![Important Notations
c(Tin) expected cost over each inspection cycle
c(Tini) expected cost over ith inspection cycle
C(Tin) total costs resulting from chosen maintenance policy
ccs clean-up cost connected with cleaning up any spillage that may result in a
failure
cin cost of inspection action performance
cp cost of preventive replacement of a unit
cr cost of failed unit replacement
din time of single inspection action performance
dp the time of preventive replacement of a system
dr time of corrective replacement of a system (after a failure)
E[x] expected value of random variable x
Ed(Tin) expected downtime in an inspection cycle of length Tin
F(t) probability distribution function of system/unit lifetime; F(t)¼1F(t)
Fh(h) probability distribution function of system delay time
fh(h) probability density function of system delay time
Fhi(h) probability distribution function of ith element delay time
fhi(h) probability density function of ith element delay time
Gh(t) probability distribution function of the initial time u, which elapses from
the beginning of operation by “as good as new” elements of a system until
the moment of first symptoms of failure occurrence
gh(t) probability density function of the initial time u, which elapses from the
beginning of operation by “as good as new” elements of a system until
the moment of first symptoms of failure occurrence
Ghi(t) probability distribution function of initial time u of ith element in a system
ghi(t) probability density function of initial time u of ith element in a system
h delay time of a defect, denoting the period between the moment of appear-
ance of the first symptoms of potential failure and the moment of an
object’s failure occurrence
ku constant rate for fault’s arrival in a technical system for any
inspection period
Mcs measure of possible impact of a failure of a piece of equipment on an
environment
Menv(Tin) environmental model of a technical object
m number of failures until element replacement
n number of elements in a system
nk minimum number of operating components to make a system function
(performance in nk-out-of-n reliability structure)
Nrin(ti–1
in, ti
in) expected number of failures over inspection interval (ti–1
in, ti
in)
pcs probability of a failure resulting in a spillage requiring clean up
pin probability of imperfect inspection occurrence
PuF(Tin) probability of a fault arising as a breakdown in a system
pw probability that, during system inspection, performance symptoms of
forthcoming failures (if they occur in a system) are identified
R(t) reliability function of a system
2 Advances in System Reliability Engineering](https://image.slidesharecdn.com/32707-250302053443-8e06b6c9/85/Advances-in-system-reliability-engineering-Davim-15-320.jpg)
![RTin(t) component reliability at time t
RTini
c
a cubic approximation of average reliability over an ith inspection
RTini
L
a linear approximation of average reliability over an ith inspection
rmr repair cost rate
rTin
(i)
(t) reliability function for a component at time t
r
˙(k)
Tin(kTin) left-hand derivative of reliability function at t¼kTin
tf random moment of failure occurrence in a system
Tin time between inspection actions’ performance
Tini time of ith inspection cycle performance (between two consecutive inspec-
tion actions’ performance)
TM(Tin) total expected length of a renewal cycle
TM(Tini) expected length of ith inspection cycle
TOP random variable denoting system’s lifetime
TOP(Tin) expected length of element/system’s lifetime
tu moment of the first symptoms of potential failure occurrence
u initial time of a defect
δh standard deviation for normal distribution of a delay time h
δr standard deviation for normal distribution of a time to failure
λh(t) intensity function of a system delay time
λu(u) random rate for fault’s arrival in a technical system for any inspection period
1 INTRODUCTION
Many real-life systems may display some symptoms of forthcoming failure.
One example may be a production process, which may start producing
defective items after some random amount of time. If the situation is not
corrected, product quality gradually deteriorates to a level where it is self-
evident to an operator that the system has failed. By inspecting the product
quality at some intervals, the operator may be able to reduce the cost
incurred with the system lifetime [1]. These types of components may ben-
efit from an inspection policy whereby a component is inspected for a defect
and consequently replaced at inspection to prevent failure [2]. Recent
reviews on inspection maintenance modeling issues are presented in the lit-
erature [3–9].
The basic inspection models are extended by using a technique called
delay-time analysis (DTA), which was developed for modeling the conse-
quences of an inspection policy for any system [10]. This approach was first
described and investigated by Christer et al. [10–16].
This maintenance concept is based on the assumption that before a sys-
tem breaks down, there will be some signs of its reduced performance. The
time between the first identification of abnormalities (called initial point)
3
Delay-Time-Based Maintenance Modeling for Technical Systems—Theory and Practice](https://image.slidesharecdn.com/32707-250302053443-8e06b6c9/85/Advances-in-system-reliability-engineering-Davim-16-320.jpg)
![and the actual failure time (called failure point) is called a delay time and
determines the best opportunity to carry out maintenance or an inspection
(Fig. 1). As a result, the delay-time concept defines a two-stage process for
three-stated systems (being either in upstate, partially upstate, or in down-
state). For more information, the author recommends reading Wang
et al. [17].
The inspection schemes in delay-time models may be periodic or based
on condition-based maintenance implementation [18]. In this chapter, the
author focuses on periodic inspection maintenance modeling issues. More
information about condition-based delay-time models may be found in
the literature [19–25].
The interest in DT modeling issues has increased significantly in recent
years. Literature reviews, in which delay-time models are investigated along
with other preventive maintenance models, are given in several studies
[26–32]. The states of art works, dedicated to DT modeling, were mostly
developed in the 1990s. One of the first literature reviews was given by
Christer et al. [19], where the authors discussed a basic DT model for a
single-unit case taking into account model parameters estimation problems.
Later, the state of art was updated in Christer et al. [33]. In this work, math-
ematical methods for updating delay-time models of industrial inspection
practice are proposed. They presented a linear delay-time update method
and model parameter variation. Moreover, the authors also provide a com-
bination of these methods in the maintenance area and defines the criteria for
choosing an updating method.
Christer later presented a basic delay-time model [34] and discussed the
main development directions, including perfect/nonperfect inspection
Initial time — during this
time interval the defect
cannot be visible
Time point when a fault
could be first identified
tu tf t
Time point when a component
fails if no maintenance
intervention during time h is
conducted
Delay time — during this
time interval the defect
can be identified by an
inspection
u h
0
Fig. 1 Time-delay-modeling concept.
4 Advances in System Reliability Engineering](https://image.slidesharecdn.com/32707-250302053443-8e06b6c9/85/Advances-in-system-reliability-engineering-Davim-17-320.jpg)
![cases. There have been simple DT models investigated for a repairable com-
ponent and for a complex plant. Moreover, the authors focused on the prob-
lems of a parameter estimation process and DT models’ implementation
possibilities. The work ends with a discussion of further developments in
delay-time modeling. The presented state of art was further extended by
Christer [35]. In this work, the author focuses on investigation of industrial
maintenance problems and presents basic maintenance models for a repair-
able component and a complex plant. The main extensions of the basic
models include, for example, nonperfect inspection case and nonsteady-state
conditions.
One of works that provides recent advances in delay-time-based main-
tenance modeling is given by Wang [36]. In this work, the author focuses on
the main development directions in delay-time-based models. As a result, he
investigates the delay-time models for optimized inspection intervals (distin-
guishing, for example, imperfect inspection and maintenance issues or mul-
tiple inspection intervals problem) and DT modeling application areas
(including e.g., the problem of spare part inventory and maintenance
modeling or maintenance outsourcing issues). The author also defines the
main directions for further research in the analyzed maintenance area.
For more information, the author recommends further reading of the
literature [2, 6, 37, 38]. Some recent developments in this area are also avail-
able for further reading [17, 21, 39].
Following these considerations, in Fig. 2, the main classification of exist-
ing delay-time models is presented. The proposed classification divides the
known models into two main groups of inspection strategies: single- and
multiunit systems.
To sum up, the author summarizes and reviews the existing literature on
periodic delay-time maintenance modeling and clusters the existing works
into several fields, mostly based on the classification on single- and multiunit
models.
DT models for multiunit
system
Models for system
with two-stage
failure process
Models for system with
three-stage failure
process
Models for complex
system
Delay time (DT) models for technical system
DT models for single-unit
system
Models for multiunit
system in nonseries
reliability structure
Fig. 2 Classification scheme of delay-time models (periodic inspection).
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![2.1.1 DTM for a Single-Defect Case
One of the first developed DT models for single-unit systems is presented by
Christer [11]. In his work, the author develops the reliability model for a
single component subject to one type of inspectable defect, which will sub-
sequently lead to a failure. In the presented model, under the policy of
inspecting every Tin time unit, the author develops a function of component
reliability at time t RTin(t).
Taking into account the following main assumptions:
• a fault may arise randomly at time u since new with pdf gh(u),
• a perfect maintenance policy,
• the time required for inspection is negligible,
• variables u and h are assumed to be independent,
• a infinite horizon case,
the required reliability function RTin(t) is given by [11]:
RTin t
ð Þ ¼ r
i
ð Þ
Tin t
ð Þ, where i1
ð ÞTin t iTin (1)
where i is a positive integer and rTin
(i)
(t) is a reliability function for the com-
ponent at time t, given by the formula [11]:
r
i
ð Þ
Tin t
ð Þ ¼
X
i1
ð Þ
j¼1
ðjTin
u¼ j1
ð ÞTin
gh u
ð ÞMh jTin u
ð Þdu
!
r
ij
ð Þ
Tin t jTin
ð Þ
#
(
+
ð∞
u¼t
gh u
ð Þdu +
ðt
u¼ i1
ð ÞTin
gh u
ð ÞM t u
ð Þdu
)
,
where i1
ð ÞTin t iTin (2)
Models for systems with a two-stage
failure process
Systems with a
single failure
mode
Systems with a
multiple
failure mode
Approximation
modeling
case
Preparedness
systems
maintenance case
Systems with
postponed
maintenance
Fig. 3 Classification scheme of delay-time models for systems with a two-stage failure
process.
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![where Mh(x) is defined as [11]:
Mh x
ð Þ ¼
ð∞
x
fh u
ð Þdu (3)
In the absence of an inspection procedure (Tin ¼∞), the reliability func-
tion is given by the following formula [11]:
R Tin
ð Þ ¼ R∞ t
ð Þ ¼ r
1
ð Þ
Tin t
ð Þ ¼ 1
ðt
u¼0
Fh t u
ð Þgh u
ð Þdu (4)
For such a model, a numerical example is provided.
DT models for single-unit systems are later investigated in works by
Redmond and Wang [2, 17]. According to these works, there is a possibility
to define the C.d.f. of time to failure, F(x), as the convolution of u and h such
that u+hx [2, 17]:
F x
ð Þ ¼
ðx
u¼0
gh u
ð ÞFh xu
ð Þdu (5)
and the reliability function, R(x)¼1F(x).
Taking into account the following additional assumptions and notation
[2, 17]:
• the system is renewed at either a failure repair or at a repair done at an
inspection if a defect is identified,
• after either a failure renewal or inspection renewal, the inspection pro-
cess restarts,
• maintenance actions restore maintained components to good as new
condition,
• failures of the system are identified immediately, and repairs or replace-
ments are made as soon as possible,
• perfect inspections case, that is, every defect is assumed to be identified
during inspection action performance,
the expected cost over each cycle, c(Tin), may be defined by the following
equation [2, 17]:
c Tin
ð Þ ¼ crF Tin
ð Þ + cp + cin
ðTin
u¼0
gh u
ð Þ 1Fh Tin u
ð Þ
ð Þdu + cin 1Gh Tin
ð Þ
ð Þ (6)
The expected downtime Ed(Tin) of an inspection cycle of length Tin is
given as [34]:
Ed Tin
ð Þ ¼ dr dp din
F Tin
ð Þ + dpGh Tin
ð Þ + din (7)
8 Advances in System Reliability Engineering](https://image.slidesharecdn.com/32707-250302053443-8e06b6c9/85/Advances-in-system-reliability-engineering-Davim-21-320.jpg)
![Assuming instantaneous inspection and replacement times, the expected
cycle length, TM(Tin), is given by [2, 34]:
TM Tin
ð Þ ¼
ðTin
x¼0
xF0
x
ð Þdx + TinR Tin
ð Þ (8)
Examples of such model implementations are given in the literature [2,
17], as well as an analysis of the problem of model parameter estimation [17].
This model is later extended by Jodejko-Pietruczuk et al. [40, 41]. In one
study, the authors focus on the development of a mathematical model for a
technical object maintenance, which extends the approach used thus far and
allows analyzing the long-term operation time period of a single-unit system
[40]. The model gives the possibility to find a constant time period between
the inspection actions’ performance that is optimal due to minimal mainte-
nance costs or maximal availability criterion satisfaction. The solution is
based on Block Inspection policy implementation. The authors present ana-
lytical and Monte Carlo simulation models with comprehensive sensitivity
analysis performance. This model is an extension of a preliminary analytical
maintenance model [42].
Another extension is presented in work by Jodejko-Pietruczuk et al.
[41], where the assumption of imperfect inspections is introduced. In this
work, it is assumed that the system inspection actions are imperfect, which
means that the defect occurrence in the system is identified with the given
probability pw. In such a situation in the analyzed system, the following
maintenance actions may occur:
• preventive replacement during inspection action performance, provided
that the defect has occurred and been identified with the probability pw,
• inspection action performance without preventive replacement, pro-
vided that the defect has not occurred nor been identified with proba-
bility (1pw).
Based on the main assumptions given earlier [40], the long-term expected
maintenance costs per unit time function is defined.
2.1.2 DTM for a Multidefect Case
One of the extensions in delay-time modeling regards consideration of more
than one type of inspectable and independent defect occurrence in a system.
The first model for single-unit systems with n different types of inspectable
defects is developed by Christer [11] and later extended by Wang [43].
Wang [43] investigated the production process subjected to two types of
deterioration. The delay-time-based inspection model was developed for
9
Delay-Time-Based Maintenance Modeling for Technical Systems—Theory and Practice](https://image.slidesharecdn.com/32707-250302053443-8e06b6c9/85/Advances-in-system-reliability-engineering-Davim-22-320.jpg)
Advances in system reliability engineering Davim
- 1. Visit ebookmass.com to download the full version and explore more ebook or textbook Advances in system reliability engineering Davim _____ Click the link below to download _____ https://ebookmass.com/product/advances-in-system- reliability-engineering-davim/ Explore and download more ebook or textbook at ebookmass.com
- 4. ADVANCES IN SYSTEM RELIABILITY ENGINEERING Edited by MANGEY RAM J. PAULO DAVIM
- 5. Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom © 2019 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-815906-4 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Matthew Deans Acquisition Editor: Brian Guerin Editorial Project Manager: Thomas Van Der Ploeg Production Project Manager: Sruthi Satheesh Cover Designer: Harris Greg Typeset by SPi Global, India
