This document describes a study on maintenance policies for a system subject to non-self-announcing failures. The system is modeled as a three-state continuous-time Markov chain representing good, poor, and failed states. Inspections occur periodically to determine the system's state, and maintenance actions of doing nothing, repairing, or replacing can be taken. The goal is to develop optimal inspection and maintenance policies to minimize long-term average costs while maintaining system availability. Transition rates and costs of actions are considered under the assumptions of an infinite time horizon and Poisson deterioration processes between states.
Comparing the two techniques Tripod Beta and Mort at a critical accident anal...IJERA Editor
Accidents are one of the leading causes of death and disability. Despite great efforts made to prevent accidents, there is still no coherent system to identify the root causes of industrial accidents. Selection of appropriate accident analysis techniques and their comparison can be useful in this regard. This research aimed to analyze a fatal accident in a power plant construction project using the two methods of MORT and Tripod-Beta, and the comparison of the analyses. First, the report of the selected accident was studied, and the accident was analyzed by the two methods of MORT and Tripod-Beta. The next step was followed by the comparison and assessment of the methods of MORT and Tripod-Beta with the measures of time, cost, training needs, the need for technical forces, the number of causes identified, quantifiable, and the need for software to conduct analysis. The Tripod-Beta accident analysis cost less and requires less time, and less technical experts. Thorough analysis of major accidents needs to identify all the possible causes of the incident, including human error and equipment failure. Therefore, the complimentary use of both techniques of industrial accident analysis is recommended.
International Refereed Journal of Engineering and Science (IRJES) irjes
International Refereed Journal of Engineering and Science (IRJES)
Ad hoc & sensor networks, Adaptive applications, Aeronautical Engineering, Aerospace Engineering
Agricultural Engineering, AI and Image Recognition, Allied engineering materials, Applied mechanics,
Architecture & Planning, Artificial intelligence, Audio Engineering, Automation and Mobile Robots
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Our 5th Impact Forum was held on September 14, 2016 at Civic Hall, NYC's foremost center for civic technology & innovation, on the topic of Elections. Election Systems are best understood by the rules and incentives constructed around them. Evaluating the United States’ and New York’s Electoral Systems reveals a series of important policy choices that will shape what our electorate will look like in the coming decades. Impact Elections will dig into key question such as: How does money predict elections? How will technology shape voting of the future? Does the current voting system fundamentally fail certain segments of our country's population? What can voting systems from other nations tell us about options for change?
Impact Experts composing the panel at Elections included: Ann Ravel (Federal Elections Commissioner), Gustavo Rivera (New York State Senator), James Bopp (Plaintiff's Attorney for Citizens United v. FEC), Richard Briffault (Professor, Columbia Law School), the legal advisor to Stephen Colbert's Presidential Campaign, the President of the NYC Campaign Finance Board, and more.
Learn more about Impact Elections here:
https://impactelections2016.splashthat.com
El documento describe los aspectos esenciales de la Navidad, incluyendo regalar felicidad a otros, compartir en familia y conocer nuevos miembros, dar gracias a Dios por las cosas que nos ha permitido hasta la fecha, compartir la cena navideña con los seres queridos, y recordar el nacimiento de Jesús y reflexionar sobre su humildad.
This resume is for Balaji Uthaiah K B, seeking a position in manufacturing engineering. He has over 4 years of experience as a New Product Development Engineer. Currently he works as a Senior Manufacturing Engineer at New Holland Fiat India, where he is responsible for agricultural and automotive parts production. Previously he held engineering roles at Nash Industries and Asha Tech, where he gained experience in sheet metal fabrication, tooling, product design and development. He has a Bachelor's degree in engineering and is proficient with CAD software, production processes, and project management.
El prolapso mitral es una valvulopatía caracterizada por el desplazamiento de una o ambas valvas de la válvula mitral hacia el atrio izquierdo durante la sístole. Está presente en el 5-10% de la población mundial y generalmente es asintomático. El diagnóstico se realiza mediante ecocardiograma que muestra el desplazamiento de las valvas. El tratamiento depende de la presencia de síntomas e incluye medicamentos, cirugía de reparación valvular o reemplazo valvular
Comparing the two techniques Tripod Beta and Mort at a critical accident anal...IJERA Editor
Accidents are one of the leading causes of death and disability. Despite great efforts made to prevent accidents, there is still no coherent system to identify the root causes of industrial accidents. Selection of appropriate accident analysis techniques and their comparison can be useful in this regard. This research aimed to analyze a fatal accident in a power plant construction project using the two methods of MORT and Tripod-Beta, and the comparison of the analyses. First, the report of the selected accident was studied, and the accident was analyzed by the two methods of MORT and Tripod-Beta. The next step was followed by the comparison and assessment of the methods of MORT and Tripod-Beta with the measures of time, cost, training needs, the need for technical forces, the number of causes identified, quantifiable, and the need for software to conduct analysis. The Tripod-Beta accident analysis cost less and requires less time, and less technical experts. Thorough analysis of major accidents needs to identify all the possible causes of the incident, including human error and equipment failure. Therefore, the complimentary use of both techniques of industrial accident analysis is recommended.
International Refereed Journal of Engineering and Science (IRJES) irjes
International Refereed Journal of Engineering and Science (IRJES)
Ad hoc & sensor networks, Adaptive applications, Aeronautical Engineering, Aerospace Engineering
Agricultural Engineering, AI and Image Recognition, Allied engineering materials, Applied mechanics,
Architecture & Planning, Artificial intelligence, Audio Engineering, Automation and Mobile Robots
Automotive Engineering….
Our 5th Impact Forum was held on September 14, 2016 at Civic Hall, NYC's foremost center for civic technology & innovation, on the topic of Elections. Election Systems are best understood by the rules and incentives constructed around them. Evaluating the United States’ and New York’s Electoral Systems reveals a series of important policy choices that will shape what our electorate will look like in the coming decades. Impact Elections will dig into key question such as: How does money predict elections? How will technology shape voting of the future? Does the current voting system fundamentally fail certain segments of our country's population? What can voting systems from other nations tell us about options for change?
Impact Experts composing the panel at Elections included: Ann Ravel (Federal Elections Commissioner), Gustavo Rivera (New York State Senator), James Bopp (Plaintiff's Attorney for Citizens United v. FEC), Richard Briffault (Professor, Columbia Law School), the legal advisor to Stephen Colbert's Presidential Campaign, the President of the NYC Campaign Finance Board, and more.
Learn more about Impact Elections here:
https://impactelections2016.splashthat.com
El documento describe los aspectos esenciales de la Navidad, incluyendo regalar felicidad a otros, compartir en familia y conocer nuevos miembros, dar gracias a Dios por las cosas que nos ha permitido hasta la fecha, compartir la cena navideña con los seres queridos, y recordar el nacimiento de Jesús y reflexionar sobre su humildad.
This resume is for Balaji Uthaiah K B, seeking a position in manufacturing engineering. He has over 4 years of experience as a New Product Development Engineer. Currently he works as a Senior Manufacturing Engineer at New Holland Fiat India, where he is responsible for agricultural and automotive parts production. Previously he held engineering roles at Nash Industries and Asha Tech, where he gained experience in sheet metal fabrication, tooling, product design and development. He has a Bachelor's degree in engineering and is proficient with CAD software, production processes, and project management.
El prolapso mitral es una valvulopatía caracterizada por el desplazamiento de una o ambas valvas de la válvula mitral hacia el atrio izquierdo durante la sístole. Está presente en el 5-10% de la población mundial y generalmente es asintomático. El diagnóstico se realiza mediante ecocardiograma que muestra el desplazamiento de las valvas. El tratamiento depende de la presencia de síntomas e incluye medicamentos, cirugía de reparación valvular o reemplazo valvular
This document describes an e-reception application that allows organizations to digitally manage visitors and activities. The app allows receptionists to input organizational information and visitors to access events, training, products and exams. It has features like Cortana, events, products, emails and notices. The app aims to easily control all organizational activities, enable smart interaction between visitors and organizations, and serve as an alternative to physical reception systems. It has modules for visitors and administrators, and can be used on Windows devices with Visual Studio 2015 and .NET Framework 4.5. When implemented, the app will help manage organizational activities and digitally connect organizations with internal and external people.
El documento describe diferentes animales vertebrados e invertebrados. Entre los animales vertebrados mencionados están la vaca, cuyos productos como la carne, leche, cuero y estiércol son útiles, y el burro que pertenece a la familia de los équidos. Los animales invertebrados descritos son el sapo, reconocido por sus patas cortas, ojos grandes y piel con verrugas, y el pulpo, un molusco acuático con ocho tentáculos.
KlientBoost and Kissmetrics Present: PPC Analytics [infographic]KlientBoost
It's hard to know which PPC metrics matter when there's so many. These insights into PPC analytics will help you generate more conversions and more revenue.
Este documento resume las actividades de un taller sobre internet. Incluye diferencias entre navegadores, portales web y motores de búsqueda, ejemplos de cada uno, y extractos de artículos sobre una ley de educación superior en Ecuador. También incluye definiciones de términos como informática, misión, visión, recursos, especialización, sistemas, bases de datos, redes y tecnología. Finalmente, presenta una noticia actual sobre libre movilidad entre Ecuador y Perú impulsada por el Ministerio de Transporte de Ecuador.
18.9. 15 ichthyodiversity of maimala river, kerala, india, jzbr 2015Mathews Plamoottil
This document summarizes a study on the ichthyodiversity of the Manimala River in Kerala, India. 97 species of fish were collected from 20 sites along the river, with cyprinid fishes being the most common. The study also discovered and described 12 new fish species found in the river. Cyprinids, percoids and silurids made up most of the catches. Many cultivable fish species were found in the lower regions of the river.
Este documento presenta la síntesis de las experiencias docentes de un profesor con 6 años de experiencia. Incluye tanto experiencias positivas como negativas. Las positivas incluyen asesorar exitosamente a estudiantes y ganar su respeto. Las negativas incluyen enfrentar a un estudiante grosero sin el apoyo de los directivos, lo que llevó a su renuncia. También describe cómo ha ayudado a estudiantes con problemas personales. Concluye que tanto las buenas como las malas experiencias le sirven para mejorar continuamente.
This document describes an e-reception application that allows organizations to digitally manage visitors and activities. The app allows receptionists to input organizational information and visitors to access events, training, products and exams. It has features like Cortana, events, products, emails and notices. The app aims to easily control all organizational activities, enable smart interaction between visitors and organizations, and serve as an alternative to physical reception systems. It has modules for visitors and administrators, and can be used on Windows devices with Visual Studio 2015 and .NET Framework 4.5. When implemented, the app will help manage organizational activities and digitally connect organizations with internal and external people.
El documento describe diferentes animales vertebrados e invertebrados. Entre los animales vertebrados mencionados están la vaca, cuyos productos como la carne, leche, cuero y estiércol son útiles, y el burro que pertenece a la familia de los équidos. Los animales invertebrados descritos son el sapo, reconocido por sus patas cortas, ojos grandes y piel con verrugas, y el pulpo, un molusco acuático con ocho tentáculos.
KlientBoost and Kissmetrics Present: PPC Analytics [infographic]KlientBoost
It's hard to know which PPC metrics matter when there's so many. These insights into PPC analytics will help you generate more conversions and more revenue.
Este documento resume las actividades de un taller sobre internet. Incluye diferencias entre navegadores, portales web y motores de búsqueda, ejemplos de cada uno, y extractos de artículos sobre una ley de educación superior en Ecuador. También incluye definiciones de términos como informática, misión, visión, recursos, especialización, sistemas, bases de datos, redes y tecnología. Finalmente, presenta una noticia actual sobre libre movilidad entre Ecuador y Perú impulsada por el Ministerio de Transporte de Ecuador.
18.9. 15 ichthyodiversity of maimala river, kerala, india, jzbr 2015Mathews Plamoottil
This document summarizes a study on the ichthyodiversity of the Manimala River in Kerala, India. 97 species of fish were collected from 20 sites along the river, with cyprinid fishes being the most common. The study also discovered and described 12 new fish species found in the river. Cyprinids, percoids and silurids made up most of the catches. Many cultivable fish species were found in the lower regions of the river.
Este documento presenta la síntesis de las experiencias docentes de un profesor con 6 años de experiencia. Incluye tanto experiencias positivas como negativas. Las positivas incluyen asesorar exitosamente a estudiantes y ganar su respeto. Las negativas incluyen enfrentar a un estudiante grosero sin el apoyo de los directivos, lo que llevó a su renuncia. También describe cómo ha ayudado a estudiantes con problemas personales. Concluye que tanto las buenas como las malas experiencias le sirven para mejorar continuamente.
1. Maintenance Policies For A Deterioration System
Subject To Non-Self-Announcing Failures
Salih Tekin
N. Onur Bakır, Bü¸sra Kele¸s
TOBB University of Economics and Technology
Ankara, Turkey
June 6, 2014
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 1 / 35
2. System Maintenance
Maintenance is crucial to improve system availability performance with
a minimum cost, especially when
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 2 / 35
3. System Maintenance
Maintenance is crucial to improve system availability performance with
a minimum cost, especially when
I the system fails stochastically
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 2 / 35
4. System Maintenance
Maintenance is crucial to improve system availability performance with
a minimum cost, especially when
I the system fails stochastically
I the degree of deterioration, in addition to
failure, are known only through inspections
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 2 / 35
5. System Maintenance
Maintenance is crucial to improve system availability performance with
a minimum cost, especially when
I the system fails stochastically
I the degree of deterioration, in addition to
failure, are known only through inspections
I optional preventive maintenance is
suggested
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 2 / 35
6. System Maintenance
Maintenance is crucial to improve system availability performance with
a minimum cost, especially when
I the system fails stochastically
I the degree of deterioration, in addition to
failure, are known only through inspections
I optional preventive maintenance is
suggested
I maintenance is not perfect (except
replacement)
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 2 / 35
7. Maintenance Categories in Literature
Table: Maintenance Categories in Literature
Maintenance policy System structure Maintenance degree
Age replacement
Block replacement
Periodic 3
Sequential
Control limit
Single-unit 3
Multi-unit
Perfect 3
Imperfect 3
Minimal 3
Worse
Maintenance cost Optimization criteria Modelling tools
Constant 3
Random
Complex
Minimize cost rate 3
Minimize availability
Minimize downtime
Renewal theory 3
Markov chain 3
Poisson process
Planning horizon Dependence System information
Infinite 3
Finite
Discrete
Continuous
Economic
Failure
Probability
State 3
Perfect 3
Imperfect
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 3 / 35
8. Outline
1 Introduction
2 Mathematical Formulations
3 Optimal Inspection Time and Optimal Policy
4 Numerical Example
5 References
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 4 / 35
9. Introduction
Problem Definition
System structure
For a stochastically failing system in which the degree of deterioration
are known only through inspections,
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 5 / 35
10. Introduction
Problem Definition
System structure
For a stochastically failing system in which the degree of deterioration
are known only through inspections,
I At each inspection epoch k⌧, k = {0, 1, 2, ...} the system occupies
one of three states: good (1), poor (2), failed (3) and the decision maker
chooses an available action: do nothing, repair and replace.
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 5 / 35
11. Introduction
Problem Definition
System structure
For a stochastically failing system in which the degree of deterioration
are known only through inspections,
I At each inspection epoch k⌧, k = {0, 1, 2, ...} the system occupies
one of three states: good (1), poor (2), failed (3) and the decision maker
chooses an available action: do nothing, repair and replace.
S = {1, 2, 3}; State space
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 5 / 35
12. Introduction
Problem Definition
System structure
For a stochastically failing system in which the degree of deterioration
are known only through inspections,
I At each inspection epoch k⌧, k = {0, 1, 2, ...} the system occupies
one of three states: good (1), poor (2), failed (3) and the decision maker
chooses an available action: do nothing, repair and replace.
S = {1, 2, 3}; State space
T = {0, 1, 2, 3, ...}; Time horizon
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 5 / 35
13. Introduction
Problem Definition
System structure
For a stochastically failing system in which the degree of deterioration
are known only through inspections,
I At each inspection epoch k⌧, k = {0, 1, 2, ...} the system occupies
one of three states: good (1), poor (2), failed (3) and the decision maker
chooses an available action: do nothing, repair and replace.
S = {1, 2, 3}; State space
T = {0, 1, 2, 3, ...}; Time horizon
A = {do nothing, repair, replace}; Action space
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 5 / 35
14. Introduction
Problem Definition
System structure
For a stochastically failing system in which the degree of deterioration
are known only through inspections,
I At each inspection epoch k⌧, k = {0, 1, 2, ...} the system occupies
one of three states: good (1), poor (2), failed (3) and the decision maker
chooses an available action: do nothing, repair and replace.
S = {1, 2, 3}; State space
T = {0, 1, 2, 3, ...}; Time horizon
A = {do nothing, repair, replace}; Action space
It is called minor repair when the repair is performed at state 2 and major
repair when it is performed at state 3.
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 5 / 35
15. Introduction
Problem Definition
Deterioration process
I The deterioration is modelled by three-state continuous time Markov
chain
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 6 / 35
16. Introduction
Problem Definition
Deterioration process
I The deterioration is modelled by three-state continuous time Markov
chain
If {Yt, t 0} represents the system state at time t, then we assume
that {Yt, t 0} is a continuous time Markov chain with transition
probabilities,
Pij(⌧) = P{Yt+⌧ = j | Yt = i}, 8 t, ⌧ 2 R (1)
when it is currently at state i = {1, 2, 3} at time t, will be at state j = {2, 3}
at time t + ⌧.
(Markovian assumption)
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 6 / 35
17. Introduction
Problem Definition
Deterioration process
Good
(1)
Poor
(2)
Failed
(3)
𝐏𝟏𝟏(𝛕) 𝐏𝟐𝟐(𝛕) 𝐏𝟑𝟑(𝛕)
𝐏𝟏𝟑(𝛕)
𝐏𝟏𝟐(𝛕) 𝐏𝟐𝟑(𝛕)
Figure: Transition Probabilities
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 7 / 35
18. Introduction
Problem Definition
Deterioration process
Good
(1)
Poor
(2)
Failed
(3)
𝐏𝟏𝟏(𝛕) 𝐏𝟐𝟐(𝛕) 𝐏𝟑𝟑(𝛕)
𝐏𝟏𝟑(𝛕)
𝐏𝟏𝟐(𝛕) 𝐏𝟐𝟑(𝛕)
Figure: Transition Probabilities
I The event that causes a transition within an inspection period from state i
to j, where i 6= j, occur after an exponential amount of time with rate ij.
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 7 / 35
19. Introduction
Problem Definition
Assumptions
I The time horizon considered is infinite and the system with good state
is put into service in time 0.
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 8 / 35
20. Introduction
Problem Definition
Assumptions
I The time horizon considered is infinite and the system with good state
is put into service in time 0.
I Transition from state i = 1, 2 to state j = 2, 3 occurs according to a Poisson
Process with rate ij.
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 8 / 35
21. Introduction
Problem Definition
Assumptions
I The time horizon considered is infinite and the system with good state
is put into service in time 0.
