Optimum Accelerated Degradation Tests for the Gamma Degradation Process Case under the Constraint of Total Cost

An accelerated degradation test (ADT) is regarded as an effective alternative to an accelerated life test in the sense that an ADT can provide more accurate information on product reliability, even when few or no failures may be expected before the end of a practical test period. In this paper, statistical methods for optimal designing ADT plans are developed assuming that the degradation characteristic follows a gamma process (GP). The GP-based approach has an advantage that it can deal with more frequently encountered situations in which the degradation should always be nonnegative and strictly increasing over time. The optimal ADT plan is developed under the total experimental cost constraint by determining the optimal settings of variables such as the number of measurements, the measurement times, the test stress levels and the number of units allocated to each stress level such that the asymptotic variance of the maximum likelihood estimator of the q-th quantile of the lifetime distribution at the use condition is minimized. In addition, compromise plans are developed to provide means to check the adequacy of the assumed acceleration model. Finally, sensitivity analysis procedures for assessing the effects of the uncertainties in the pre-estimates of unknown parameters are illustrated with an example.


Introduction
Strong pressure from customers and intense global competition among manufacturers have resulted in the production of highly reliable products. Reliability inferences based on the results of life tests or OPEN ACCESS accelerated life tests can be inaccurate when used to evaluate highly reliable products since few or no failures may be expected before the end of a practical test period. In contrast, if a degradation characteristic related to the failure mechanism exists, a test which monitors the behavior of the characteristic may provide more accurate information on product reliability. In order to accelerate the degradation process for highly reliable products, such a test is usually conducted under stress levels higher than the normal use condition. This type of test is called an accelerated degradation test (ADT) in the literature [1]. For statistical modeling and analysis of various degradation phenomena, the reader is referred to the papers in Escobar et al. [2].
An ADT, as all other reliability tests, must be carefully designed beforehand to obtain the estimates of the quantities of interest as precisely as possible. A recent review of the literature by Yum et al. [3] on designing ADT plans shows that many researchers have designed ADT plans based on the general degradation path (GDP) model. For instance, see Boulanger and Escobar [4], Park and Yum [5], Yu [6,7], Park and Yum [8], and Shi et al. [9]. The GDP model consists of an actual degradation path and an error term. The actual degradation path is represented as a deterministic function of time, and the error term usually represents the measurement error assumed to be independent over time. Given this formulation, most of the GDP models developed to date do not consider the time-dependent error structure (Tseng and Peng [10]). For this reason, a stochastic process (SP) model that naturally incorporates the correlation among degradation measurements over time can represent a useful alternative.
Currently used SP models for degradation include the Wiener process (WP), geometric Brownian motion (GBM), and gamma process (GP) models, among others. Tang et al. [11], Liao and Tseng [12], and Lim and Yum [13] developed optimal ADT plans under the assumption of a WP model for degradation. Liao and Elsayed [14] designed optimal ADT plans based on an accelerated geometric Brownian motion degradation rate (AGBMDR) model. In a WP model, the degradation may take negative values and does not always increase with time, while in the GBM model the degradation is always positive, but not strictly increasing over time. However, in certain physical situations where the measurement error is relatively small, the degradation should always be nonnegative and strictly increasing over time. In such situations, a GP model which always gives nonnegative, strictly increasing degradation over time is considered to be a more adequate model. Bagdonavicius and Nikulin [15] and Lawless and Crowder [16] modeled degradation as a GP which allows covariates. Park and Padgett [17,18] organized a basic framework for the degradation model using the GP as well as the WP and GBM.
Despite the appropriateness of GP models for degradation, little research has been conducted on designing optimal ADT plans based on a GP model (Yum et al. [3]). The notable exceptions is Tseng et al. [19], and Pan and Sun [20] in which optimal step-stress ADT plans are developed for a gamma degradation process by determining the sample size, measurement frequency and termination time such that the asymptotic variance of the estimated mean time to failure or q-th quantile of the lifetime distribution is minimized under the total cost constraint, and Tsai et al. [21] in which optimal ADT plans with two stress variable are developed. In this paper, optimal ADT plans are developed based on the assumptions that a single stress variable is considered for ease of conducting the test and interpreting the result, the constant-stress loading method is employed and the degradation characteristic follows a GP. The number of measurements, the measurement times, the test stress levels and the number of units allocated to each stress level are determined under the total experimental cost constraint such that the asymptotic variance of the maximum likelihood estimator (MLE) of the q-th quantile of the lifetime distribution at the use condition is minimized. This paper is organized as follows: Section 2 introduces the assumed accelerated degradation model, and derives the lifetime distribution. In Section 3, the asymptotic variance of the MLE of the q-th quantile of the lifetime distribution at the use condition and the total experimental cost are described, and the corresponding optimization problem is formulated. The procedure for obtaining the optimal ADT plan is presented in Section 4. In Section 5, a compromise plan with three stress levels is developed for checking the adequacy of the assumed acceleration function. Sensitivity analyses of the test plans with respect to the uncertainties involved in the pre-estimates of unknown parameters are given in Section 6 with an example. Finally, conclusions and future research directions are presented in Section 7.

Gamma Process Degradation Model
In the present investigation, ( ) y t , the degradation characteristic at time t , is assumed to follow a GP with shape coefficient

Acceleration Function and Standardization
In this paper, it is assumed that the relationship between the shape coefficient α′ of a GP and the stress variable s′ can be described by any one of the following (Nelson [1]):  It is assumed that the use stress level 0 s′ and maximum stress level M s′ are pre-specified. For simplicity and without loss of generality, the stress level is standardized as follows:

Lifetime Distribution
When the degradation characteristic ( ) y t follows a GP with shape coefficient ( ) is the incomplete gamma function defined as: In addition, the pdf of T is obtained as follows (Park and Padgett [17]):

Optimization Criterion and Constraint
Various optimization criteria have been proposed for designing ADT plans. The most frequently used criteria include the (asymptotic) variance and mean squared error (MSE) of the estimator of the q-th quantile (or the mean) of the lifetime distribution at the use condition (Park and Yum [5]; Yu [6,7]; Li and Kececioglu [22,23]; Park and Yum [8]; Liao and Tseng [12]; Shi et al. [9]; and Tseng et al. [19]). In addition to these criteria, Boulanger and Escobar [4] and Liao and Elsayed [14] considered the generalized variance, namely, the determinant of the Fisher information matrix of the MLEs of unknown parameters, and Tang et al. [11] considered the total experimental cost. These optimization criteria have been also used as constraint. Among constraints, the total experimental cost is frequently used (Li and Kececioglu [22]; Liao and Tseng [12]; and Tseng et al. [19]). In addition to the constraint, Tang et al. [11] considered the precision constraint using the asymptotic variance of the estimator of the mean lifetime at the use condition. In this paper, the asymptotic variance of the MLE of the q-th quantile of the lifetime distribution at the use condition is adopted as an optimization criterion and the total experimental cost as a constraint.

The Asymptotic Variance of the MLE of the q-th Quantile of the Lifetime Distribution
In the following, we consider a constant-stress loading ADT where i n test units among follows a gamma distribution with the following pdf: From Equations (1) and (3), the log likelihood function for n test units is given by:  . The MLEs of 1 2 , δ δ and β , which is 1 2, δ δ and β , can be obtained by solving the simultaneous equations: Hence, the corresponding MLE of the q-th quantile of the lifetime distribution at the use condition ( ,0 Let F be the Fisher information matrix obtained by taking expectations of the negative second partial derivatives of ln L with respect to unknown parameters 1 2 , and δ δ β (Lawless [24]). Then, it is shown in Appendix A that: is the trigamma function defined as Then, by using the delta method, the asymptotic variance of the MLE of the q-th quantile of the lifetime distribution at the use condition is obtained as follows (Liao and Tseng [12]): where the superscript t represents a transposition.

