Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 9283295, 11 pages
http://dx.doi.org/10.1155/2016/9283295

Research Article

Optimal Constant-Stress Accelerated Degradation Test Plans Using Nonlinear Generalized Wiener Process

Zhen Chen, Shuo Li, and Ershun Pan

Department of Industrial Engineering & Management, Shanghai Jiao Tong University, 800 Dongchuan RD, Minhang District, Shanghai 200240, China

Received 11 April 2016; Accepted 26 July 2016

Academic Editor: David Bigaud

Copyright © 2016 Zhen Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Accelerated degradation test (ADT) has been widely used to assess highly reliable products’ lifetime. To conduct an ADT, an appropriate degradation model and test plan should be determined in advance. Although many historical studies have proposed quite a few models, there is still room for improvement. Hence we propose a Nonlinear Generalized Wiener Process (NGWP) model with consideration of the effects of stress level, product-to-product variability, and measurement errors for a higher estimation accuracy and a wider range of use. Then under the constraints of sample size, test duration, and test cost, the plans of constant-stress ADT (CSADT) with multiple stress levels based on the NGWP are designed by minimizing the asymptotic variance of the reliability estimation of the products under normal operation conditions. An optimization algorithm is developed to determine the optimal stress levels, the number of units allocated to each level, inspection frequency, and measurement times simultaneously. In addition, a comparison based on degradation data of LEDs is made to show better goodness-of-fit of the NGWP than that of other models. Finally, optimal two-level and three-level CSADT plans under various constraints and a detailed sensitivity analysis are demonstrated through examples in this paper.

1. Introduction

Reliability assessment is a critical task for improving the products’ quality and reliability. Traditional life tests only record time-to-failure data to assess the lifetime distribution of the products. In presence of the quick development of technology and more and more demands from customers, some products need to be designed for highly reliable performance. Therefore, no failures are likely to occur over a reasonable test time and we would not obtain enough time-to-failure data. In such a case, if there exists a quality characteristic whose degradation is associated with the reliability of a product, we could obtain more information by collecting degradation data [1]. However, some highly reliable products degrade very slowly and it is impossible to access their lifetime within a reasonable test time. To overcome this problem, the accelerated degradation test (ADT) is usually adopted to collect the quality characteristic data under higher stress levels. The failure is defined as the event that the degradation path exceeds a fixed threshold.

In a common constant-stress ADT (CSADT), a number of units are allocated to several stress levels, and the degradation process is measured, analyzed, and extrapolated to the failure threshold in order to estimate the lifetime of the products under normal operating conditions. Consequently, before conducting an ADT, the degradation model and test plan should be developed first. The Wiener Process is often used for modelling ADT data because of its physical interpretations, nonmonotonic property, and infinite divisibility property, which could fit the products’ dynamic characteristic well. Reference [2] gave an example of modelling the degradation process by the Wiener Process. Reference [3] studied the Wiener Process for handling the degradation data of LEDs.

Setting stress levels is an important part of designing an ADT. Several types of stress levels ADT have been proposed including CSADT, step-stress ADT (SSADT), and progressive-stress ADT (PSADT). Reference [4] designed a CSADT with a reciprocal Weibull degradation rate. Reference [5] proposed optimal SSADT plans for Gamma degradation processes, while [6] proposed optimal SSADT plans for Wiener degradation processes. Reference [7] discussed PSADT for highly reliable products under nonlinear Wiener Process. As the failure mechanisms of some products may drift along with stress increasing in SSADTs and PSADTs, we cannot use a stable model to describe the degradation paths. Thus CSADTs tend to be used more in practical applications. Moreover, if the CSADT had set only one stress level as in [4], no enough effective degradation data can be collected and the lifetime distribution of products cannot be estimated precisely [8].

