Optimum Accelerated Life Testing Models with Time-Varying Stresses

The following sections are included:

• CONTENTS
• PREFACE
• ACRONYMS
• LIST OF FIGURES
• LIST OF TABLES
• ABOUT THE AUTHOR

The following sections are included:

• Introduction
  • Difference between ALT and ADT
  • Fully and Partially Accelerated Tests
  • Inspection Modes in Accelerated Testing
• Types of Stress Loadings
  • Constant-Stress Loading
  • Step-Stress Loading
  • Progressive-Stress Loading
  • Cyclic-Stress Loading
  • Random-Stress Loading
  • Ramp-Constant Stress Loading
  • Ramp-Step Stress Loading
• Types of Stresses
  • Mechanical Stresses
  • Electrical Stresses
  • Environmental Stresses
  • Usage Rate Acceleration
• A Product’S Life Distribution
  • Reliability Metrics
  • Life Distribution Models
    • Exponential Distribution
    • Normal Distribution
    • Lognormal Distribution
    • Weibull Distribution
    • Extreme Value Distribution
    • Logistic Distribution
    • Log-Logistic Distribution
    • Truncated Logistic Distribution
    • Burr Type-XII Distribution
• Fully ALT Models
  • Life–Stress Relationship
    • Arrhenius Relationship
    • Inverse Power Law Relationship
    • Eyring Relationship
    • Temperature–Humidity Relationship
    • Temperature–Nonthermal Relationship
    • Multivariable Relationship
• Constant-Stress ALT Models
• Proportional Hazards Based ALT Models
• Proportional Odds (PO) Based ALT Models
• ALT Models Under Time-Varying Stress
  • Step-Stress ALT Model
  • Progressive-Stress ALT Model
• Time-Varying Stress ALT Models with Log-Location-Scale Distribution
• Relationship Between Constant-Stress, Step-Stress, and Progressive-Stress Loadings
• PALT Models
  • Constant-Stress PALT Models
  • Time-varying Stress PALT Models: Step-Stress Set-Up
    • Tampered Failure Rate (TFR) Model: Step-Stress Set-Up
• Data Types
• Data Analysis Methods
• Optimum Test Plans
  • Variance Optimality
  • D-Optimality
  • A-Optimality
• Likelihood Function Formulation
  • Formulation of Likelihood Function with Time-Censored Data
  • Formulation of Likelihood Function with Failure-Censored Data
• Sensitivity Analysis
• Design of Fully ALT Models Under Continuous Inspection: A Review
  • Constant-Stress ALT Models
  • Step-Stress ALT Models
  • Progressive-Stress ALT Models
  • Modified-Stress ALT Models
  • ALT Models with Competing Causes of Failure
• Design of PALT Models: A Review
  • Constant-Stress PALT Models
  • Step-Stress PALT Models
• Design of ALT Models Under Periodic Inspection: A Review

The following sections are included:

• Introduction
  • Notation
• An Optimum Simple Log-Logistic Step-Stress ALT Model for Complete Data Set
  • Assumptions
  • Test Procedure
  • Life Distribution under Step-Stress
  • Maximum Likelihood Estimation
  • Fisher Information Matrix
  • Optimality Criterion
• An Optimum Simple Log-Logistic Step-Stress ALT Model for Type-I Censored Data Set
  • Assumptions
  • Test Procedure
  • Life Distribution under Step-Stress
  • Maximum likelihood Estimation
  • Fisher Information Matrix
  • Optimum Plan
• Testing of Hypotheses
  • For Complete Data Set
  • For Time-Censored Data Set
• Numerical Examples
  • For Complete Data Set
  • For Time-Censored Data Set
• Sensitivity Analysis
• Summary

The following sections are included:

• Introduction
  • Notation
  • Truncated Logistic Distribution under Step-Stress PALT
• Optimum Time-Censored Step-Stress PALT Model
  • Assumptions
  • Test Procedure
  • Log-Likelihood Function
  • Fisher Information Matrix
  • Optimal Test Plan
• Optimum Failure-Censored Step-Stress PALT Model
  • Case I: Number of Failures Pre-Specified
    • Assumptions
    • Test Procedure
    • Log-Likelihood Function
    • Fisher Information Matrix
    • Optimal Test Plan
  • Case II: Proportion of Units Failing Before Censoring Pre-Specified
    • Assumptions
    • Test Procedure
    • Log-Likelihood Function
    • Fisher Information Matrix
    • Optimal Test Plan
  • Numerical Examples
    • Type-I Censoring
    • Optimal Plan
    • Simulated Data
    • MLEs of the Design Parameters
    • Confidence Intervals
    • Graphical Goodness of Fit
    • Coverage Probabilities
  • Type-II Censoring: Case I — Number of Failures Pre-Specified
    • Optimal Plan
    • Simulated Data
    • MLEs of the Design Parameters
    • Confidence Intervals
    • Graphical Goodness of Fit
    • Coverage Probabilities
  • Type-II Censoring: Case II — Proportion of Censoring Pre-Specified
    • Optimal Plan
    • Simulated Data
    • MLEs of the Design Parameters
    • Confidence Intervals
  • Sensitivity Analysis
    • Type-I Censoring and Type-II Censoring (Case: I — Number of Failures Pre-Specified)
    • Type-II Censoring (Case II: Proportion of Items Failing before Censoring Pre-Specified)
  • Comparative Study for Type-II Censoring (Case II — Proportion of Items Failing Before Censoring Pre-Specified)
  • Summary

