Conducting an accelerated life testing (ALT) requires the determination or develop- ment of a reliability inference model that relates the failure data at stress conditions with design or operating conditions. Moreover, an accelerated test plan needs to be developed to obtain appropriate and sufficient information in order to estimate reliability performance accurately at operating conditions. A test plan requires the identification of the type of stresses to be applied, stress levels, methods of stress application (constant, ramp, cyclic), number of units at every stress level, minimum number of failures at every stress level, optimum test duration, frequency of test data collection and other test parameters. Indeed, without an optimum test plan, it is likely that a large sequence of expensive and time consuming tests be conducted that might cause delays in product release or in some cases the termination of the entire product.
In this section, we describe the procedure for designing an optimum test plan based on the proportional hazards model followed by a numerical example. Opti- mum test plans based on other ALT models can be developed in a similar fashion.
7.4.1 Design of ALT Plans
An ALT plan requires the determination of the type of stress, method of applying stress, stress levels, the number of units to be tested at each stress level and an applicable accelerated life testing model that relates the failure times at accelerated conditions to those at normal conditions.
When designing an ALT, we need to address the following issues: (a) select the stress types to use in the experiment, (b) determine the stress levels for each stress type selected, (c) determine the proportion of devices to be allocated to each stress level (Elsayed and Jiao 2002). We refer the reader to Meeker and Escobar (1998) and Nelson (2004) for other approaches for the design of ALT plans.
We consider the selection of the stress level zi and the proportion of devices pi to allocate for each zi such that the most accurate reliability estimate at use conditions zD can be obtained. We consider two types of censoring: type I censoring involves running each test unit until a prespecified time. The censoring times are fixed and the number of failures is random. Type II censoring involves simultaneously testing units until a prespecified number of them fails. The censoring time is random while the number of failures is fixed. We use the following notations:
ln Natural logarithm ML Maximum likelihood n Total number of test units
zH, zM, zL High, medium, low stress levels respectively zD Specified design stress
1, ,2 3
p p p Proportion of test units allocated to zL, zM and zL, respectively T Pre-specified period of time over which the reliability estimate is of
interest
R(t; z) Reliability at time t, for given z f(t; z) Pdf at time t, for given z F(t; z) Cdf at time t, for given z
( ; )t z
Λ Cumulative hazard function at time t, for given z
0( )t
λ Unspecified baseline hazard function at time t
We assume the baseline hazard function λ0( )t to be linear with time:
0( )t 0 1t λ =γ +γ
Substituting λ0( )t into the PH model given by Equation 7.5, we obtain,
0 1
( ; ) (t t)exp( ) λ z = γ +γ βz
We obtain the corresponding cumulative hazard function Λ( ; )t z , and the variance of the hazard function as
12
( ; ) ( 0 ) 2
t γ t γt e
Λ z = + βz
2
2 2
ˆ 2 2( [ ] )ˆ
0 1 0 1
ˆ ˆ
2 [ ] [ ] 2
0 1
ˆ ˆ ˆ ˆ
[( ) ] ( [ ] [ ] )
( 1)( )
ZD z Var z
z Var z Var z
Var t e Var Var t e
e e t
β β β
β β β
γ γ γ γ
γ γ
+ +
+ = +
+ − +
7.4.1.1 Formulation of the Test Plan
Under the constraints of available test units, test time and specification of minimum number of failures at each stress level, the problem is to allocate stress levels and test units optimally so that the asymptotic variance of the hazard rate estimate at normal conditions is minimized over a prespecified period of time T. If we consider three stress levels, then the optimal decision variables (z z p p p*L, *M, , ,1* *2 3*) are obtained by solving the following optimization problem with a nonlinear objective function and both linear and nonlinear constraints:
Min 0 1 ˆ
0
ˆ ˆ
[( ) D]
T z
Var γ +γt eβ dt
∫
subject to F−1
Σ =
0< pi<1, i=1,2,3
3
1 i 1
i p
= =
∑
D L M H
z <z <z <z
Pr[ | ] , 1,2,3
i i
np t≤τ z ≥MNF i=
where, MNF is the minimum number of failures and Σ
is the inverse of the Fisher's information matrix.
Other objective functions can be formulated which result in different design of the test plans. These functions include the D-Optimal design that provides efficient estimates of the parameters of the distribution. It allows relatively efficient deter- mination of all quantiles of the population, but the estimates are distribution depen- dent.
7.4.1.2 Numerical Example
An accelerated life test is to be conducted at three temperature levels for MOS capacitors in order to estimate its life distribution at design temperature of 50°C.
The test needs to be completed in 300 h. The total number of items to be placed under test is 200 units. To avoid the introduction of failure mechanisms other than those expected at the design temperature, it has been decided, through engineering judgment, that the testing temperature should not exceed 250°C. The minimum number of failures for each of the three temperatures is specified as 25. Further- more, the experiment should provide the most accurate reliability estimate over a 10-year period of time.
Consider three stress levels; then the formulation of the objective function and the test constraints follow the same formulation given in the above section. The optimum plan derived (Elsayed and Jiao 2002) that optimizes the objective function and meets the constraints is shown as follows:
o o o
160 , 190 , 250
L M H
z = C z = C z = C
The corresponding allocations of units to each temperature level are:
1 0.5, 0.4, 0.12 3
p = p = p =
7.4.1.3 Concluding Remarks
Design of ALT plans plays a major role in providing accurate estimates of reliability, mean time to failure and the variance of failure time at normal operating conditions. These estimates have a major impact on many decisions during the product life cycle such as maintenance schedules, warranty and repair policies and replacement times. Therefore, the test plans should be robust (Pascual 2006), i.e., it should be:
1. Robust to planning values of the model parameters. This implies that ALT conducted at three or more stresses are more robust than those conducted at two stresses. Allocating more units at the low stress level will also improve the robustness of the plan.
2. Robust to the type of the underlying distribution. In other words, misspeci- fication of the underlying distribution should not result in significant errors in calculating reliability characteristics.
3. Robust to the underlying stress-life relationship. The commonly used concept that higher stresses result in more failures might result in the
“wrong” stress-life relationship. For example, testing circuit packs at higher temperature reduces humidity which in turn results in fewer failures than those at field conditions. In essence, this is a deceleration test (higher stresses show fewer failures).