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

Sandip Barui, Shovan Chowdhury

TL;DR

This paper addresses optimal accelerated life testing plan design when lifetime characteristics respond nonlinearly to a single accelerating stress. It introduces a piecewise linear approximation to link stress to Weibull-location and -scale parameters, enabling non-identical lifetimes across stress levels with a Type-I censoring framework. The acceptability criterion is grounded in the Fisher information matrix, and two constrained optimization problems minimize aggregate costs or the variance of log-lifetime quantiles, ensuring producer and consumer risk constraints. A simulated example demonstrates that PLA-based models provide a superior fit and more efficient plans than linear-link alternatives, highlighting practical benefits for reliable warranty-based acceptance testing.

Abstract

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.

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

TL;DR

This paper addresses optimal accelerated life testing plan design when lifetime characteristics respond nonlinearly to a single accelerating stress. It introduces a piecewise linear approximation to link stress to Weibull-location and -scale parameters, enabling non-identical lifetimes across stress levels with a Type-I censoring framework. The acceptability criterion is grounded in the Fisher information matrix, and two constrained optimization problems minimize aggregate costs or the variance of log-lifetime quantiles, ensuring producer and consumer risk constraints. A simulated example demonstrates that PLA-based models provide a superior fit and more efficient plans than linear-link alternatives, highlighting practical benefits for reliable warranty-based acceptance testing.

Abstract

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.
Paper Structure (17 sections, 2 theorems, 70 equations, 1 figure, 5 tables, 2 algorithms)

This paper contains 17 sections, 2 theorems, 70 equations, 1 figure, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model Framework
  3. Product lifetime distribution
  4. Piecewise Linear Approximation of Lifetime Characteristics based on Accelerating Stress
  5. Choice of Knots
  6. Example of PLA-based Model
  7. Likelihood Function under Type-I Censoring and Associated Fisher Information Matrix
  8. Lot Acceptance Criterion and Asymptotic Distribution
  9. Asymptotic Mean and Variance of $W$
  10. Acceptability Constant and Optimization Constraint
  11. Optimal Sampling Plan
  12. Aggregate Cost Minimization
  13. Variance of Aggregate Quantile of Log-Lifetime Minimization
  14. Numerical Illustration
  15. Case Illustration and Comparison
  16. ...and 2 more sections

Key Result

Theorem 2.1

The Fisher information matrix for $\bm \theta$ is given by $F(\bm \theta)=n \sum_{i=0}^m \pi_i l_i(\bm \theta)$ where $\pi_i$ is the proportion of allocation to the stress level $s_i$ and $\bm \theta=(\bm \gamma_{\mu}^{\intercal}, \bm \gamma_{\sigma}^{\intercal})^{\intercal}$.

Figures (1)

  • Figure 1: A graphical illustration of the goodness of fit of the piecewise linear approximation model

Theorems & Definitions (4)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof