reliability and maintenance of medical devices · reliability and maintenance of medical devices...

RELIABILITY AND MAINTENANCE

OF MEDICAL DEVICES

by

Sharareh Taghipour

A thesis submitted in conformity with the requirements for the degree of Doctor of

Philosophy

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

© Copyright by Sharareh Taghipour, 2011

ii

RELIABILITY AND MAINTENANCE

OF MEDICAL DEVICES

Sharareh Taghipour

Doctor of Philosophy

Department of Mechanical and Industrial Engineering

University of Toronto

2011

ABSTRACT

For decades, reliability engineering techniques have been successfully applied in many

industries to improve the performance of equipment maintenance management. Numerous

inspection and optimization models are developed and widely used to achieve maintenance

excellence, i.e. the balance of performance, risk, resources and cost to reach to an optimal

solution. However, the application of all these techniques and models to medical devices is new.

Hospitals, due to possessing a large number of difference devices, can benefit significantly if the

optimization techniques are used properly in the equipment management processes. Most

research in the area of reliability engineering for medical equipment mainly considers the

devices in their design or manufacturing stage and suggests some techniques to improve the

reliability. To this point, best maintenance strategies for medical equipment in their operating

context have not been considered.

We aim to address this gap and propose methods to improve current maintenance

strategies in the healthcare industry. More specifically, we first identify or propose the criteria

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which are important to assess the criticality of medical devices, and propose a model for the

prioritization of medical equipment for maintenance decisions. The model is a novel application

of multi-criteria decision making methodology to prioritize medical devices in a hospital

according to their criticality. The devices with high level of criticality should be included in the

hospital’s maintenance management program.

Then, we propose a method to statistically analyze maintenance data for complex medical

devices with censoring and missing information. We present a classification of failure types and

establish policies for analyzing data at different levels of the device. Moreover, a new method for

trend analysis of censored failure data is proposed. A novel feature of this work is that it

considers dependent failure histories which are censored by inspection intervals. Trend analysis

of this type of data has not been discussed in the literature.

Finally, we introduce some assumptions based on the results of the analysis, and develop

several new models to find the optimal inspection interval for a system subject to hard and soft

failures. Hard failures are instantaneously revealed and fixed. Soft failures are only rectified at

inspections. They do not halt the system, although they reduce its performance or productivity.

The models are constructed for two main cases with the assumption of periodic inspections, and

periodic and opportunistic inspections, respectively. All numerical examples and case studies

presented in the dissertation are adapted from the maintenance data received from a Canadian

hospital.

iv

To My Parents

v

ACKNOWLEDGEMENTS

I offer my sincerest gratitude to my co-supervisors, Professor Andrew K.S. Jardine and

Dr. Dragan Banjevic for giving me the opportunity to do my PhD, and for their excellent advice,

insight, support and encouragement during the course of my doctoral studies. This thesis would

not have been completed without them. Words are inadequate to express my special appreciation

to Dr. Dragan Banjevic, who helped me patiently with his excellent knowledge all throughout

these years. His enthusiasm and encouragement made me eager to succeed.

I gratefully thank Professor Dionne Aleman and Professor Tom Chau (from the

University of Toronto) who served as the members of my research committee, and provided me

with valuable comments on and insights into both the research and the presentation of the thesis.

I am very thankful to Professor David Coit (from Rutgers the State University of New

Jersey) and Professor Timothy Chan (from the University of Toronto) who served as the

examination committee members, and for their great comments.

Many thanks go to Mr. John Leung from Toronto General Hospital, who gave me access

to the maintenance data of medical devices and devoted his time and expertise to answer my

vi

questions about the data. The data he provided me made it possible to present the application of

the proposed models in the real case studies of medical equipment.

I wish to express my appreciation to Dr. Elizabeth Thompson from C-MORE lab, for her

editorial assistance and comments, which exceptionally helped me to improve my research

papers submitted or published during my PhD studies.

Thanks also to Mr. Neil Montgomery from C-MORE lab for his useful comments on my

papers, and constructive discussions during my time at the University of Toronto.

I also benefited from the advice, experience, and friendship of a great group of friends

and colleagues at the C-MORE lab during these past years. My thanks especially go to Dr. Ali

Zuashkiani, research associate, and Dr. Nima Safaei, postdoctoral fellow at C-MORE.

I wish to acknowledge the Natural Sciences and Engineering Research Council (NSERC)

of Canada, the Ontario Centre of Excellence (OCE), and the C-MORE Consortium members for

their financial support, which has made this research possible.

Last, but not least, I would like to express my deepest gratitude to my family, especially

my parents, who always believed in me and supported me. I would have achieved for less

without their continuous support. They have been my greatest mentors, and I owe all my success

to them.

