Accelerated Life Test Plans: Analytical Solution and Comparison with Simulation

Accelerated Life Test Plans:
Analytical Solution and Comparison with Simulation

Test planners are often asking two questions before they conduct an accelerated life test. The first question is: What stress levels should I use? The second question is: How many units should I test in each stress level? This article will describe methods that can be used to provide answers to these two questions.

In fact, these two questions are not independent. They are affected by each other. An optimal test plan consists of two parts, which are the answers to the above two questions. Much research has been conducted to search for a proper combination of stress levels and number of test units at each stress level. Nelson (1990), Meeker and Escobar (1998) and Nelson (2005) provide a substantial review of the state-of-the-art methods. In this article, we will briefly review some of the methods and use a single-stress accelerated test as an example to illustrate how to find an optimum test plan. The procedure is similar when designing plans for tests with two or more stresses.

Any accelerated life test must have an objective. A common objective is to accurately predict the BX life at the use stress level by using the failure data obtained from the accelerated stress test. For example, the objective could be to accurately estimate the B10 life of a product, the time at which 10% of the units are expected to have failed. Here, accuracy is measured in terms of the width of the confidence bounds at a given confidence level. A result with narrow confidence bounds is more accurate than a result with wide confidence bounds (i.e. there is less uncertainty).

In many cases, the test time and the total number of test units for an accelerated life test are selected based on available test equipment, budget, time to market, development cycle, etc. Constrained by available resources, test planners need to define the stress levels and number of units at each stress level, or the proportion of the total units in the test. From the study of published materials, we found five commonly used plans for single-stress accelerated tests.

1. Two Level Statistically Optimum Plan

2. Three Level Best Standard Plan

3. Three Level Best Compromise Plan

4. Three Level Best Equal Expected Number Failing Plan

5. Three Level 4:2:1 Allocation Plan

The Two Level Statistically Optimum Plan is the foundation for all of the test plans. The method to obtain this plan can be extended easily to the other four plans. Therefore, in this article, we will focus on the design of a Two Level Statistically Optimum Test Plan.

Analytical Solution for an Accelerated Life Test Plan
In this section, the theoretical background for designing an accelerated life test plan will be provided. More details can be found in Nelson (1990) and Meeker and Escobar (1998).

General Assumptions
Most accelerated life testing plans use the following assumptions that correspond to many practical quantitative accelerated life testing problems.

1) The log-time-to-failure for each unit follows a location-scale distribution such that:

Where m and s are the location and scale parameters respectively and F(?) is the standard form of the location-scale distribution. Commonly used distributions such as the Weibull, Lognormal, Normal and Smallest Extreme Value (SEV) distributions are in the location-scale distribution family. The logarithm transformations of Weibull distributed random variables follow an SEV distribution. Similarly, the logarithm transformation of a Lognormal distribution is a Normal distribution.

2) Failure times for all test units, at all stress levels, are statistically independent.

3) The location parameter m is a linear function of stress. Specifically, it is assumed that:

4) The scale parameter s does not depend on the stress levels. All units are tested until a pre-specified test time. For the Weibull distribution, s = 1/b. s is the scale parameter of the SEV distribution. Similarly, s is also the scale parameter of a Normal distribution.

5) The SEV and Normal distributions are used in the modeling. Denoting as Y the logarithm transformed failure time, the model for the Weibull distribution is:

The Lognormal model is given by:

Accelerated Test Planning Criteria
A common purpose of an accelerated life test experiment is to estimate a particular percentile T_p (life corresponding to unreliability value of p) in the lower tail of the failure distribution at use stress. Thus a natural criterion is to design a test such that, using the failure data obtained from this test, we can predict T_p with a relatively small Var(T_p), or Var(Y_p) such that Y_p=ln(T_p). Var(Y_p) measures the precision of the p quantile estimator; smaller values mean more confidence in the value of Y_p in repeated samplings. Hence a "good" test plan should yield a relatively small, if not the minimum, Var(Y_p) value.

Analytical Solution for Two Level Statistically Optimum Plan
Since the procedure to obtain a solution for the Two Level Statistically Optimum Plan is the foundation, we will give the problem formulation for the plan in this section.

Without loss of generality, a stress can be standardized as follows:

Qalt Plan
Equation Stress Standardized

Where:

x_Dis the use stress or design stress at which product life is of primary interest.
x_His the highest test stress level.