- 6. CONTRIBUTORS Mohini Agrawal School of Business, Galgotias University, Greater Noida, India Adarsh Anand Department of Operational Research, University of Delhi, New Delhi, India Alessandro Barbiero Department of Economics, Management and Quantitative Methods, Università degli Studi di Milano, Milan, Italy Navneet Bhatt Department of Operational Research, University of Delhi, New Delhi, India F. De Caro Department of Engineering, University of Sannio, Benevento, Italy Lirong Cui School of Management & Economics, Beijing Institute of Technology, Beijing, China Ant^ onio Carlos Lopes da Costa Centro de Desenvolvimento da Tecnologia Nuclear—CDTN, Belo Horizonte, Brazil Vanderley de Vasconcelos Centro de Desenvolvimento da Tecnologia Nuclear—CDTN, Belo Horizonte, Brazil Chen Fang School of Management & Economics, Beijing Institute of Technology, Beijing, China Soufiane Gasmi Optimization, Modeling and Decision Support (OMAD), University of Tunis—Tunis National Higher School of Engineering, Tunis, Tunisia Miroslav Kvassay Department of Informatics, University of Zilina, Zilina, Slovakia Dongjin Lee School of Computing, Informatics, and Decision Systems Engineering, Arizona State University, Tempe, AZ, United States of America Preeti Malik Department of Computer Science and Engineering, Graphic Era Deemed to be University, Dehradun, India Monika Manglik Department of Mathematics, University of Petroleum & Energy Studies, Dehradun, India Lata Nautiyal Department of Computer Science and Engineering, Graphic Era Deemed to be University, Dehradun, India ix
- 7. Rong Pan School of Computing, Informatics, and Decision Systems Engineering, Arizona State University, Tempe, AZ, United States of America Brian A. Polin Jerusalem College of Technology, Jerusalem, Israel Ludmila Pustylnik Department of Mechanical Engineering, Afeka Tel-Aviv Academic College of Engineering, Tel Aviv, Israel Jan Rabcan Department of Informatics, University of Zilina, Zilina, Slovakia Mangey Ram Department of Mathematics, Computer Science & Engineering, Graphic Era University, Dehradun, India Amanda Laureano Raso Centro de Desenvolvimento da Tecnologia Nuclear—CDTN, Belo Horizonte, Brazil Alexander Rotshtein Department of Industrial Engineering, Jerusalem College of Technology, Jerusalem, Israel Patrik Rusnak Department of Informatics, University of Zilina, Zilina, Slovakia Luis Mejia Sanchez Cummins Inc., Columbus, IN, United States of America Wellington Antonio Soares Centro de Desenvolvimento da Tecnologia Nuclear—CDTN, Belo Horizonte, Brazil A. Vaccaro Department of Engineering, University of Sannio, Benevento, Italy D. Villacci Department of Engineering, University of Sannio, Benevento, Italy Sylwia Werbi nska-Wojciechowska Department of Operation and Maintenance of Logistic Systems, Transportation Systems and Hydraulic Systems, Wroclaw University of Science and Technology, Wroclaw, Poland Petek Yontay Intel, Inc., Chandler, AZ, United States of America x Contributors
- 8. EDITORS BIOGRAPHY Dr. Mangey Ram received a Ph.D. degree, majoring in Mathematics and minoring in Computer Science, from G. B. Pant University of Agriculture and Technology, Pantnagar, India. He has been a faculty member for around 10 years and has taught several core courses in pure and applied mathematics at undergraduate, postgraduate, and doctorate levels. He is currently a pro- fessor at Graphic Era Deemed to be University, Dehradun, India. Before joining Graphic Era, he was a deputy manager (probationary officer) with Syndicate Bank for a short period. He is editor-in-chief of International Jour- nal of Mathematical, Engineering and Management Sciences and the guest editor and member of the editorial board of various journals. He is a regular reviewer for international journals, including IEEE, Elsevier, Springer, Emerald, John Wiley, Taylor Francis, and many other publishers. He has published 131 research publications in IEEE, Taylor Francis, Springer, Elsevier, Emerald, World Scientific, and many other national and interna- tional journals of repute, and also presented his works at national and inter- national conferences. His fields of research are reliability theory and applied mathematics. He is a senior member of the IEEE, life member of Opera- tional Research Society of India, Society for Reliability Engineering, Qual- ity and Operations Management in India, Indian Society of Industrial and Applied Mathematics, member of International Association of Engineers in Hong Kong, and Emerald Literati Network in the UK. He has been a member of the organizing committee of a number of international and national conferences, seminars, and workshops. He has been conferred with “Young Scientist Award” by the Uttarakhand State Council for Science and Technology, Dehradun, in 2009. He has been awarded the “Best Faculty Award” in 2011 and recently Research Excellence Award in 2015 for his significant contribution in academics and research at Graphic Era. xi
- 9. Dr. J. Paulo Davim received a Ph.D. degree in Mechanical Engineering in 1997, an M.Sc. degree in Mechanical Engineering (materials and manufacturing processes) in 1991, a Mechanical Engineering degree (5 years) in 1986 from the University of Porto (FEUP), the Aggregate title (Full Habilitation) from the University of Coimbra in 2005, and a D.Sc. from London Metropolitan University in 2013. He is Eur Ing by FEANI-Brussels and senior chartered engineer by the Portuguese Institution of Engineers with an MBA and Specialist title in Engineering and Industrial Management. Currently, he is Professor at the Department of Mechanical Engineering of the University of Aveiro, Portugal. He has more than 30 years of teaching and research experience in Manufacturing, Materials and Mechanical Industrial Engineering with special emphasis in Machining Tribology. He also has an interest in Management, Engineering Education, and Higher Education for Sustainability. He has guided large numbers of postdoctorate, Ph.D., and Masters students as well as coordinated and participated in several research projects. He has received several scientific awards. He has worked as evaluator of projects for international research agencies as well as examiner of Ph.D. theses for many universities. He is the editor-in-chief of several international journals, guest editor of journals, book editor, book series edi- tor, and scientific advisory for many international journals and conferences. Presently, he is an editorial board member of 25 international journals and acts as reviewer for more than 80 prestigious Web of Science journals. In addition, he has also published as editor (and coeditor) more than 100 books and as author (and coauthor) more than 10 books, 80 book chapters, and 400 articles in journals and conferences (more than 200 articles in journals indexed in Web of Science core collection/h-index 45+/6000+ citations and SCOPUS/h-index 52+/8000+ citations). xii Editors Biography
- 10. PREFACE Advances in modeling and simulation is arguably one of the most multidi- mensional topics that one can face in system reliability engineering today. This rapid development also creates many opportunities and challenges for both industrialists and academics, and has completely changed the global design and system engineering environment. More modeling tasks can now be undertaken within a computer environment using simulation and virtual reality technologies. Through this book, Recent Advances in System Reliability Engineering, engineers and academicians will gain the knowledge to help them in system reliability engineering. This book is meant for those who are pursuing reliability engineering as a subject of study. The material is intended for an audience at the level of postgraduate or senior undergraduate students. That’s why system reliability engineering is now a well-recognized and rapidly developing branch of engineering. Topics of focus include: • Delay-time-based maintenance modeling for technical systems • Probabilistic and deterministic safety analysis • Reliability-game theory • Integrating reliability models and adaptive algorithms for wind power forecasting • Time-dependent analysis of series-parallel multistate systems using struc- ture function and Markov processes • Modeling-correlated counts in reliability engineering • Statistical inference of an imperfect repair model with uniform distrib- uted repair degrees • Method of fuzzy perfectness in human reliability analysis; selection of performance conditions • System reliability assessment through Bayesian network modeling • Multistate multifailures system analysis with reworking strategy and imperfect fault coverage • Software patch scheduling policy incorporating functional safety standards • Tools and techniques in software reliability modeling Throughthisbook,theundergraduateandpostgraduatestudentsofengineer- ing, engineers, research scientists, and academicians will gain the knowledge xiii
- 11. to help them in their reliability courses. Thisbookis meant forthosewho take reliability and safety as a subject of study. The editors acknowledge Elsevier for this opportunity and for their professional support. Finally, we would like to thank all the chapter authors for their availability for this work. Mangey Ram Dehradun, India J. Paulo Davim Aveiro, Portugal xiv Preface
- 12. Visit https://ebookmass.com today to explore a vast collection of ebooks across various genres, available in popular formats like PDF, EPUB, and MOBI, fully compatible with all devices. Enjoy a seamless reading experience and effortlessly download high- quality materials in just a few simple steps. Plus, don’t miss out on exciting offers that let you access a wealth of knowledge at the best prices!
- 13. ACKNOWLEDGMENTS The editors acknowledge Elsevier and the editorial team for their adequate and professional support during the preparation of this book. Also, we would like to acknowledge all of the chapter authors and reviewers for their avail- ability for work on this book project. Mangey Ram Graphic Era Deemed to University, Dehradun, India J. Paulo Davim University of Aveiro, Aveiro, Portugal xv
- 14. CHAPTER 1 Delay-Time-Based Maintenance Modeling for Technical Systems— Theory and Practice Sylwia Werbi nska-Wojciechowska Department of Operation and Maintenance of Logistic Systems, Transportation Systems and Hydraulic Systems, Wroclaw University of Science and Technology, Wroclaw, Poland Abstract This chapter will be a valuable resource for understanding the latest developments in maintenance modeling issues in the field of delay-time approach implementation. This chapter presents the literature review on delay-time modeling for single- and multiunit systems. The maintenance models for single-unit systems assume two-stage or three- stage failure processes. In the case of complex systems, the discussed problems are with regard to models’ parameters, estimation issues, case studies analysis, or hybrid model- ing approach implementation. The maintenance models for multiunit systems examine the known models for nonseries systems. A case study for maintenance modeling for multiunit systems based on a delay-time approach is provided. The directions for further research work are defined. Keywords: Delay-time concept, Multiunit systems, Literature review, Maintenance modeling, Optimization Abbreviations AIC Akaike information criterion BI Block Inspection CM corrective maintenance DT delay time DTA delay-time approach DTM delay-time models ELECTRE elimination and choice expressing the reality HPP homogeneous Poisson process MAUT multiattribute utility theory MC Monte Carlo MLE maximum likelihood estimation MRT mean repair time MTBF mean time between failures NHPP nonhomogeneous Poisson process PAR proportional age reduction PM preventive maintenance PROMETHEE preference ranking organization method for enrichment evaluations 1 Advances in System Reliability Engineering © 2019 Elsevier Inc. https://doi.org/10.1016/B978-0-12-815906-4.00001-4 All rights reserved.
- 15. Important Notations c(Tin) expected cost over each inspection cycle c(Tini) expected cost over ith inspection cycle C(Tin) total costs resulting from chosen maintenance policy ccs clean-up cost connected with cleaning up any spillage that may result in a failure cin cost of inspection action performance cp cost of preventive replacement of a unit cr cost of failed unit replacement din time of single inspection action performance dp the time of preventive replacement of a system dr time of corrective replacement of a system (after a failure) E[x] expected value of random variable x Ed(Tin) expected downtime in an inspection cycle of length Tin F(t) probability distribution function of system/unit lifetime; F(t)¼1F(t) Fh(h) probability distribution function of system delay time fh(h) probability density function of system delay time Fhi(h) probability distribution function of ith element delay time fhi(h) probability density function of ith element delay time Gh(t) probability distribution function of the initial time u, which elapses from the beginning of operation by “as good as new” elements of a system until the moment of first symptoms of failure occurrence gh(t) probability density function of the initial time u, which elapses from the beginning of operation by “as good as new” elements of a system until the moment of first symptoms of failure occurrence Ghi(t) probability distribution function of initial time u of ith element in a system ghi(t) probability density function of initial time u of ith element in a system h delay time of a defect, denoting the period between the moment of appear- ance of the first symptoms of potential failure and the moment of an object’s failure occurrence ku constant rate for fault’s arrival in a technical system for any inspection period Mcs measure of possible impact of a failure of a piece of equipment on an environment Menv(Tin) environmental model of a technical object m number of failures until element replacement n number of elements in a system nk minimum number of operating components to make a system function (performance in nk-out-of-n reliability structure) Nrin(ti–1 in, ti in) expected number of failures over inspection interval (ti–1 in, ti in) pcs probability of a failure resulting in a spillage requiring clean up pin probability of imperfect inspection occurrence PuF(Tin) probability of a fault arising as a breakdown in a system pw probability that, during system inspection, performance symptoms of forthcoming failures (if they occur in a system) are identified R(t) reliability function of a system 2 Advances in System Reliability Engineering
- 16. RTin(t) component reliability at time t RTini c a cubic approximation of average reliability over an ith inspection RTini L a linear approximation of average reliability over an ith inspection rmr repair cost rate rTin (i) (t) reliability function for a component at time t r ˙(k) Tin(kTin) left-hand derivative of reliability function at t¼kTin tf random moment of failure occurrence in a system Tin time between inspection actions’ performance Tini time of ith inspection cycle performance (between two consecutive inspec- tion actions’ performance) TM(Tin) total expected length of a renewal cycle TM(Tini) expected length of ith inspection cycle TOP random variable denoting system’s lifetime TOP(Tin) expected length of element/system’s lifetime tu moment of the first symptoms of potential failure occurrence u initial time of a defect δh standard deviation for normal distribution of a delay time h δr standard deviation for normal distribution of a time to failure λh(t) intensity function of a system delay time λu(u) random rate for fault’s arrival in a technical system for any inspection period 1 INTRODUCTION Many real-life systems may display some symptoms of forthcoming failure. One example may be a production process, which may start producing defective items after some random amount of time. If the situation is not corrected, product quality gradually deteriorates to a level where it is self- evident to an operator that the system has failed. By inspecting the product quality at some intervals, the operator may be able to reduce the cost incurred with the system lifetime [1]. These types of components may ben- efit from an inspection policy whereby a component is inspected for a defect and consequently replaced at inspection to prevent failure [2]. Recent reviews on inspection maintenance modeling issues are presented in the lit- erature [3–9]. The basic inspection models are extended by using a technique called delay-time analysis (DTA), which was developed for modeling the conse- quences of an inspection policy for any system [10]. This approach was first described and investigated by Christer et al. [10–16]. This maintenance concept is based on the assumption that before a sys- tem breaks down, there will be some signs of its reduced performance. The time between the first identification of abnormalities (called initial point) 3 Delay-Time-Based Maintenance Modeling for Technical Systems—Theory and Practice
- 17. and the actual failure time (called failure point) is called a delay time and determines the best opportunity to carry out maintenance or an inspection (Fig. 1). As a result, the delay-time concept defines a two-stage process for three-stated systems (being either in upstate, partially upstate, or in down- state). For more information, the author recommends reading Wang et al. [17]. The inspection schemes in delay-time models may be periodic or based on condition-based maintenance implementation [18]. In this chapter, the author focuses on periodic inspection maintenance modeling issues. More information about condition-based delay-time models may be found in the literature [19–25]. The interest in DT modeling issues has increased significantly in recent years. Literature reviews, in which delay-time models are investigated along with other preventive maintenance models, are given in several studies [26–32]. The states of art works, dedicated to DT modeling, were mostly developed in the 1990s. One of the first literature reviews was given by Christer et al. [19], where the authors discussed a basic DT model for a single-unit case taking into account model parameters estimation problems. Later, the state of art was updated in Christer et al. [33]. In this work, math- ematical methods for updating delay-time models of industrial inspection practice are proposed. They presented a linear delay-time update method and model parameter variation. Moreover, the authors also provide a com- bination of these methods in the maintenance area and defines the criteria for choosing an updating method. Christer later presented a basic delay-time model [34] and discussed the main development directions, including perfect/nonperfect inspection Initial time — during this time interval the defect cannot be visible Time point when a fault could be first identified tu tf t Time point when a component fails if no maintenance intervention during time h is conducted Delay time — during this time interval the defect can be identified by an inspection u h 0 Fig. 1 Time-delay-modeling concept. 4 Advances in System Reliability Engineering
- 18. cases. There have been simple DT models investigated for a repairable com- ponent and for a complex plant. Moreover, the authors focused on the prob- lems of a parameter estimation process and DT models’ implementation possibilities. The work ends with a discussion of further developments in delay-time modeling. The presented state of art was further extended by Christer [35]. In this work, the author focuses on investigation of industrial maintenance problems and presents basic maintenance models for a repair- able component and a complex plant. The main extensions of the basic models include, for example, nonperfect inspection case and nonsteady-state conditions. One of works that provides recent advances in delay-time-based main- tenance modeling is given by Wang [36]. In this work, the author focuses on the main development directions in delay-time-based models. As a result, he investigates the delay-time models for optimized inspection intervals (distin- guishing, for example, imperfect inspection and maintenance issues or mul- tiple inspection intervals problem) and DT modeling application areas (including e.g., the problem of spare part inventory and maintenance modeling or maintenance outsourcing issues). The author also defines the main directions for further research in the analyzed maintenance area. For more information, the author recommends further reading of the literature [2, 6, 37, 38]. Some recent developments in this area are also avail- able for further reading [17, 21, 39]. Following these considerations, in Fig. 2, the main classification of exist- ing delay-time models is presented. The proposed classification divides the known models into two main groups of inspection strategies: single- and multiunit systems. To sum up, the author summarizes and reviews the existing literature on periodic delay-time maintenance modeling and clusters the existing works into several fields, mostly based on the classification on single- and multiunit models. DT models for multiunit system Models for system with two-stage failure process Models for system with three-stage failure process Models for complex system Delay time (DT) models for technical system DT models for single-unit system Models for multiunit system in nonseries reliability structure Fig. 2 Classification scheme of delay-time models (periodic inspection). 5 Delay-Time-Based Maintenance Modeling for Technical Systems—Theory and Practice
- 19. The articles referred to in this study were found using Google Scholar as a search engine and ScienceDirect, JStor, SpringerLink, SAGEJournals, and Taylor Francis as online databases. The author primarily searched the rel- evant literature based on keywords, abstracts, and titles. The following main terms and/or a combination of them were used for searching the literature: delay time maintenance or inspection maintenance. As a result, 130 papers from 1982 to 2018 are reviewed in this study. The applicability of the given maintenance approach is presented in a case study. The case study is developed based on historical data of mainte- nance and operation processes of four production plants (injection molding machines) operated by an international company located in Poland. These production plants have defined inspection policies based on ser- vice regulations defined by a producer. However, a company finds that some developments in this area may be useful to make further decisions. Following this, this chapter is organized as follows. Sections 2 and 3 examine various types of delay-time modeling approaches for single- and multiunit systems, which are the most widely known in the literature. In Section 4, a case study is proposed to investigate the optimal inspection interval for a four-unit system performing in a series reliability structure. Section 5 concludes the chapter and highlights future research in DT modeling. 2 DELAY-TIME MODELS FOR SINGLE-UNIT SYSTEMS First, there are delay-time maintenance models for one-unit stochastically failing or deteriorating system that were investigated in which the system’s defective state can be detected by actual inspection. In this area, the known models have been explored in two main dimensions: optimization of inspec- tion policy for systems with a two-stage failure process and for systems with a three-stage failure process. Moreover, some research works can be defined that are aimed at investigation and development of special problems (e.g., case studies, comparative studies). 2.1 Delay-Time Models for Single-Unit Systems With a Two-Stage Failure Process The basic delay-time concept considers that the degenerative process of a sys- tem canbedividedintotwostages:anormalworkingstageandadelayeddefect stage. For this modeling case, the main developments are given in Fig. 3. 6 Advances in System Reliability Engineering