Sharareh Taghipour

Toronto, February 2011

vii

PREFACE

Most of the chapters presented in this dissertation are the extensions of the following

journal and conference papers, which are published, accepted or submitted during my PhD

studies:

Journal Papers

1. Taghipour S, Banjevic D and Jardine A K S (2011). Reliability Analysis of Maintenance

Data for Complex Medical Devices. Quality and Reliability Engineering International 27(1):

71-84. This paper won the Best Student Paper Award 2010 of the American College of

Clinical Engineering (ACCE). [Used in Chapter 4]

2. Taghipour S, Banjevic D and Jardine A K S (2010). Periodic Inspection Optimization Model

for a Complex Repairable System. Reliability Engineering and System Safety 95(9): 944-952.

[Used in Chapter 5]

3. Taghipour S, Banjevic D and Jardine A K S (2010). Prioritization of Medical Equipment for

Maintenance Decisions. Journal of Operational Research Society. (published online at

viii

dx.doi.org/10.1057/jors.2010.106). This paper received the 3rd

place at the Best Student

Paper Award 2011 of the American College of Clinical Engineering (ACCE). [Used in

Chapter 3]

4. Taghipour S and Banjevic D (2011). Periodic Inspection Optimization Models for a

Repairable System Subject to Hidden Failures. IEEE Transactions on Reliability. 60(1): 275-

285. [Used in Chapter 5]

5. Taghipour S and Banjevic D (2011). Trend Analysis of the Power Law Process Using

Expectation-Maximization Algorithm for Data Censored by Inspection Intervals. Reliability

Engineering and System Safety (accepted in April 2011). [Used in Chapter 4]

6. Taghipour S and Banjevic D (2010). Optimum Inspection Interval for a System Under

Periodic and Opportunistic Inspections. IIE Transactions (submitted in December 2010).

[Used in Chapter 5]

7. Taghipour S and Banjevic D (2011). Maximum likelihood estimation from interval censored

recurrent event data. Computational Statistics & Data Analysis (submitted in March 2011).

[Used in Chapter 4]

8. Taghipour S and Banjevic D (2011). Optimal inspection of a complex system subject to

periodic and opportunistic inspections and preventive replacemenets. European Journal of

Operational Research (submitted in March 2011). [Used in Chapter 5]

ix

Papers Published in the Conference Proceedings

9. Taghipour S, Banjevic D and Jardine A K S (2008). Risk-based Inspection and Maintenance

for Medical Equipment. Proceedings of the 2008 International Industrial Engineering

Research Conference. 104-109. [Used in Chapter 3]

10. Taghipour S, Banjevic D, and Jardine A K S (2010). An Inspection Optimization Model for a

System subject to Hidden Failures. Proceedings of the 2010 ICOMS Asset Management

Conference. [Used in Chapter 5]

11. Taghipour S and Banjevic D (2011). Trend Analysis of the Power Law Process with

Censored Data. Proceedings of the Annual Reliability and Maintainability Symposium 2011.

[Used in Chapter 4]

x

TABLE OF CONTENTS

1. INTRODUCTION....................................................................................................................1

2. INSPECTION AND MAINTENANCE OF MEDICAL DEVICES ...................................6

2.1. Essential Medical Equipment .............................................................................................6

2.2. Medical Equipment Management .......................................................................................9

2.2.1. Life Cycle of Medical Equipment ............................................................................9

2.2.2. Joint Commission Standards for Medical Equipment ............................................13

2.3. Preventive Maintenance of Medical Equipment ...............................................................15

2.4. Computerized Maintenance Management Systems ..........................................................17

2.4.1. CMMS Core Modules ............................................................................................18

2.4.2. Scheduled and Non-Scheduled Work Orders .........................................................21

2.5. Concluding Remarks .........................................................................................................23

3. PRIORITIZATION OF MEDICAL DEVICES .................................................................24

3.1. Literature Review ..............................................................................................................24

3.2. Model for Prioritization of Medical Devices ....................................................................27

3.2.1. Proposed Criticality Assessment Model for Medical Equipment ..........................29

TABLE OF CONTENTS

xi

3.2.2. Descriptions of Criteria and Sub-criteria ................................................................33

3.2.3. Determining Weighting Values for Criteria and Sub-Criteria ..............................37

3.2.4. Setting Up Grades and Intensities for Each Criterion ...........................................39

3.2.5. Ranking Medical Devices .....................................................................................40

3.2.6. Numerical Example and Discussion ......................................................................42

3.2.7. Classification and Maintenance Strategies ............................................................53


4. RELIABILITY AND TREND ANALYSIS OF FAILURE DATA ...................................60

4.1. Literature Review ..............................................................................................................60

4.2. A Model Proposed for Reliability Analysis of Medical Devices’ Failure Data ...............63

4.2.1. Preliminary Analysis of Failure Data ....................................................................63

4.2.2. Hard and Soft Failures ............................................................................................67

