statistical methods in reliability testing - Statistics and Actuarial

65 Pages·2012·1.26 MB·English

STATISTICAL METHODS IN RELIABILITY TESTING
by
Alexey Antonitsin
Spct., Vologda State Dairy Academy, 2009
a project submitted in partial fulfillment
of the requirements for the degree of
Master of Science
in the
Department of Statistics and Actuarial Science
Faculty of Science
© Alexey Antonitsin 2012
SIMON FRASER UNIVERSITY
Spring 2012
All rights reserved. However, in accordance with the Copyright Act of
Canada, this work may be reproduced without authorization under the
conditions for Fair Dealing. Therefore, limited reproduction of this
work for the purposes of private study, research, criticism, review and
news reporting is likely to be in accordance with the law, particularly
if cited appropriately.
Approval
Name: Alexey Antonitsin
Degree: Master of Science
Title of Project: Statistical Methods in Reliability Testing
Examining Committee:
Chair: Richard Lockhart
Professor
Steven K. Thompson
Senior Supervisor
Professor
Carl Schwarz
Professor
Derek Bingham
Internal Examiner
Associate Professor
Date Approved: March 23, 2012
ii
Abstract
Inthisprojectweexaminedifferentsamplingplansaspartofreliabilitycontrolprocedures
inmanufacturing. Thesystematicsamplingconventionallyusedbymanycompaniescanbere-
placedbyaskip-lotsamplingplanwhichprovidesthesamelevelofprotectionagainstdefective
products while requiring fewer resources. In this project we investigate skip-lot sampling pro-
cedureanditsstatisticalproperties. Wewillalsocomparethemediumrankregressionmethod
of fitting Weibull distribution parameters with the maximum likelihood method. Finally, we
conduct a simulation study to empirically investigate the performance of the proposed method
and its properties. This project considers a wide range of reliability sampling procedures and
makes extensive use of simulation methods.
Keywords: Reliability testing, Skip-lot sampling, Weibull analysis.
iii
To my mother
iv
Acknowledgments
More than two years that I have spent at the Department of Statistics and Actuarial
Science at SFU stand particularly prominent in my life. I really enjoyed being a graduate
student at a department with such a great faculty with strong passion for Statistics, wonderful
teaching skills, and ability to create the right atmosphere for learning. I want to thank my
great classmates for providing a lot of help in understanding the material and for just being
great people to spend time with.
MyspecialthanksgotoStevenK.Thompsonforbeingmysupervisorandprovidinggreat
guidance throughout these two years. I am very grateful for all the support, ideas, and wise
advices Professor Thompson gave me during my Master’s Degree.
Finally I want to thank my family and friends for their help and encouragement. I am
especially grateful to Svetlana Smirnova for her assistance and inspiration.
v
Contents
Approval ii
Abstract iii
Dedication iv
Acknowledgments v
Contents vi
List of Tables viii
List of Figures ix
1 Introduction 1
2 Reliability Testing 3
3 Parameter Estimation 7
3.3 Estimating parameters of Weibull distribution . . . . . . . . . . . . . . . . . . . 7
3.3.1 Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3.2 Weibull Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . 11
3.3.3 Goodness of fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3.4 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Sampling units for reliability analysis 20
4.1 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Current Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 Proposed Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3.1 Skip-lot Sampling Plan with Double Sampling as the Reference Plan . . 23
vi
4.3.2 Power Series Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.4 Markov Chain Approach to Determine P . . . . . . . . . . . . . . . . . . . . . 30
a
5 Simulation 36
5.1 Frequency (f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Number of accepted lots that triggers skipping inspection (i) . . . . . . . . . . 38
5.3 Probability of lot acceptance (P) . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Conclusions 41
Appendix A Simulation Study for Estimating β and η 44
Appendix B Partial R Code for the Simulation Study 47
B.1 Main Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
B.2 CDF fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Bibliography 55
vii
List of Tables
2.1 Life Characteristics for two failure distributions . . . . . . . . . . . . . . . . . . 6