I Transition from state i = 1, 2 to state j = 2, 3 occurs according to a Poisson
Process with rate ij.
I The system state is monitored only through inspections and inspection time
is negligible.
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 8 / 35
22. Introduction
Problem Definition
Assumptions
I The time horizon considered is infinite and the system with good state
is put into service in time 0.
I Transition from state i = 1, 2 to state j = 2, 3 occurs according to a Poisson
Process with rate ij.
I The system state is monitored only through inspections and inspection time
is negligible.
I Time to inspect is negligible; however, inspection cost is assumed to be a
monotonically non-increasing function of time interval between two
consecutive inspections [4] to prevent frequent inspections.
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 8 / 35
23. Introduction
Problem Definition
Assumptions
I If the system is identified as good, "do nothing" action is required;
otherwise decision maker may choose other actions.
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 9 / 35
24. Introduction
Problem Definition
Assumptions
I If the system is identified as good, "do nothing" action is required;
otherwise decision maker may choose other actions.
I Minor (Major) repair takes the system to state 1, with probability pmn (pmj),
and to state 2 with probability qmn = 1 pmn (qmj = 1 pmj) after a fixed time
tmn (tmj) with a fixed cost cmn (cmj).
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 9 / 35
25. Introduction
Problem Definition
Assumptions
I If the system is identified as good, "do nothing" action is required;
otherwise decision maker may choose other actions.
I Minor (Major) repair takes the system to state 1, with probability pmn (pmj),
and to state 2 with probability qmn = 1 pmn (qmj = 1 pmj) after a fixed time
tmn (tmj) with a fixed cost cmn (cmj).
I Replace always returns the system to state 1 with a fixed amount cost crpl
after a fixed maintenance time trpl.
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 9 / 35
26. Introduction
Problem Definition
Assumptions
I If the system is identified as good, "do nothing" action is required;
otherwise decision maker may choose other actions.
I Minor (Major) repair takes the system to state 1, with probability pmn (pmj),
and to state 2 with probability qmn = 1 pmn (qmj = 1 pmj) after a fixed time
tmn (tmj) with a fixed cost cmn (cmj).
I Replace always returns the system to state 1 with a fixed amount cost crpl
after a fixed maintenance time trpl.
I The system is not in working condition during repair or replace and hence,
the inspection cost does not occur.
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 9 / 35
27. Introduction
Problem Definition
Assumptions
I If the system is identified as good, "do nothing" action is required;
otherwise decision maker may choose other actions.
I Minor (Major) repair takes the system to state 1, with probability pmn (pmj),
and to state 2 with probability qmn = 1 pmn (qmj = 1 pmj) after a fixed time
tmn (tmj) with a fixed cost cmn (cmj).
I Replace always returns the system to state 1 with a fixed amount cost crpl
after a fixed maintenance time trpl.
I The system is not in working condition during repair or replace and hence,
the inspection cost does not occur.
I The system state is known instantaneously after each repair.
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 9 / 35
28. Introduction
Problem Definition
Maintenance Policies
Five available maintenance policies are proposed shown to emphasise how
the maintenance parameters affect the optimum inspection period and policy.
Table: Policy Description
Policy State 1 State 2 State 3
⇧1 Do nothing Do nothing Replacement
⇧2 Do nothing Do nothing Major repair
⇧3 Do nothing Minor repair Major repair
⇧4 Do nothing Minor repair Replacement
⇧5 Do nothing Replacement Replacement
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29. Introduction
Problem Definition
Optimization criteria
Optimizing the maintenance policies is based on Renewal Theory.
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30. Introduction
Problem Definition
Optimization criteria
Optimizing the maintenance policies is based on Renewal Theory.
I A cycle ends (a renewal) after a maintenance of the failed system,
while the maintenance brings the system to "good" state.
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31. Introduction
Problem Definition
Optimization criteria
Optimizing the maintenance policies is based on Renewal Theory.
I A cycle ends (a renewal) after a maintenance of the failed system,
while the maintenance brings the system to "good" state.
I Then cost rate is the ratio of the expected total cost occurred during
the renewal cycle over the expected cycle length.
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 11 / 35
32. Introduction
Problem Definition
Optimization criteria
Optimizing the maintenance policies is based on Renewal Theory.
I A cycle ends (a renewal) after a maintenance of the failed system,
while the maintenance brings the system to "good" state.
I Then cost rate is the ratio of the expected total cost occurred during
the renewal cycle over the expected cycle length.
I The objective is when to inspect the system periodically and what
action to be made by minimizing the expected cost rate in the long run.
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33. Mathematical Formulations
Mathematical Calculations
Transition probabilities
Let Xij be the holding time between state i = {1, 2, 3} and state j = {2, 3} .
By Poisson arrivals assumption leads to:
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 12 / 35
34. Mathematical Formulations
Mathematical Calculations
Transition probabilities
Let Xij be the holding time between state i = {1, 2, 3} and state j = {2, 3} .