In this article, a Nonlinear Generalized Wiener Process (NGWP) model is proposed to describe the degradation paths of LEDs. Next, a CSADT with the constraints of prefixed budget, test duration, and sample size is developed by minimizing the asymptotic variance of the estimated mean time to failure (MTTF) of the products under normal operation conditions. Then an optimization algorithm is used to determine the stress levels, the number of units allocated to each level, inspection frequency, and measurement times simultaneously. Finally, a comparison is conducted for measuring the goodness-of-fit of various Wiener Process models, and optimal two-level and three-level CSADT plans under various constraints are obtained.

The rest of this paper proceeds as follows. A motivating example and literature review for this study are provided in Section 2. Section 3 develops a NGWP model, derives the lifetime distribution, and proposes a maximum likelihood estimation (MLE) method to estimate the unknown parameters of the model. Section 4 presents the optimization model for CSADT plan. An example is presented to illustrate the proposed method in Section 5. Finally, Section 6 concludes the paper.

2. Motivating Example and Literature Review

2.1. Motivating Example

Nowadays, light-emitting diodes (LEDs) have been applied to many fields (e.g., traffic signals and full color displays) and are desirable, because of their high brightness, low power consumption, and high reliability. A LED fails when the LED relative luminosity drops to 0.5, that is, 50% of initial luminosity. Hamada et al. [9] give a degradation data of relative luminosity (proportion of initial luminosity for LEDs). The data consist of three accelerating levels of thermal stress, 25°C, 65°C, and 105°C. At each level, the light intensity of 25 LEDs was measured at 29 inspection times. Figure 1 shows the degradation of light intensity of the LEDs.

(a)

(b)

(c)

Figure 1: Plot of LED luminosity data over time in hours at (a) 25°C, (b) 65°C, and (c) 105°C.

Obviously, the degradation paths are nonlinear and the light intensity degrades more slowly at a lower thermal stress level. Reference [10] used the LED degradation data to optimize sample allocation for ADT based on Wiener Process. The Wiener Process, which modelled the degradation paths in [10], only took nonlinearity and the effects of stress level into account. However, Figure 1 indicates that the degradation paths of different products are also different. It means that the product-to-product variability among different products exists, owing to variation of materials, manufacturing, and environment [11]. Models with random effects can be used to represent the product-to-product variability [12, 13]. Moreover, it is inevitable that some measurement errors may be introduced during the imperfect observation process in practical applications, and the external environment will also result in measurement errors [14]. The degradation models in [12, 13] did not take measurement errors into consideration, while the degradation model in [14] did not consider product-to-product variability and the effects of stress level. Therefore, for modelling the degradation paths of LEDs more accurately and generally, a generalized Wiener Process model with a consideration of all these influence factors is urgently required for their potential importance.

In this paper, the NGWP is developed. The first advantage is that the NGWP which considers nonlinearity, the effects of stress level, product-to-product variability, and measurement errors simultaneously has higher estimation accuracy and better goodness-of-fit. The second advantage is that the NGWP covers the constant models as its special case. In practical applications, the specific form of the model can be selected according to the actual situation. Clearly, the NGWP can describe more complex and diverse degradation processes of many products and can be widely applied to highly reliable products.

Furthermore, if the NGWP is used to model the degradation paths of a CSADT, then how to conduct the test should be investigated. For this problem, there exist some issues worthy of further consideration:

(i)Typically, the asymptotic variance of the estimated MTTF (Avar) of the lifetime distribution of the product is used to judge whether a CSADT plan is optimal. So how can we derive the Avar of the NGWP?(ii)Can we possibly determine the stress levels, the number of units allocated to each level, inspection frequency, and measurement times simultaneously, by optimizing the CSADT plan with the constraints on sample size, test duration, and test cost?

2.2. Literature Review

For the ADT modelling, [15] used the nonlinear Wiener Process to model PSADT, and [8] used the nonlinear Wiener Process to optimize sample size allocation for ADT. In addition, [12, 16, 17] discussed the residual life estimation based on nonlinear Wiener Process. Reference [13] investigated nonlinear Wiener Process with random effects for ADT data. So the nonlinear Wiener Process has an excellent applicability. Reference [14] showed that the nonlinear Wiener Process with measurement errors can be widely used to describe the degradation processes of various products. Similarly, [18] also indicated that the nonlinear Wiener Process with measurement errors performed better than other models in degradation data analysis. Furthermore, [11, 15, 19] predicted the real time remaining useful life based on the nonlinear Wiener Process with measurement errors. Reference [20] presented an accelerated-stress acceptance test based nonlinear Wiener Process.