- 20. 2.1.1 DTM for a Single-Defect Case One of the first developed DT models for single-unit systems is presented by Christer [11]. In his work, the author develops the reliability model for a single component subject to one type of inspectable defect, which will sub- sequently lead to a failure. In the presented model, under the policy of inspecting every Tin time unit, the author develops a function of component reliability at time t RTin(t). Taking into account the following main assumptions: • a fault may arise randomly at time u since new with pdf gh(u), • a perfect maintenance policy, • the time required for inspection is negligible, • variables u and h are assumed to be independent, • a infinite horizon case, the required reliability function RTin(t) is given by [11]: RTin t ð Þ ¼ r i ð Þ Tin t ð Þ, where i1 ð ÞTin t iTin (1) where i is a positive integer and rTin (i) (t) is a reliability function for the com- ponent at time t, given by the formula [11]: r i ð Þ Tin t ð Þ ¼ X i1 ð Þ j¼1 ðjTin u¼ j1 ð ÞTin gh u ð ÞMh jTin u ð Þdu ! r ij ð Þ Tin t jTin ð Þ # ( + ð∞ u¼t gh u ð Þdu + ðt u¼ i1 ð ÞTin gh u ð ÞM t u ð Þdu ) , where i1 ð ÞTin t iTin (2) Models for systems with a two-stage failure process Systems with a single failure mode Systems with a multiple failure mode Approximation modeling case Preparedness systems maintenance case Systems with postponed maintenance Fig. 3 Classification scheme of delay-time models for systems with a two-stage failure process. 7 Delay-Time-Based Maintenance Modeling for Technical Systems—Theory and Practice
- 21. where Mh(x) is defined as [11]: Mh x ð Þ ¼ ð∞ x fh u ð Þdu (3) In the absence of an inspection procedure (Tin ¼∞), the reliability func- tion is given by the following formula [11]: R Tin ð Þ ¼ R∞ t ð Þ ¼ r 1 ð Þ Tin t ð Þ ¼ 1 ðt u¼0 Fh t u ð Þgh u ð Þdu (4) For such a model, a numerical example is provided. DT models for single-unit systems are later investigated in works by Redmond and Wang [2, 17]. According to these works, there is a possibility to define the C.d.f. of time to failure, F(x), as the convolution of u and h such that u+hx [2, 17]: F x ð Þ ¼ ðx u¼0 gh u ð ÞFh xu ð Þdu (5) and the reliability function, R(x)¼1F(x). Taking into account the following additional assumptions and notation [2, 17]: • the system is renewed at either a failure repair or at a repair done at an inspection if a defect is identified, • after either a failure renewal or inspection renewal, the inspection pro- cess restarts, • maintenance actions restore maintained components to good as new condition, • failures of the system are identified immediately, and repairs or replace- ments are made as soon as possible, • perfect inspections case, that is, every defect is assumed to be identified during inspection action performance, the expected cost over each cycle, c(Tin), may be defined by the following equation [2, 17]: c Tin ð Þ ¼ crF Tin ð Þ + cp + cin ðTin u¼0 gh u ð Þ 1Fh Tin u ð Þ ð Þdu + cin 1Gh Tin ð Þ ð Þ (6) The expected downtime Ed(Tin) of an inspection cycle of length Tin is given as [34]: Ed Tin ð Þ ¼ dr dp din F Tin ð Þ + dpGh Tin ð Þ + din (7) 8 Advances in System Reliability Engineering
- 22. Assuming instantaneous inspection and replacement times, the expected cycle length, TM(Tin), is given by [2, 34]: TM Tin ð Þ ¼ ðTin x¼0 xF0 x ð Þdx + TinR Tin ð Þ (8) Examples of such model implementations are given in the literature [2, 17], as well as an analysis of the problem of model parameter estimation [17]. This model is later extended by Jodejko-Pietruczuk et al. [40, 41]. In one study, the authors focus on the development of a mathematical model for a technical object maintenance, which extends the approach used thus far and allows analyzing the long-term operation time period of a single-unit system [40]. The model gives the possibility to find a constant time period between the inspection actions’ performance that is optimal due to minimal mainte- nance costs or maximal availability criterion satisfaction. The solution is based on Block Inspection policy implementation. The authors present ana- lytical and Monte Carlo simulation models with comprehensive sensitivity analysis performance. This model is an extension of a preliminary analytical maintenance model [42]. Another extension is presented in work by Jodejko-Pietruczuk et al. [41], where the assumption of imperfect inspections is introduced. In this work, it is assumed that the system inspection actions are imperfect, which means that the defect occurrence in the system is identified with the given probability pw. In such a situation in the analyzed system, the following maintenance actions may occur: • preventive replacement during inspection action performance, provided that the defect has occurred and been identified with the probability pw, • inspection action performance without preventive replacement, pro- vided that the defect has not occurred nor been identified with proba- bility (1pw). Based on the main assumptions given earlier [40], the long-term expected maintenance costs per unit time function is defined. 2.1.2 DTM for a Multidefect Case One of the extensions in delay-time modeling regards consideration of more than one type of inspectable and independent defect occurrence in a system. The first model for single-unit systems with n different types of inspectable defects is developed by Christer [11] and later extended by Wang [43]. Wang [43] investigated the production process subjected to two types of deterioration. The delay-time-based inspection model was developed for 9 Delay-Time-Based Maintenance Modeling for Technical Systems—Theory and Practice
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- 30. REDAKCIO: BUDAPEST, IX, ÜLLŐI-UT 59. ARTPLENAJN FOTOGRAFAĴOJN FARAS DISKAY BUDAPEST VII., RÁKÓCZY-UT 72 SENZACIO DE NIA LITERATURO! La poemaro de JULIO BAGHY aperos. ABONPREZOJ: Simpla bela eldono Sv. franko 1. Luksa eldono kun portreto kaj subskribo de la poeto Sv. franko 2. Abonu kaj abonigu tuj! Sendu la abonprezon laŭeble en naciaj bankbiletoj aŭ iel ajn al la LITERATURA MONDO, BUDAPEST VI, ANDRÁSSY-ÚT 81
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