By Poisson arrivals assumption leads to:
P11(⌧) = P{min{X12, X13} > ⌧} = e 1⌧
(2)
P12(⌧) = P{X12 ⌧, X12 < X13, X12 + X23 > ⌧}
=
8
>><
>>:
12 e 2⌧ ⌧, 1 2 = 0
12
1 2
e 2⌧ e 1⌧ , 1 2 6= 0
(3)
P13(⌧) = P{X12 ⌧, X12 < X13, X12 + X23 ⌧}
=
8
>><
>>:
1 e 1⌧
12 e 2⌧ ⌧, 1 = 2
1 e 1⌧ 12
1 2
e 2⌧ + e 1⌧ , 1 6= 2
(4)
where the total transition rate out of a state 1 is 1 = 12 + 13 and of state 2
is 2 = 23.
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35. Mathematical Formulations
Mathematical Calculations
Expected downtime
I How long ago could the failure have first been noticed
by an inspection?
1
Inspection time
kτ
At the time system fails
τ' Downtime
3
Inspection time
(k+1)τ
Figure: Downtime
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36. Mathematical Formulations
Mathematical Calculations
Expected downtime
D1(⌧) : the expected downtime within an inspection period, ⌧ , of the system
initially found at state 1
D2(⌧) : the expected downtime within an inspection period, ⌧ , of the system
initially found at state 2
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37. Mathematical Formulations
Mathematical Calculations
Expected downtime
D1(⌧) : the expected downtime within an inspection period, ⌧ , of the system
initially found at state 1
D2(⌧) : the expected downtime within an inspection period, ⌧ , of the system
initially found at state 2
D1(⌧) = DQ
(⌧) Q(⌧) + DH
(⌧) H(⌧) (5)
where
1
Inspection period, τ
32
Q(τ)
H(τ)
3
𝐏𝟑𝟑(τ)
Figure: Downtime
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38. Mathematical Formulations
Mathematical Calculations
Expected downtime
DQ
(⌧) = ⌧ E[X13 | X13 ⌧, X13 < X12 ] =
⌧
1 e 1⌧
1
1
(6)
DH
(⌧) = ⌧ E[X12 | X12 ⌧, X12 < X13, X12 + X23 ⌧]
E[X23 | X12 ⌧, X12 < X13, X12 + X23 ⌧]
=
8
>>>><
>>>>:
⌧ 2
1
+e 1⌧
⇣
2
1
+⌧
⌘
1 e 1⌧
1e 1⌧ ⌧
, 1 2 = 0
( 1 2)
⇣
⌧ 1
1
1
2
⌘
1 2 1e 2⌧ + 2e 1⌧ +
e 2⌧
⇣
1
2
⌘
e 1⌧
⇣
2
1
⌘
1 2 1e 2⌧ + 2e 1⌧ , 1 2 6= 0
(7)
D2(⌧) = ⌧
1 e 2⌧
2
(8)
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39. Mathematical Formulations
Mathematical Calculations
Average cost rate
Let i(⌧) be the cost rate for ⇡i, i = 0, .., 5,
The first aim is to calculate each i(⌧) via optimizing the inspection
interval time ⌧.
The second is to determinate the optimal policy with the correspon-
ding minimum cost rate.
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40. Mathematical Formulations
Mathematical Calculations
Average cost rate
For 8 ⇡i, i = 1, .., 5, we define
Li
j(⌧) as the expected time length the system spends to state 1 from
state j = {1, 2, 3}, and
Ci
j(⌧) as the expected maintenance cost the system spends to state
1 from state j = {1, 2, 3}.