Reference [21] developed an algorithm for determination of inspection frequency and unit allocation of ADT plans with multiple stress levels. Reference [22] set several stress levels to fix the stress levels and the number of samples for each stress level of ADT plans. Reference [23] developed an ADT method of luminous flux degradation for LEDs. Reference [24] suggested an analytical optimal CSADT design method for reliability demonstration by minimizing the asymptotic variance of decision variable in reliability demonstration under the constraints of sample size, test duration, test cost, and predetermined decision risks. To conduct a CSADT or SSADT, [25] determined the optimal decision variables based on C/D/A-optimality criteria. Reference [26] planed constant-stress accelerated life tests for acceleration model selection. Reference [27] established an optimal ADT procedure to minimize the asymptotic variance of the MLE of the MTTF of a product, given a budget for the total cost.

3. Degradation Process Description

3.1. Acceleration Degradation Process Modelling

To solve the problem about degradation model raised above, we propose the NGWP model considering the effects of stress level, product-to-product variability, and measurement errors, as follows:

where is the drift parameter related to the stress level , is the diffusion parameter, is the drift function, is the standard Brownian motion representing a time-correlated structure, is the measurement error with , and is the error coefficient.

Referring to (1), the drift parameter reflects the effect of stress level on the performance, and it determines the degradation rate of the NGWP. Acceleration model [28] in CSADT for products is commonly assumed as

where and are unknown coefficients and is a function of . If , acceleration model is inverse power law. If , it is the Arrhenius equation. Besides, to account for the random variation of the performance caused by variation of materials, manufacturing, and environment, the drift parameter is assumed as a random parameter and it is s-independent from stress levels. Then, we have

Note that if in (1), then the NGWP model turns to the Linear Wiener Process. If in (3) is set to , then the random model turns to the conventional acceleration model. This is as expected since any properly developed model should cover the constant model as its special case. Obviously, the NGWP model can describe more complex and diverse degradation processes of many products and have a wider range of applicability.

3.2. Derivation of the Lifetime Distribution

Let denote the true degradation path of the product under stress (); then the product’s lifetime can be defined as the first passage time when crosses the critical value under a normal operation stress (). Hence, we have

The lifetime conditioning on follows a transformation-inverse Gaussian distribution, whose cumulative distribution function (CDF) is

where is the CDF of the standard normal distribution.

Considering the accelerated model in (3), that is, , , the CDF of by integrating out of (5) becomes

From (6), the product’s MTTF under the normal operation stress can be approximated by using

3.3. Parameters Estimation

In this subsection, the issue of estimating the model parameter is addressed by using the MLE method. Suppose that test units are available for conducting a CSADT under the following conditions:

 (i)The CSADT uses -stress levels, . (ii)We assign items for a degradation test at a stress level , where , . (iii)For each stress level, the inspections are made times and the measurements of each unit are available at time . (iv)The inspection frequency is and satisfies , , . Thus . (v)For , , , let denote the sample degradation path of th test unit at time under the stress level .

 where the measurement errors are assumed to be i.i.d. realizations of . (vi)The Arrhenius equation is adopted to describe the relationship between and :

For simplicity, let , , and . Clearly, follows a multivariate normal distribution with mean and variance , where is an identity matrix of order , and

The log-likelihood function of is

where , and .

By differentiating the log-likelihood function in (11) with respect to , we have

For specified values of , , , and , the MLE of can be obtained by equaling (12) to ; that is,

Then, the profile log-likelihood function of , , , and can be given by substituting for in (12). Subsequently, the MLEs of , , , and can be obtained by maximizing the profile log-likelihood function through a multiple-dimensional search. Here, we made use of the MATLAB function “fmincon” for this purpose. By substituting the MLE of , , , and into (13), the MLE of can be obtained.