Total Experimental Cost
The total experimental cost comprises three parts:

Formulation of the Problem
The ADT design problem with two stress levels is formulated in this section. For the case of two stress levels, the objective function is given by (see Appendix B):   (5) is a scaling factor, the objective function can be reduced to:

P n A B s n A B s n A n A n A s n A s
Finally, the ADT design problem with two stress levels is formulated as follows:

Pre-Estimation
The objective function v in Equation (6) depends on the unknown parameters 1 2 , δ δ and c β , which need to be estimated in advance. Pre-estimates of these parameters could be obtained based on engineering judgement and/or preliminary experiments. In this paper, it is suggested to conduct a preliminary experiment at the maximum stress level ( 1 s = ), and in addition guess the value of the failure probability ( 0 p ) until a specific time point (τ ) at the use condition. Using the MLE method for the data from the preliminary experiment at the maximum stress level, the three unknown parameters, 1 2 , δ δ and c β cannot be estimated separately, but 1 2 δ δ + and c β can be estimated. For separation, the information on 0 p can be utilized. First, 0 p can be expressed as follows using Equation (2): Then, 1 δ can be numerically estimated by inserting the pre-estimate of c β obtained from the preliminary experiment into (7), and 2 δ can be estimated by subtracting the pre-estimate of 1 δ from that of 1 2 δ δ + .

Optimization
It is difficult to obtain the analytic expression of the optimal solution since v is highly complicated. There is simulation method such as simulated annealing to search the optimal solution, but it takes relatively long time to obtain the global optimum. In this paper, considering the simplicity in the constraint structure and the integer restriction on the decision variables except the stress levels, the optimal solution can be fast determined through the following algorithm (as part of which a simple grid search method is employed for determining the optimal stress levels where the distance between adjacent grid points is 0.01) and Figure 2 shows the flow chart of the algorithm (see Tseng et al. [19])

Algorithm:
Step1. Obtain the upper bound for the possible number of test units where x     is the largest integer less than x .
Step 3. Obtain the upper bound of the measurement time interval for fixed n .
Step 5. Set the number of measurements ( ) m as large as possible since v is the decreasing function of m .
Step 6. Find the combination of 1 2 , n n N ∈ such that 1 2 n n n + = .
Calculate v for all possible combinations of grid values of each stress level which satisfy Step 8. Set 1 t t Δ = Δ + , and repeat step 5 through 7 until b t t Δ = Δ .
Step 9. Set 1 n n = + , and repeat step 3 through 8 until b n n = .
Step 10. The optimal solution is determined as the combination of the decision variables (

Compromise Plans
Since the optimal ADT plan developed in Section 4 involves two stress levels, it does not provide means to check the validity of the assumed acceleration model. For the case of three stress levels, the asymptotic variance of ,0q t can be obtained in a similar manner as for the case of two stress levels as follows: where:

R n n A A B B s s n n A A B B s s n n A A B B s s
Then, a compromise plan with three stress levels is developed as follows: 3. For given 2 π , the decision variables 1 3 1 , , , n n s t Δ and m are determined such that c v is minimized.