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41. Mathematical Formulations
Mathematical Calculations
Average cost rate
The cost rate of the system is then
i(⌧) =
Ci
1(⌧)
Li
1(⌧)
+ (⌧) (9)
where (⌧) is the inspection cost per time which is a non increasing
function of time interval between two consecutive inspections such that
lim
⌧!0
(⌧) = K < 1 ve lim
⌧!1
(⌧) = 0 (10)
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42. Mathematical Formulations
Mathematical Calculations
Average cost rate
Cycle Length
Li
1(⌧) =
1
1 P11(⌧)
⇥
⌧ + P12(⌧)Li
2(⌧) + P13(⌧)Li
3(⌧)
⇤
(11)
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43. Mathematical Formulations
Mathematical Calculations
Average cost rate
Cycle Length
Li
1(⌧) =
1
1 P11(⌧)
⇥
⌧ + P12(⌧)Li
2(⌧) + P13(⌧)Li
3(⌧)
⇤
(11)
where
Li
2(⌧) =
8
>>>><
>>>>:
1
1 P22(⌧)
⇥
⌧ + P23(⌧)Li
3(⌧)
⇤
i = 1, 2
tmn
pmn
i = 3, 4
trpl i = 5
(12)
Li
3(⌧) =
8
<
:
tmj + 1 pmj Li
2(⌧) i = 3, 4
trpl i = 5
(13)
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44. Mathematical Formulations
Mathematical Calculations
Average cost rate
Cycle Cost
Ci
1(⌧) =
1
1 P11(⌧)
⇥
P12(⌧)Ci
2(⌧) + P13(⌧) cdD1(⌧) + Ci
3(⌧)
⇤
(14)
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 20 / 35
45. Mathematical Formulations
Mathematical Calculations
Average cost rate
Cycle Cost
Ci
1(⌧) =
1
1 P11(⌧)
⇥
P12(⌧)Ci
2(⌧) + P13(⌧) cdD1(⌧) + Ci
3(⌧)
⇤
(14)
where
Ci
2(⌧) =
8
>>><
>>>:
cdD2(⌧) + Ci
3(⌧), i = 1, 2
cmn+tmn[cd (⌧)]
pMn
, i = 3, 4
crpl + trpl [cd (⌧)], i = 5
(15)
Ci
3(⌧) =
8
>>><
>>>:
cmj + tmj [cd (⌧)]
+(1 pmj)Ci
2(⌧), i = 3, 4
crpl + trpl [cd (⌧)], i = 5
(16)
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46. Optimal Inspection Time and Optimal Policy
Optimization Criteria
For any ⌧ 2 (0, 1), the parameters in (11-16) are all positive and finite,
then the cost rate is both positive and finite.
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47. Optimal Inspection Time and Optimal Policy
Optimization Criteria
For any ⌧ 2 (0, 1), the parameters in (11-16) are all positive and finite,
then the cost rate is both positive and finite.
We need to examine the behaviour of the cost rate as
⌧ ! 0 (continuous inspection) and ⌧ ! 1 (never inspect)
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48. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Continuous inspection
I Continuously inspection ensures that the system state is always
known.
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49. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Continuous inspection
I Continuously inspection ensures that the system state is always
known.
I Failure is detected as soon as the system fails.
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50. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Continuous inspection
I Continuously inspection ensures that the system state is always
known.
I Failure is detected as soon as the system fails.
I The cost rate becomes
i(0) =
Ci
1(0)
Li
1(0)
+ K (17)
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51. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Continuous inspection
The cycle length leads to:
Li
1(0) =
12
1
1
12
+ Li
2(0) +
13
1
1
13
+ Li
3(0) (18)
where
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52. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Continuous inspection
The cycle length leads to:
Li
1(0) =
12
1
1
12
+ Li
2(0) +
13
1
1
13
+ Li
3(0) (18)
where
Li
2(0) =
8
>>><
>>>:
1
2
+ Li
3(0), i = 1, 2
tmn
pmn
, i = 3, 4
trpl, i = 5
(19)
Li
3(0) =
8
<
:
tmj + (1 pmj)Li
2(0), i = 3, 4
trpl, i = 5
(20)
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53. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Continuous inspection
The cycle maintenance cost leads to:
Ci
1(0) =
12
1
Ci
2(0) +
13
1
Ci
3(0) (21)
where
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54. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Continuous inspection
The cycle maintenance cost leads to:
Ci
1(0) =
12
1
Ci
2(0) +
13
1
Ci
3(0) (21)
where
Ci
2(0) =
8
>>><
>>>:
Ci
3(0), i = 1, 2
cmn+tmn(cd K)
pmn
, i = 3, 4
crpl + trpl(cd K), i = 5
(22)
Ci
3(0) =
8
<
:
cmj + tmj(cd K) + (1 pmj) Ci
2(0), i = 3, 4
crpl + trpl(cd K), i = 5
(23)
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55. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Never inspect
When ⌧ ! 1, the failure of the system occurs with probability 1 which
leads that the cycle cost consists of only downtime cost and inspection
cost.
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56. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Never inspect
When ⌧ ! 1, the failure of the system occurs with probability 1 which
leads that the cycle cost consists of only downtime cost and inspection
cost.
The cost rate becomes
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57. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Never inspect
When ⌧ ! 1, the failure of the system occurs with probability 1 which
leads that the cycle cost consists of only downtime cost and inspection
cost.
The cost rate becomes
lim
⌧!1
i(⌧) = lim
⌧!1
hcdD1(⌧)
⌧
+ (⌧)
i
= cd + lim
t!1
(⌧)
= cd (24)
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58. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Never inspect
When ⌧ ! 1, the failure of the system occurs with probability 1 which
leads that the cycle cost consists of only downtime cost and inspection
cost.
The cost rate becomes
lim
⌧!1
i(⌧) = lim
⌧!1
hcdD1(⌧)
⌧
+ (⌧)
i
= cd + lim
t!1
(⌧)
= cd (24)
I i(⌧) is a continuous function and bounded on [0, min{ (0), i(1)}]
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59. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Corollary 1
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60. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Corollary 1
If there exists 9 ⌧ 2 [0, 1) such that i(⌧) < cd, then an optimal
inspection interval of each policy exists such that
⌧⇤
i 2 1
i ([0, min{ i(0), i(1)}])
otherwise , all policies converge each other (cd) in the long run.