The values of x, x_Dand x_H refer to the actual values of stress or to the transformed values in the case where a transformation is used (e.g. the reciprocal transformation to obtain the Arrhenius relationship or the log transformation to obtain the power relationship). Therefore, x = 0at the design stress and x = 1 at the highest test stress. x_Dand x_H are given before planning the test.

Searching for a better testing plan is an optimization problem. The objective of the optimization problem is to minimize the value of Var(Y_p). For a two level statistically optimum plan, there are two decision variables in the objective

function. They are x_L (the standardized stress value at low stress in the test) and p_L (the percentage of the total test units allocated at the low stress level). The optimization problem can be formulated as follows.

Minimize:

Subject to:

f(x_L,p_L) is a function from the Fisher Information Matrix, which can be used to estimate the value of Var(Y_p). The final test plan will be (x_L,1; p_L,1-p_L). This means we will allocate 100p_L% of the total test units to the low stress x_L and the rest of the units will be tested at the highest stress level.

In order to calculate the function f(x_L,p_L), we also need the following information to be determined beforehand:

The test duration (or censoring time). Assume the test durations are the same at all the stress levels.

The probability of failure at x_D and at x_H by the end of the test. These two probabilities of failure are used to get the plan value of g₀ and g₁ in the life-stress relationship.

The following example will demonstrate an application of this test planning method.

Example
A reliability engineer is planning an accelerated test for a mechanical component. Torque is the only factor in the test. The purpose of the experiment is to estimate the B10 life of the components (i.e. Time at Unreliability = 0.1). The reliability engineer wants to use a Two Level Statistically Optimum Plan. 40 units are available for the test. The mechanical component is assumed to follow a Weibull distribution with b = 3.5 and a Power model is assumed for the life-stress relationship. The test is planned to last for 10,000 cycles. Based on experience, the engineer has estimated that there is a 0.0006 probability that a unit will fail by 10,000 cycles at the use stress level of 60Nm. The highest stress level allowed in the test is 120Nm and a unit is estimated to fail with a probability of 0.99999 at that level.

Solution:
Based on the above information, one can use any optimization tool to solve the decision variables (x_L,p_L) and get the optimum test plan. Figure 1 shows the setup to generate the test plan in ReliaSoft’s ALTA 7 software. Figure 2 shows the resulting accelerated test plan.

From Figure 2, we can see that the low stress level is 95.3982Nm (or about 95.4Nm) with 28 test units allocated. The high stress level is 120Nm and has 12 units on test. In the "Evaluate Test Plan" utility at the right side of the Test Plan Results window shown in Figure 2, we can see that with 40 test units and with this test plan, the ratio of the upper confidence bound to the lower confidence bound of the estimated B10 life is 2.946, at the 90% confidence level. We will validate this ratio in the next section by using a simulation method.

If this confidence bounds ratio is wider than what the test planners want, they need to increase the sample size. For example, if they want the ratio to be reduced to 2 at the 90% confidence level, then by using this optimum plan, they need to increase the sample size up to 97 units, as shown in the inset to Figure 2.

Alta Test Plan
Figure 1: Test plan setup for a single stress test

Alta Test Plan Results
Figure 2: The Two Level Statistically Optimum test plan for the settings
in Figure 1 and the sample size required for a confidence bounds ratio = 2.

Comparison of the Analytical Solution with the Simulation Results
Simulation results are often used to validate analytical solutions. Unlike the analytical solution, which assumes that the logarithm transformation of the B10 life is normally distributed, the simulation results are obtained not based on any assumptions. From the example described in the previous section, we can get the following two equations:

Qalt Plan Equation Etadh

From these equations, we can solve h_D (at use stress 60Nm) = 83270.5 and h_H (at high stress 120Nm) = 4975.

We also know that the life-stress relationship is Inverse Power Law (IPL), which is given by:

(2)

Using h_D and h_H, we can solve K = 7.10108E-13 and n = 4.064999866. In fact, Eqn. (2) is the same as Eqn. (1) if we define m = ln(h), g₀ = -ln(K), g₁ = -n and x = ln(V). K and n also can be solved automatically by ALTA’s Parameter Experimenter, as shown in Figure 3.

Alta Parameter
Experimenter
Figure 3: Using ALTA’s Parameter Experimenter to solve for K and n

The following steps describe the simulation procedure:

Step 1: Set i = 1 and total number of simulation = N.

Step 2: Set the test duration (censor time) to 10,000 and the use stress to 60Nm.

Step 3: Use the model parameters b = 3.5, K = 7.10108E-13 and n = 4.064999866 to generate 28 random numbers at stress = 95.4Nm. If the value of a random number is greater than 10,000, this unit is treated as a suspension. Do the same thing for the 12 units at stress = 120Nm.

Step 4: Use the generated failure and suspension data at stress levels = 95.4Nm and = 120Nm to estimate the model parameters b, K and n.

Step 5: Use the model parameters estimated in Step 4 to predict the B10 life, recorded as T₁ (.90).

Step 6: If i < N, set i = i + 1 and go back to Step 2.

At the end of the simulation, you will have N values of the B10 life from the N simulations. In order to get the two-sided confidence bounds at the 90% confidence level, all the N values are sorted in ascending order. The value at the position of 0.05N will be the lower bound and the value at the position of 0.95N will be the upper bound of the B10 life.

Fortunately, you do not need to perform these simulation steps manually. The upper and lower bounds of the B10 life can be obtained automatically using the SimuMatic utility that is integrated with ALTA 7. Figures 4 and 5 show part of the setup for the simulation while Figure 6 shows the results after 1,000 simulation runs. Please note that your results may vary since they were obtained through Monte Carlo simulation.

Alta Simumatic Model
Figure 4: Settings for the Model page of ALTA’s SimuMatic,
specifying the model parameters and the number of points to generate at each stress level

Alta Simumatic
Reliabilities
Figure 5: Settings for the Reliabilities & Times page of ALTA’s SimuMatic,
specifying that a sorted list of B10 Life values will be generated

As shown in Figure 6, the two-sided lower bound at the 90% confidence level is 27779.27. The two-sided upper bound is 80649.82. Therefore, the ratio of the two bounds is:

(3)

From the analytical solution described in the previous section, we know that the analytical bounds ratio is 2.946, which is very close to the result obtained from simulation, 2.90.

In order to test whether the optimum test plan really can provide the narrowest bounds ratio, the next step would be to use simulation to get the bounds ratios for other possible test plans.

For example, if we perform the simulation with 20 units at 95.4Nm and 20 units at 120Nm, the two-sided lower bound at the 90% confidence level is 26919.75 and the two-sided upper bound is 86494.12. The ratio of the two bounds is:

(4)

Then, if we perform the simulation with 20 units at 100Nm and 20 units at 120Nm, the two-sided lower bound at the 90% confidence level is 24847.80. The two-sided upper bound is 83042.23. The ratio of the two bounds is:

(5)

And so on. You will notice that both ratios in Eqn. (4) and Eqn. (5) are larger than the ratio obtained from Eqn. (3), which, together with many more simulations to test other combinations, serves to confirm that testing 28 units at 95.4Nm and 12 units at 120Nm is the optimum accelerated test plan for this example.

From the comparison of the analytical solution and simulation results, you can see that by using the analytical solution method, you can get a result that is very close to the one obtained from the simulation method. However, in order to get a statistically optimum test plan from simulation, you have to try many different combinations of the unit allocations and stress level settings. On the other hand, the analytical solution is based on the approximated Normal distribution of a large sample size. For small sample sizes, it may not yield an accurate result and the simulation method may be preferred.

Alta Simumatic Results
Figure 6: Simulation results with the
90% two-sided lower confidence bound highlighted

Discussion
In this article, we discussed how to design an optimum test plan using both analytical and simulation methods. In reality, though, designing a good accelerated test plan is more complicated than applying pure statistical methods. Strong engineering knowledge is always helpful. For example, planning tests may involve a compromise between "efficiency" and "extrapolation." More failures correspond to better estimation efficiency, requiring higher stress levels but more extrapolation to the use condition. Efforts to choose the "best" plan must take into account the trade-offs between efficiency and extrapolation. Test plans with more stress levels are more robust than plans with fewer stress levels because they rely on the validity of the life-stress relationship assumption. This is one of the reasons why the test planning method described in this article is called "statistically optimum" instead of "optimum."

References
This section presents a brief bibliography of the literature regarding accelerated life test planning. More information on the test planning utility in the ALTA software is available on the Web from http://www.weibull.com/acceltestwebcontents.htm and http://ALTA.ReliaSoft.com.

1. Meeker, W. Q., and Escobar, L. A. (1998), Statistical Methods for Reliability Data, John Wiley & Sons, New York.

2. Nelson, W. (1990), Accelerated Testing: Statistical Models, Test Plans, and Data Analysis, John Wiley & Sons, New York.

3. Nelson, W., (2005), "A Bibliography of Accelerated Test Plans," IEEE Transaction on Reliability, vol. 54, no. 2, pg. 194-197.