Optimal design of ALT plans based on proportional hazards models

Accelerated life tests are used to quickly obtain failure time data under high stress levels in order to predict product life and performance under the design stress. Optimal accelerated life test plans yield the most accurate ML estimates of product life measurements at the design stress. In this paper, the optimization criterion is to minimize the asymptotic variance of the reliability estimate over a specified period of time at the design stress. The optimal design determines the optimum stress levels and corresponding sample sizes of test units associated with those stress levels. This optimal design is based on Cox's Proportional Hazards Models. In this paper we consider the selection of constant stress levels z _j and the proportion of devices p _i to allocate for each z _j such that the most accurate reliability estimate at design condition z _D can be obtained. We assume the baseline hazard function λ ₀(t) to be linear with time. We obtain the corresponding cumulative hazard function Λ(t; z), reliability function R(t; z) respectively as: Λ(t; z) = ∫ ₀ ^t λ(u)du = ∫ ₀ ^t (γ ₀ + γ ₁u)e ^βzdu = (γ ₀t + γ ₁t ²/2)e ^βz, R(t; z) = exp(-(γ ₀t + γ ₁t ² / 2)e ^βz). The pdf is obtained as: f(t; z) = (γ ₀ + γ ₁t)e ^βz exp(-(γ ₀t + γ ₁t ² / 2)e ^βz). We develop the log likelihood function of an observation t (failure time) at a single stress level z and Type I censoring as: l = ln L(t; z) = ∑ _i=1 ⁿ{I(t _i ≤ τ)[ln(γ ₀ + γ ₁t ₁) + βz] - (γ ₀t _i + γ ₁t ² / 2)e ^βz}, where I(t ≤ τ) = 1, if t ≤ τ ; or 0, if t ≥ τ. The maximum likelihood estimates γ̂ ₀, γ̂ ₁, and β̂ are the parameter values that maximize the above log likelihood function. We plan an accelerated life test to obtain the most accurate reliability estimate under the limited testing conditions (time, cost, test units, etc.). Here, the optimal criterion is to minimize the asymptotic variance of the reliability estimate over a pre-specified period of time T at the design stress z _D, that is, to minimize ∫ ₀ ^T Var(R̂(t; z _D))dt, and Var(R̂(t; z _D)) = [∂R̂(t; z _D)/∂γ ₀, ∂R̂(t; z _D)/ ∂γ ₁, ∂R̂(t; z _D)/ ∂γ _β]∑[∂R̂(t; z _D)/ ∂γ ₀, ∂R̂(t; z _D) /∂γ ₁, ∂R̂(t; z _D)/ ∂γ _β] ^T, and ∑ is the variance-covariance matrix of the parameter estimates [γ̂ ₀, γ̂ ₁, β̂], and is the inverse of Fisher information matrix F. Assumptions: 1. There is a single accelerating stress z; 2. Test will be run at three levels of stress z _L < z _M < z _H; 3. Total test time is limited to τ units times; 4. A total of n units are available for testing; 5. z _H is the highest level of stress at which the failure mechanisms remain the same; 6. Initial baseline values for the model parameters are given. Under the constraints of available test units, test time and specification of minimum number of failures at each stress level, the problem is to optimally allocate stress levels and test units so that the asymptotic variance of the reliability estimate at normal conditions is minimized over a pre-specified period of time T. The optimal decision variables (z* _L, z* _M, p* ₁, p* ₂, p* ₃) are chosen by solving the following optimization problem with a nonlinear objective function and both linear and nonlinear constraints: Objective function: min ∫ ₀ ^T Var(R̂(t; z _D))dt, subject to ∑ = F ^-1; 0 < p _i < 1, i = 1,2,3; ∑ _i=1 ³p _i = 1; z _D < z _L < z _M < z _H; n p _i Pr[t ≤ τ | z _i] ≥ MNF, i = 1,2,3. Where, MNF is the minimum number of failures.