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61. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Corollary 1
If there exists 9 ⌧ 2 [0, 1) such that i(⌧) < cd, then an optimal
inspection interval of each policy exists such that
⌧⇤
i 2 1
i ([0, min{ i(0), i(1)}])
otherwise , all policies converge each other (cd) in the long run.
Corollary 2
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62. Optimal Inspection Time and Optimal Policy
Optimization Criteria
Corollary 1
If there exists 9 ⌧ 2 [0, 1) such that i(⌧) < cd, then an optimal
inspection interval of each policy exists such that
⌧⇤
i 2 1
i ([0, min{ i(0), i(1)}])
otherwise , all policies converge each other (cd) in the long run.
Corollary 2
If cd = 1 and other cost parameters set to zero, the average cost rate
gives the system unavailability.
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63. Numerical Example
Numerical Example
We present several numerical examples which illustrate how system
parameters (transition rates, maintenance cots and times) could affect the
optimum inspection period and the optimum maintenance policy as well.
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64. Numerical Example
Numerical Example
We present several numerical examples which illustrate how system
parameters (transition rates, maintenance cots and times) could affect the
optimum inspection period and the optimum maintenance policy as well.
Consider the deterioration model with inspection cost function as
(⌧) = Ke 0.1⌧
. Other input parameters are shown below of illustrated figures.
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65. Numerical Example
Numerical Example
Transition rates
Replacement
Minor
Repair
Major
Repair
Fixed cost 750 300 500
Time to
maintenance
1 0.5 2
Probability of repair-
success
1 0.9 0.6
Inspection cost
parameter
1000
Figure: Relationships between transition rates and inspection interval
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69. Numerical Example
Numerical Example
Maintenance probabilities
Probability of Major
repair-success, 𝐩 𝐦𝐣
1
0,8
0,6
0,4
0,2
0,2 0,4 0,6 0,8 1 0,2 0,4 0,6 0,8 1
1
0,8
0,6
0,4
0,2
0,2 0,4 0,6 0,8 1 0,2 0,4 0,6 0,8 1
𝛑 𝟎
𝛑 𝟏
𝛑 𝟐
𝛑 𝟑
𝛑 𝟒
𝛑 𝟓
Probability of Minor
repair-success, 𝐩 𝐦𝐧
I. 𝐜 𝐫𝐩𝐥 = 𝟓𝟎𝟎 II. 𝐜 𝐫𝐩𝐥 = 𝟔𝟓𝟎
III. 𝐜 𝐫𝐩𝐥 = 𝟕𝟓𝟎 IV. 𝐜 𝐫𝐩𝐥 = 𝟏𝟎𝟎𝟎
Replace
ment
Minor
Repair
Major
Repair
Fixed
cost
𝐜 𝐫𝐩𝐥 150 500
Time to
maintenance
1 0.5 2
Probability
of repair-
success
1 𝐩 𝐦𝐧 𝐩 𝐦𝐣
Inspection cost
parameter
1000
Down Time Cost 400
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70. Numerical Example
Numerical Example
Maintenance times
Maintenance time
for major repair, 𝐭 𝐦𝐣
Replace
ment
Minor
Repair
Major
Repair
Fixed
cost
750 150 500
Time to
maintenance
𝐭 𝐫𝐩𝐥 𝐭 𝐦𝐣 𝐭 𝐦𝐧
Probability
of repair-
success
1 0.9 0.6
Maintenance time
for minor repair,
𝐭 𝐦𝐧
Inspection cost
parameter
1000
Downtime Cost 400
𝛑 𝟎
𝛑 𝟏
𝛑 𝟐
𝛑 𝟑
𝛑 𝟒
𝛑 𝟓
I. 𝐭 𝐫𝐩𝐥 = 𝟎. 𝟎𝟏
II. 𝐭 𝐫𝐩𝐥 = 𝟎. 𝟏
III. 𝐭 𝐫𝐩𝐥 = 𝟎. 𝟓
IV. 𝐭 𝐫𝐩𝐥 = 𝟏
2,5
2
1,5
1
0,5
0,5 1 1,5 2 2,5 0,5 1 1,5 2 2,5
2,5
2
1,5
1
0,5
0,5 1 1,5 2 2,5 0,5 1 1,5 2 2,5
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 33 / 35
71. Numerical Example
Thank you!
Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 34 / 35
72. References
References
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Salih Tekin N. Onur Bakır, Bü¸sra Kele¸s (TOBB University of Economics and Technology Ankara, Turkey)Maintenance Policies For A Deterioration System Subject To Non-Self-Announcing FailuresJune 6, 2014 35 / 35