The following sections are included:

• Introduction
  • Notation
  • Burr Type-XII Life Distribution
• Optimum Multi-Level Ramp-Stress ALT Plan Without Stress Upper Bound
  • Assumptions
  • Test Procedure
  • Life Distribution under Multi-Level Ramp-Test without Stress Upper Bound
  • Log-Likelihood Function
  • Parameter Estimation
  • Fisher Information Matrix
  • Asymptotic Variance of the MLE of Logarithm of Quantile at Design Stress
• Optimum Multi-Objective Ramp-stress ALT Plan with Stress Upper Bound
  • Assumptions
  • Test Procedure
  • Life Distribution under Ramp-Stress with Stress Upper Bound
  • Log-Likelihood Function
  • Parameter Estimation
  • Fisher Information Matrix
  • Asymptotic Variance of the MLE of Logarithm of Quantile at Design Stress
• Formulation of a Multi-Objective Optimization Problem
  • Optimization Problem For Multi-Level Ramp-Stress ALT Plan without Stress Upper Bound
  • Optimization Problem For Ramp-Stress ALT Plan with Stress Upper Bound
• Numerical Examples
  • Multi-Level Ramp-Stress ALT Plan without Stress Upper Bound
    • Optimal Plan
    • Simulated Data
    • MLEs of the Design Parameters
    • Confidence Intervals
  • Ramp-Stress ALT Plan with Stress Upper Bound
    • Optimal Plan
    • Simulated Data
    • MLEs of the Design Parameters
    • Confidence Intervals
  • Sensitivity Analysis
  • Comparative Study
    • Multi-Level Ramp-Stress ALT Plan without Stress Upper Bound
    • Ramp-Stress ALT Plan with Stress Upper Bound
  • Summary

The following sections are included:

• Introduction
  • Notation
  • Burr Type-XII Lifetime Distribution
• Optimum Multi-Objective Modified Step-Stress ALT Plan
  • Assumptions
  • Test Procedure
  • Life Distribution Under Modified Step-Stress
  • Log-Likelihood Function
  • Fisher Information Matrix
  • Asymptotic Variance of the MLEs of Logarithm of Quantile q at Design Stress
• Optimum Multi-Objective Modified Constant-Stress ALT Plan
  • Assumptions
  • Test Procedure
  • Life Distribution under Modified Constant-Stress
  • Log-Likelihood Function
  • Fisher Information Matrix
  • Asymptotic Variance of the MLEs of the Logarithm of Quantile q at Design Stress
• Formulation of a Multi-Objective Optimization Problem
  • Modified Step-Stress ALT Plan
  • Modified Constant-Stress ALT Plan
• Numerical Examples
  • Modified Step-Stress ALT Plan
    • Optimal Plan
    • Simulated Data
    • MLEs of the Design Parameters
    • Confidence Intervals
  • Modified Constant-Stress ALT Plan
    • Optimal Plan
    • Simulated Data
    • MLEs of the Design Parameters
    • Confidence Intervals
  • Sensitivity Analysis
  • Comparative Study
    • Modified Step-Stress ALT Plan
    • Modified Constant-Stress ALT Plan
  • Summary

The following sections are included:

• Introduction
  • Notation
• Assumptions
• Test Procedure
• Life Distribution Under Step-Stress with Competing Failure Modes
• Log-Likelihood
• Fisher Information Matrix
• Optimal Plan
• Testing of Hypotheses
• A Numerical Example
• Sensitivity Analysis
• Summary

The following sections are included:

• Introduction
• Product Control Metrics
• Types of LSPs (RASPs)
• LSPs (RASPs) by Attributes
  • Single Sampling RASP
  • Double Sampling Plan
  • Other LSPs By Attributes
• LSPs by Variables
  • Variable Sampling Plans for a Process Parameter
    • Single Sampling Plan for a Process Parameter
  • Single Sampling by Variables for Proportion Nonconforming
    • -method, k-method and M-method
  • Variable Repetitive Group Sampling Plan (VRGSP) for Proportion Nonconforming
  • Other VRASPs for Proportion Nonconforming
• LSPs Based on a Cost Model
  • Types of Warranty Policies
  • Warranty Cost Under a Free Replacement Policy
  • Warranty Cost Under a Pro-Rata Replacement Policy
  • Warranty Cost Under a Combination — Free and Pro-Rata Replacement Policy
  • Expected Total Cost for the Sampling Plan
  • Optimal LSP
• Optimum Accelerated Life Tests Sampling Plans (ALTSPs)
• Sensitivity Analysis
• Design of ALTSP and PALTSP Models: A Review
  • Constant-Stress ALTSP Models
  • Time-Varying Stress ALTSP Models
  • RASP With Acceleration Factor
  • Constant-Stress PALTSP Models
  • Time-Varying Stress PALTSP Models