Example and Sensitivity Analysis
LEDs are widely used as a light source for optical fiber transmission systems and consumer electronics due to their high brightness, low power consumption and high reliability. An ADT is employed in order to estimate the 0.1-th quantile of the lifetime distribution of the LEDs at the use condition. A failure-related degradation characteristic ( ) y t of the LED is the percent decrease of its light intensity over time. It is assumed that the degradation characteristic follows a GP since the degradation of LED progresses monotonically as shown as the real degradation data in Liao and Elsayed [14]. The current is considered as the accelerating stress variable in the ADT, and the maximum and use stress levels are specified as 40 mA and 10 mA, respectively. In addition, the power model is assumed between the shape coefficient of the GP and current. The failure time of the LED is defined as the time when its light intensity degrades below 50% from its initial value. In other words, 0.5 c y = .
In order to pre-estimate the unknown parameters for optimally designing the ADT, part of the data in Liao and Elsayed [14] is regarded as the data obtained from a preliminary experiment. Liao and Elsayed [14] tested 20 units for 250 h using two stress variables, temperature and current. In this paper, only the current is considered as a stress variable, and the data obtained at 40 mA (and 413 K) for 50, 100 and 150 h from 5 units are regarded as the preliminary experimental data. These data are analyzed using the MLE method, and the estimates of 1 2 δ δ + and c β are respectively obtained as −2.74 and 7.17. In addition, the failure probability until 4 months at the use condition is estimated as 5ⅹ10 −5 . Then, the pre-estimate of 1 δ is numerically obtained as −9.32 from Equation (7), and 2 δ is estimated as 6.58 , the optimal ADT plan is given by: That is, the optimal plan is to allocate 6 units to the use current (= 10 mA) and 13 units to the high current (= 40 mA) levels, and measure 26 times every 7 h the degradation characteristic for each unit.  The pre-estimated values of 1 δ , 2 δ and c β used to design the above optimal plan may be different from the true values. It is therefore desirable to assess the sensitivity of the optimal plan to the uncertainties in 1 δ , 2 δ and c β . In this example, sensitivity analyses are conducted for misspecifications of ±10% in the pre-estimated values of 1 δ , 2 δ and c β , and the results are summarized in Table 2 where 0 v and * v respectively denote v values for the plan in (8) and for the optimal plan obtained using the true values of 1 2 , δ δ and c β . The ratios ( Table 2 indicate that the plan in (8) is insensitive to the plausible departures of the true 1 2 , δ δ and c β valules from their pre-estimated ones.
A compromise plan with three stress levels is needed if we want to check the adequacy of the assumed acceleration model. The compromise plans under various budgets for 1 9.32 δ = − , 2 6.58 δ = , 7.17 c β = and 2 0.2 π = are shown in Table 3. When 2000 b C = , the compromise plan with three stress levels is given by: That is, the compromise plan with three stress levels is that 1 2 5, 3 n n = = and 3 11 n = are allocated to the use (= 10 mA), middle (= 20 mA) and high (= 40 mA) current levels, respectively, and measure 26 times every 7 h the degradation characteristics for each unit.
Sensitivity analyses for the compromise plan are conducted in a similar manner as for the case of the optimal plan and the results are summarized in Table 4 where 0 c v and * c v are respectively c v values for the plan in (9) and for the compromise plan obtained using the true values of 1 2 , δ δ and c β . The ratios ( Table 4 indicate that the plan in (9) is not sensitive to the plausible departures of the true 1 2 , δ δ and c β values from their pre-estimated ones.

Conclusions
In this paper, optimal ADT plans are developed based on the assumption that the degradation characteristica follow a GP. Under the constraint that the total experimental cost does not exceed a predetermined budget, the decision variables such as the number of measurements, the measurement times, the test stress levels and the number of units allocated to each stress level are optimally determined by minimizing the asymptotic variance of the MLE of the q-th quantile of the lifetime distribution at the use condition. Compromise plans are also developed to provide means to check the adequacy of the assumed acceleration model. Lastly, sensitivity analyses for assessing the effects of the uncertainties in the pre-estimates of unknown parameters on the optimal and compromise plans are illustrated with an example.
Unlike previous works on GDP-based ADT planning, the present SP-based approach is able to take into account the correlation among degradation measurements over time. Furthermore, compared to the WP-or GBM-based approach, the proposed GP-based approach can deal with more frequently encountered situations in which the degradation characteristic is always nonnegative and strictly increasing over time.
The constant-stress loading method is assumed in the present study. A fruitful area of future research would be to extend the present study to the cases of other stress loading methods (e.g., progressive-stress is the digamma function defined as