Finally, we obtain all MLEs of the model’s unknown parameters, . The MLE of MTTF under a normal operation stress is

4. The Optimal CSADT Plans

In this section, to design an efficient CSADT for a typical highly reliable product, we consider the optimization problem of determining the allocation of the units (), inspection frequency (), and measurement times () by minimizing the asymptotic variance of under normal operating conditions subject to a prefixed budget.

4.1. Objective Function

From (14), we find that the value of determines the accuracy of extrapolation. The smaller asymptotic variance of is, the more efficient CSADT plan will be. Thus, we set the asymptotic variance of as an objective function by using the delta method

where , , and is the Fisher information matrix. The detailed expressions for calculating are listed in the Appendix.

4.2. Constraints

The constraints in the design of CSADT plan usually include the following:

(i)The test time should not exceed the specified test duration .(ii)The sample size should not exceed the number of test units available .(iii)The total test cost TC should not exceed the prefixed budget .

The total cost of conducting a CSADT can be expressed as

where denotes the unit cost for operation per time, denotes the unit cost for each measurement, denotes the unit cost for each test device.

4.3. Optimization Model

From the expressions given above, the optimization problem can be formulated as follows:

where .

Due to the complex form of the objective function, an analytic expression for solution of this problem seems impossible. However, with the simplicity in the structure of the constraint and the integer restriction on these decision variables, the optimal solution can be easily determined by a complete enumeration method in a finite number of steps. The detailed algorithm is described below in nine steps.

Step 1. Set , where is a truncated integer; is the largest possible number for , when , and for .

Step 2. Set .

Step 3. Set ; is the largest possible number for when for and fixed .

Step 4. Set .

Step 5. Find such that , .

Step 6. Calculate by .

Step 7. Set , and repeat Steps 5 and 6 until .

Step 8. Set , and repeat Steps 3 and 7 until .

Step 9. The optimal solution is then obtained as .

5. Illustrative Example

In this section, we illustrate the proposed procedure with a numerical example based on the degradation data of LEDs. In order to capture the curvature by the NGWP model, the data of LEDs is a logarithm transformation and the general function . The transformation degradation paths which have clearly linear characteristic are presented in Figure 2. Then the parameters of the degradation model, that is, , are estimated by the proposed method from the 75 LEDs, respectively, and shown in Table 1. By using the estimates, the optimal CSADT plans can be obtained.

Table 1: Estimated parameters of the degradation model.

1.2889
0.3120
1831.7
0.0071
0.0270
4.2416

(a)

(b)

(c)

Figure 2: Plot of transformation degradation paths of LEDs luminosity data over time in hours at (a) 25°C, (b) 65°C, and (c) 105°C.

5.1. Comparison with Other Degradation Models

For further illustrating the rationality and applicability of the proposed model, this subsection compares some degradation models with the degradation data of LEDs. For simplicity, the NGWP model in (1) is referred to as .

where , .

If the degradation path is assumed to be linear in (18), that is, , then the NGWP model turns into a Linear Wiener Process model.

where , .

Similarly, if the measurement errors are not considered, then the NGWP model becomes a Wiener Process model without measurement errors.

where , .

If the degradation path is treated as a fixed-effect model, then the following Wiener Process model without random effects can be used to describe the LED degradation paths.

where and and are both constants.

To measure the goodness-of-fit of these different Wiener Process models above, the Akaike information criterion (AIC) is employed. AIC, which is frequently used in engineering and statistical literature for the purpose of model selection, is defined as

where is the number of model parameters and is the maximized value of the log-likelihood function of the estimated model. When there are several potential available models, the one with the smallest AIC among these could be selected as the best fitting model.

Table 2 shows the estimation results of the parameters, the log-likelihood function value, and the AIC. From Table 2, it can be found that the model obtains the highest and the lowest AIC compared to other models. This implies that the proposed model has better model fit than other models. Therefore, a model considering nonlinearity, the effects of stress level, product-to-product variability, and measurement errors simultaneously has a better and wider range of practical applicability.