The following sections are included:

• Introduction
  • Notation
• Assumptions
• Log-Logistic Life Distribution Under Ramp-Stress Test
  • Log-Lifetime Distribution
• Life Test Procedure
• Lot Acceptance Sampling Procedure
• Optimum Time-Censored Ramp-Stress ALTSP Under Variable SSP for Proportion Nonconforming
  • Log-Likelihood
  • Fisher Information Matrix
  • Asymptotic Variance of the Test Statistic
  • Operating Characteristic (OC) Function
  • Optimal Plans
• Optimum Time-Censored Ramp-Stress ALTSP Under Variable SSP for the Process Parameter Based on Median Lifetime
  • Log-Likelihood
  • Asymptotic Variance of the Test Statistic
  • OC Function
  • Optimal Plan
• Cost Structure
  • Cost Structure under Variable SSP for Proportion Nonconforming
  • Cost Structure under Variable SSP for the Process Parameter based on Median Lifetime
• Design of Optimal Sampling Plans
  • Design of Optimization Problem for Time-Censored Ramp-Stress ALTSP under Variable SSP for Proportion Nonconforming
  • Design of Optimization Problem for Time-Censored Ramp-Stress ALTSP under Variable SSP for the Process Parameter based on Median Lifetime
• Numerical Examples
  • Numerical Example on Time-Censored Ramp-Stress ALTSP under Variable SSP for Proportion Nonconforming
  • Numerical Example on Time-Censored Ramp-Stress under Variable SSP for the Process Parameter based on Median Lifetime
• Sensitivity Analyses
• Summary

The following sections are included:

• Introduction
  • Notation
• The Model
  • Assumptions
  • Burr Type-XII Life Distribution
  • Log-Lifetime Distribution
  • Test Procedure
  • Lot Acceptance Sampling Procedure
  • Life Distribution under TFR Model
• Parameter Estimation
• Fisher Information matrix
• Asymptotic Variance of Test Statistic
• Operating Characteristic (OC) Function
• Cost Consideration
  • Expected Warranty Cost Per Item
  • Expected Total Cost Per Lot
• Formulation of an Optimization Problem
  • Algorithm
• Numerical Example
• Sensitivity Analysis
• Summary

The following sections are included:

• Introduction
  • Notation
• Model
  • Assumptions
  • Test Procedure
  • Model Formulation
• Lot Acceptance Procedure
  • SSP
  • VRGSP
• Log-Likelihood
• Fisher Information matrix
• Operating Characteristic (OC) Function
• Formulation of an Optimization Problem
  • Bilevel Programming: VRGSP
  • Bilevel Programming: SSP
• Illustrative Example
• Sensitivity Analyses
• Summary

The following sections are included:

• Appendix A
  • Truncated Distributions and Order Statistics
    • Truncated Distributions
    • Order Statistics
  • Exponential Distribution as a Limiting Case of the Burr Type-XII Distribution
  • Weibull Distribution as a Limiting Case of the Burr Type-XII Distribution
  • Calculations of the Elements of the Fisher Information Matrix in 3.2.4
  • Calculations of Elements of the Fisher Information Matrix Given in 3.3.1.4
  • Calculations of the Elements of the Fisher Information Matrix Given in Section 3.3.2.4
  • Justification for the Use of Linear Cumulative Exposure Model in the Burr Type-XII Distribution in Section 4.2.3
  • Justification for the Use of Linear Cumulative Exposure Model in the Burr Type-XII Distribution in Section 4.3.3
  • Calculations of the Elements of the Fisher Information Matrix Given in Section 4.2.6
  • Calculations of the Elements of the Fisher Information Matrix in Section 4.3.6
  • Justification for the Use of Linear Cumulative
  • Exposure Model in the Burr Type-XII Distribution in Section 5.2.3
  • Calculations of the Elements of the Fisher Information Matrix in Section 5.2.5
  • Justification for the Use of Linear Cumulative Exposure Model in the Burr Type-XII Distribution in Section 5.3.3
  • Calculation of the Elements of the Fisher Information Matrix Given in Section 5.3.5
• Appendix B
  • Calculations of the Elements of the Fisher Information Matrix Given in Section 8.6.2
  • Calculations of the Asymptotic Variance of the Test Statistic Given in Section 8.6.3 for Time-Censored Case
  • Calculations of the Elements of the Fisher Information Matrix Given in Section 9.4
  • Calculation of the Asymptotic Variance of the Test Statistic Given in Section 9.5
  • Calculations of the Second Partial Derivatives and Elements of the Fisher Information Matrix Given in Section 10.5
  • Calculation of the Asymptotic Variance of the Estimated Value of Mean Lifetime, , Given in Sections 10.3.1 and 10.3.2
• Appendix C
  • Bilevel Programming
• Appendix D
  • Tampered Failure Rate (TFR) Model under Constant-Stress Setup

The following sections are included:

• REFERENCES
• INDEX