Optimal Accelerated Life Testing Sampling Plan Design with Piecewise Linear Function based Modeling of Lifetime Characteristics

Researchers have widely used accelerated life tests to determine an optimal inspection plan for lot acceptance. All such plans are proposed by assuming a known relationship between the lifetime characteristic(s) and the accelerating stress factor(s) under a parametric framework of the product lifetime distribution. As the true relationship is rarely known in practical scenarios, the assumption itself may produce biased estimates that may lead to an inefficient sampling plan. To this endeavor, an optimal accelerating life test plan is designed under a Type-I censoring scheme with a generalized link structure similar to a spline regression, to capture the nonlinear relationship between the lifetime characteristics and the stress levels. Product lifetime is assumed to follow Weibull distribution with non-identical scale and shape parameters linked with the stress factor through a piecewise linear function. The elements of the Fisher information matrix are computed in detail to formulate the acceptability criterion for the conforming lots. The decision variables of the sampling plan including sample size, stress factors, and others are determined using a constrained aggregated cost minimization approach and variance minimization approach. A simulated case study demonstrates that the nonlinear link-based piecewise linear approximation model outperforms the linear link-based model.

💡 Research Summary

The paper addresses a fundamental limitation in accelerated life testing (ALT) – the reliance on a pre‑specified, often linear or log‑linear, relationship between accelerating stress and lifetime characteristics. Recognizing that the true stress‑life relationship is rarely known, the authors propose a flexible, spline‑like modeling framework in which the Weibull shape (α) and scale (λ) parameters of product lifetimes are linked to a single accelerating stress factor through piecewise linear (PL) functions.

The methodology begins by assuming that product lifetimes Xij follow independent Weibull distributions with potentially different shape and scale parameters at each stress level si. By taking logarithms, the lifetimes are transformed to Tij = ln Xij, which follow an extreme‑value (EV) distribution with location µi = –ln λi and scale σi = αi⁻¹. The authors then model µi and σi as PL functions of the standardized stress ξi = (si – s0)/(sm – s0). User‑defined knots (cut‑points) ξµ,q and ξσ,q partition the stress range into Q1 and Q2 intervals, respectively, allowing distinct linear segments with separate intercepts and slopes. This yields a globally nonlinear but piecewise linear link that can approximate arbitrary stress‑life relationships.

Under a Type‑I censoring scheme (fixed test time τ0), the observed data consist of failure times for units that fail before τ0 and censored observations for those that survive. The authors derive the full log‑likelihood for the PL‑linked Weibull model and analytically compute the Fisher information matrix I(θ), where θ = (γµ⊤, γσ⊤) aggregates all PL parameters. The matrix elements are expressed as sums over stress levels and intervals, reflecting the non‑identical Weibull parameters across stresses. This detailed information matrix enables precise asymptotic variance calculations for the maximum likelihood estimators.

Using the Fisher information, the paper constructs an operating characteristic (OC) function. An acceptance statistic W is defined based on a tolerance constant k and target variances σ²µ0, σ²σ0 for the estimated location and scale. By invoking normal approximations, the authors control producer’s risk (α) and consumer’s risk (β), thereby establishing a statistically sound acceptance criterion for lot quality.

The core contribution lies in the optimal design of the ALT sampling plan (AL‑TSP). Two competing objectives are considered simultaneously: (1) minimization of an aggregate cost function C_T that aggregates product cost, inspection cost, and warranty cost (including free‑replacement and pro‑rata warranty periods), and (2) minimization of the variance V_Q of a chosen quantile of the log‑lifetime distribution, reflecting product reliability. Decision variables include total sample size n, allocation proportions πi to each stress level, and the positions of the PL knots. Constraints enforce budget limits, minimum detection power, warranty periods, and risk limits (α, β). The resulting problem is a constrained nonlinear optimization, solved numerically.

A simulated case study evaluates six scenarios with varying numbers of knots (Q1 = Q2 = 2 versus 3) and different cost/variance weightings. Results show that the PL‑based model consistently yields lower estimator variance and higher power than a traditional linear link model, while achieving comparable or lower total cost. The flexibility of the PL link allows the optimal stress levels to adapt to the underlying nonlinear relationship, reducing bias and over‑fitting risk.

In summary, the paper makes three major contributions: (i) introduction of a piecewise linear, spline‑like link for Weibull parameters that captures complex stress‑life relationships; (ii) derivation of a complete Fisher information matrix under Type‑I censoring for this non‑identical Weibull framework; and (iii) formulation of a dual‑objective, constrained optimization for ALT sampling plan design that balances cost and reliability variance. The authors suggest future extensions to multiple stress factors, hybrid censoring schemes, and Bayesian estimation to further broaden applicability.