3.1 Types of failures corresponding to β values . . . . . . . . . . . . . . . . . . . . 10
3.2 Intermediate values obtained from an experiment set . . . . . . . . . . . . . . . 17
3.3 MRR and MLE estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Simulation study results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.5 Input set for analysis of differences between MLE and MRR methods . . . . . . 18
4.1 Example of inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Sequence of lot samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Distribution of Failure Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
viii
List of Figures
3.1 General and special case distributions . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Weibull curves for different values of β . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Weibull curves for different values of η . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Median rank regression method on Weibull paper. . . . . . . . . . . . . . . . . 15
4.1 Example of Operating Characteristic Curve: OR=9. . . . . . . . . . . . . . . . 21
4.2 Ideal OC Curve (defective acceptance proportion=0.4): OR=1 . . . . . . . . . 21
4.3 Algorithm of the current method . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4 Algorithm of the proposed method . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.1 OC curve for sampling plan (n = 20, n = 20, f = 1, f = 3, c = 1, c = 0,
1 2 1 2 1 2
i = 5) and (n = 20, n = 20, f = 2, f = 5, c = 1, c = 0, i = 5) . . . . . . . 37
1 2 1 2 1 2
5.2 OC curve for sampling plan (n = 20, n = 20, f = 1, f = 3, c = 1, c = 0,
1 2 1 2 1 2
i = 5) and (n = 20, n = 20, f = 1, f = 3, c = 1, c = 0, i = 15) . . . . . . . 38
1 2 1 2 1 2
5.3 OC curve for sampling plan (n = 20, n = 20, f = 1, f = 3, c = 1, c = 0,
1 2 1 2 1 2
i = 5) and (n = 40, n = 20, f = 1, f = 3, c = 1, c = 0, i = 5) . . . . . . . 39
1 2 1 2 1 2
5.4 OC curve for SkSP-2DSP sampling plan (n = 20, n = 20, f = 2, f = 4,
1 2 1 2
c = 1, c = 0, i = 5) and systematic sampling (f = 3, n = 20, c = 0) . . . . . . 40
1 2
A.1 Simulation study for estimating β using MLE . . . . . . . . . . . . . . . . . . . 44
A.2 Simulation study for estimating η using MLE . . . . . . . . . . . . . . . . . . . 45
A.3 Simulation study for estimating β using MRR . . . . . . . . . . . . . . . . . . . 45
A.4 Simulation study for estimating η using MRR . . . . . . . . . . . . . . . . . . . 46
ix
Chapter 1
Introduction
In recent years the popularity and usage of reliability tests and analysis have significantly
grown. Many companies outsource the production process of product components or even
the whole product to second party companies. In order to stay competitive on the market
these companies require procedures to control the quality and reliability of outsourced prod-
ucts. Even if the manufacturing line at the outsourcing company is automated, there are still
variations in the characteristics of outgoing items due to the material used, line operators,
transportation and other controllable and non-controllable factors.
The terms “reliability” and “quality” are often used interchangeably. In fact, these terms
have different meanings: reliability is concerned with the performance of a product through
some desired period (product lifetime), whereas quality is concerned with the performance of
a product at a particular point in time. Therefore, reliability tests are usually more complex
proceduresthanqualitytests. Nevertheless,qualitytestingandreliabilitytestingbothmeasure
the “goodness” of a product and product reliability depends on the initial product quality.
Most reliability tests are designed to answer one or both of the following questions:
1. Does the product comply with the reliability criteria set by the company?
2. Is one version of the product better than the other one?
If the product has different failure types and each tested item needs to be checked after
everytimeunitofthetest,thereliabilitytestingcouldbeveryexpensive. Therefore,companies
often try to optimize the cost of reliability. As a result of this, it is common practice to test
only a sample of the produced items and design an efficient test procedure.
This thesis is specially focused on the sampling procedure used to select items from a
population of produced items. The theoretical and simulation study results show that the
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