6181 13 204 2
9.9% -9.9% 0 0.0660 0.1164 13 62 6
2.7% 2.7% -2.3% 0.2616 0.8181 13 204 2
-0.12 -0.12 0.12 0.0917 0.0931 13 62 6
Table 3: Sensitivity analysis for and different values of , and .

The optimal design depends on the initial estimates of the parameters. Hence, it is appropriate for the optimization algorithm to be rather robust for departures from the real values of the parameters. In order to check the effect of variation of on the optimal setting , we consider a pilot run on the SSADT plan for various combinations of . Table 3 presents some values of , and which cause a change in the optimal setting and the corresponding , for .

From the values of Table 3, it is clear that the optimization algorithm is quite robust for small departures from , since the changes in decision variables occur for relatively large values of . In fact, the variation of affect on the value of , and this leads to a more robust structure for the remaining decision variables .

To examine the stability of the optimal test plan, we perform a simulation study with taken as the true parameter of the SSADT model. A Monte Carlo simulation study with 10,000 iterations is performed and the ML estimate of the is obtained for each iteration using the log-likelihood function in (17), for some values of the decision vector . The bias and the mean square error (MSE) of the estimates are obtained and tabulated in Table 2. From the values of Table 2 it can be observed that the estimates of the parameters are quite stable, for small departures from the optimal plan.

6 Conclusion

This paper presents an approach for performing an SSADT experiment in which the stress level is elevated on the basis of the degradation value. In the proposed method, there is no need to observe the value of the degradation continually. The stress level is elevated as soon as a new measurement of the degradation exceeds the threshold value. We consider this threshold value as a decision variable in the optimization problem of the plan as well as the usual variables such as sample size, measurement frequency and number of measurements. An algorithm is proposed for minimizing the approximated variance of the ML estimator of the quantile of the lifetime of the products, under the budget constraints. It is observed that the proposed optimization algorithm is quite robust for departures from the values of the parameters. Also the estimates of the parameters are quite stable for small departures from the optimized plan.

Appendix [The Fisher information matrix]

Since, the likelihood function in (17) satisfies the regularity conditions, specially that the paramater space does not depend on the observations, the Fisher information matrix of the SSADT data about the parameter is as follows

where, is the log-likelihood of which, for the case of (17), is

in which and .

For the sake of brivity we define the following notations:

for any function , and

Also, let

and

denote the differentiation of and with respect to , respectively.

Since we have for and

we obtain

where and are the digamma and trigamma functions, respectively, and