Table 2: Comparisons of four degradation models with the degradation data of LEDs.

AIC

1.2889
0.3120
1831.7
0.0071
0.0270
4241.6
−8473.2

0.0150
0.0018
1878.9
0.0021
0.0160
3685.3
−7360.6

1.3781
0.0001
1853.1
0.026
—
3691.2
−7374.4

1.3629
—
1851.2
0.010
0.0260
4156.9
−8305.8

Remark : “—” means that the estimate does not exist in that case.

5.2. Optimal CSADT Plans

We first consider a two-level CSADT plan (), where , , and . And the lifetime is the first passage time when degradation path crosses . Suppose ; the optimal CSADT plans under various constraints which were determined by using the algorithm presented earlier are shown in Table 2. Since there are three constraints, the optimal plans are obtained by fixing two of the three and varying the last one. For example, Table 3(a) displays the optimal solutions by fixing the test duration and the sample size and varying the budget. When , the optimal test plan turned out to be . That is, the optimal sample sizes for stress level and are 31 and 19, respectively, and the total test time for the CSADT is 1200 hours. Under such a test plan, the total cost is 2500.

Table 3: (a) Optimal two-level CSADT plans under various prefixed budget constraints. (b) Optimal two-level CSADT plans under various duration constraints. (c) Optimal two-level CSADT plans under various sample size constraints.

(a)

Avar
Budgets

1000
13
7
320
2
0.8933
1000

1500
18
10
328
3
0.4805
1500

2000
26
15
349
3
0.3212
1999.5

2500
31
19
300
4
0.2363
2500

3000
30
19
314
6
0.1987
3000

3500
31
19
333
6
0.1921
3099

4000
31
19
333
6
0.1921
3099

Remark : the test duration constraint and the sample size constraint .

(b)

Avar
Budgets

500
32
18
100
5
0.3001
2250

1000
31
19
200
5
0.2382
2500

1500
31
19
300
4
0.2363
2500

2000
31
19
300
4
0.2363
2500

2500
31
19
300
4
0.2363
2500

3000
31
19
300
4
0.2363
2500

Remark : the prefixed budget constraint and the sample size constraint .

(c)

Avar
Budgets

20
12
8
85
23
0.4054
2497.5

30
18
12
200
10
0.2974
2500

40
25
15
211
7
0.2515
2498.5

50
31
19
300
4
0.2362
2500

60
32
19
281
4
0.2362
2500

70
32
19
281
4
0.2362
2500

Remark : the prefixed budget constraint and the test duration constraint .

It is interesting to observe that the three constraints all have a significant impact on the results. As one of them is increasing and the other two remain constant, the change of the optimal test plan is getting smaller and smaller. This result is true because a constraint will be out of action when it becomes lager enough. In this case, the optimal plan is completely determined by the other two constraints. Therefore, the constraints should be developed reasonably in practical application.

Moreover, the asymptotic variance of was gradually decreasing. This means the test accuracy becomes higher with the constraints relaxing. Besides, we could observe that the magnitude of the reduction of was also getting smaller slowly. Therefore, we only need to select an appropriate constraints condition rather than a more relaxed one within the requirement of test accuracy.

5.3. Sensitivity Analysis

In practice, the estimated parameter would depart from the true parameter . Without loss of generality, we assume that denote the estimation bias for , , (as the values of and are too small, we do not consider their estimation bias).

Under the same cost configuration , Table 4 displays the optimal plan under various combinations of , , and . From these results, we can see that the optimal test plan tends to be robust to estimation bias, given that the bias is not too large.

Table 4: Optimal CSADT plan considering the estimation bias for , , .

Avar
Budgets

5%
5%
5%
29
16
13
280
5
0.4925
2500

5%
0
0
31
19
12
300
4
0.1944
2500

5%
−5%
−5%
31
19
12
300
4
0.8060
2500

0
5%
0
31
19
12
300
4
0.2485
2500

0
0
−5%
30
20
10
300
4
0.1045
2500

0
−5%
5%
29
16
13
280
5
0.5514
2500

−5%
5%
−5%
30
20
10
300
4
0.1368
2500

−5%
0
5%
29
17
12
264
5
0.7048
2500

−5%
−5%
0
31
19
12
300
4
0.2759
2500

0
0
0
31
19
12
300
4
0.2363
2500

5.4. Optimal CSADT Plans with -Stress, Where

We have already presented the optimal CSADT plans with two stress levels. When stress levels , it is not easy to get results of test plans. Instead, we select a condition of three stress levels to show the optimal CSADT plans. Suppose three stress levels are , , and . The cost configuration is . Table 5 lists the optimal CSADT plans under various constraints.

Table 5: (a) Optimal three-level CSADT plans under various prefixed budget constraints. (b) Optimal-three level CSADT plans under various duration constraints. (c) Optimal three-level CSADT plans under various sample size constraints.

(a)

Avar
Budgets

1000
7
5
4
282
3
1.926
999

1500
13
10
7
280
3
1.031
1500

2000
16
12
10
278
4
0.6934
2000

2500
21
16
13
300
4
0.5150
2500

3000
21
16
13
300
4
0.4303
3000

3500
21
16
13
200
10
0.3908
3500

4000
21
16
13
133
15
0.3719
3997.5

Remark : the test duration constraint and the sample size constraint .

(b)

Avar
Budgets

500
22
16
12
166
3
0.7538
2049

1000
21
16
13
200
5
0.5190
2500

1500
21
16
13
300
4
0.5150
2500

2000
21
16
13
300
4
0.5150
2500

2500
21
16
13
300
4
0.5150
2500

3000
21
16
13
300
4
0.5150
2500

Remark : the prefixed budget constraint and the sample size constraint .

(c)

Avar
Budgets

20
8
7
5
90
22
0.8951
2470

30
12
10
8
200
10
0.6518
2500

40
17
13
10
211
7
0.5497
2498.5

50
21
16
13
300
4
0.5150
2500

60
21
16
13
300
4
0.5150
2500

70
22
16
13
281
4
0.5148
2500

Remark : the prefixed budget constraint and the test duration constraint .

No matter how the constraints vary, the optimal plans demonstrate that the lowest stress levels were allocated to more units. Comparing Tables 3 and 5, we can find that the of the optimal two-level CSADT plan is smaller than that of the optimal three-level CSADT plan under the same conditions. It means that the two-level CSADT plans have higher test accuracy. However, this result does not intend to suggest that the optimal plans are the only choice for conducting a CSADT. A degradation test may need degradation data of more than two stress levels, so as to verify the validity of the model in (1) and ensure that the research can be generalized.

6. Conclusion

We have investigated the optimal CSADT plans based on the NGWP model. The NGWP model which considers nonlinearity, the effects of stress level, the product-to-product variability, and measurement errors has higher estimation accuracy and better goodness-of-fit. By minimizing the asymptotic variance of the reliability estimation of the products under normal operation conditions subject to sample size, test duration, and test cost, the objective of CSADT plans is to properly determine the stress levels, the number of units allocated to each level, inspection frequency, and measurement times, simultaneously. An optimization algorithm is proposed to determine the decision variables. Moreover, the MLE method to estimate unknown parameters and MTTF of products is presented in this study. Then, comparison based on degradation data of LEDs is conducted to show better goodness-of-fit of the NGWP than that of other models. Finally, optimal two-level CSADT plans and optimal three-level CSADT plans under various constraints are demonstrated. A detailed sensitivity analysis for the estimated parameters is also conducted in this study.

When the stress levels are more than three, a new algorithm which is more efficient should be developed. If the sample size is only moderate or even small, it is necessary to investigate other methods for designing a CSADT. Overall, many interesting issues about degradation models and accelerated test plans require further study.

Appendix

Detailed Expressions of and in (15)

The expression of Fisher information is

where

The expression of is .

By the delta method, the asymptotic variance of can be calculated by using the following formulation:

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to acknowledge great support by National Natural Science Foundation of China (51475289).

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