4.2.3. Proposed Policy for Analyzing Soft and Hard Failures .........................................69

4.2.4. Trend Analysis – The Laplace Trend Test .............................................................72

4.2.5. Trend Analysis and Distribution Fitting of Failure Data .......................................73

4.3. Basic Assumptions for the System Under Study ..............................................................78

4.4. A Proposed Method for Trend Analysis of Failure Data Subject to Censoring ...............81

4.4.1. Parameter Estimation and Trend Analysis .............................................................83

4.4.2. Case Studies ...........................................................................................................90


5. INSPECTION AND MAINTENANCE OPTIMIZATION MODELS ...........................100

5.1. Literature Review ............................................................................................................100

5.2. Case 1: A System Under Periodic Inspections ...............................................................107

TABLE OF CONTENTS

xii

5.2.1. Minimal Repair of Soft Failures ...........................................................................108

5.2.1.1. Model Considering Downtime and Repair of Components .......................109

5.2.1.2. Model Considering the Number of Failures ..............................................117

5.2.1.3. The Combined Model ................................................................................126

5.2.2. Minimal Repair and Replacement of Soft Failures ..............................................128

5.2.2.1. Model Over a Finite Time Horizon ...........................................................128

5.2.2.2. Model Over Infinite Time Horizon ............................................................136

5.3. Case 2: A System Under Periodic and Opportunistic Inspections ..................................141

5.3.1. Minimal Repair of Soft and Hard Failures ...................................................143

5.3.2. Minimal Repair and Replacement of Soft Failures .......................................154

5.3.3. Minimal Repair and Replacement of Soft and Hard Failures .......................160

5.3.4. Preventive Replacement of Components with Hard Failures at

Periodic Inspections .....................................................................................168

5.4. Concluding Remarks .......................................................................................................175

6. CONCLUSION AND FUTURE RESEARCH ..................................................................178

6.1. Conclusion ......................................................................................................................178

6.2. Future Research ..............................................................................................................183

7. REFERENCES .....................................................................................................................185

8. APPENDICES ......................................................................................................................203

APPENDIX A- Qualitative grades and intensities for criteria/sub-criteria ..........................204

APPENDIX B- Calculating the lower bound for the total score value .................................209

APPENDIX C- Soft and hard failures ..................................................................................211

APPENDIX D- Rules for analyzing failure data at the system level ....................................213

TABLE OF CONTENTS

xiii

APPENDIX E- Confidence limits estimation for (the Delta method) .............................215

APPENDIX F- Analysis of failure data ................................................................................216

APPENDIX G - Proof of the recursive equation (4.11) .......................................................219

APPENDIX H- Proof of the recursive equation (4.12) ........................................................221

APPENDIX I- Proof of recursive equations (5.17) and (5.18) .............................................222

APPENDIX J- Proof of recursive equations (5.23) and (5.24) ............................................223

APPENDIX K- Upper bound for 1

( ) ( )k

k

V t V t

...............................................................224

APPENDIX L- Proof of recursive equations (5.31) and (5.32) ............................................228

APPENDIX M- Simulation results for the case of opportunistic maintenance ....................231

APPENDIX N- Proof of recursive equations (5.34) and (5.35) ...........................................236

APPENDIX O- Proof of recursive equations (5.37) .............................................................238

APPENDIX P- Audible Signal and Housing/Chassis Datasets ............................................241

APPENDIX Q- Source Codes ..............................................................................................245

INDEX .........................................................................................................................................307

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LIST OF TABLES

Table 3.1. Criteria/sub-criteria weighting values ..........................................................................38

Table 3.2a. Assessment of the devices with respect to “Function”, “Mission”, “Age” and

“Risk” .................................................................................................................. 43-44

Table 3.2b. Assessment of the devices with respect to “Recalls” and “Maintenance

requirement”, and total, normalized and transformed criticality scores .............. 44-45

Table 3.3. Assessment of the devices with respect to “Mission criticality” sub-criteria ........ 45-46

Table 3.4. Risk assessment of the failure modes..................................................................... 46-49

Table 3.5. Consequence assessment of the failure modes ....................................................... 49-52

Table 3.6. Proposed classes and the thresholds .............................................................................54

Table 3.7. Heuristic reasoning for maintenance decisions ............................................................57

Table 4.1. The total number of scheduled and non-scheduled work orders and findings at

inspection .....................................................................................................................64

Table 4.2. Qualitative tests results extracted from scheduled and non-scheduled work orders ....65

Table 4.3. Quantitative tests results extracted from scheduled and non-scheduled work orders ..66

Table 4.4. PM checks results extracted from scheduled and non-scheduled work orders ............66

Table 4.5. Soft and hard failure frequencies at scheduled and non-scheduled work orders .........69

LIST OF TABLES

xv

Table 4.6. Estimated parameters of Weibull distribution for times between events –

system level ..................................................................................................................75

Table 4.7. Estimated 95% confidence limits for the Weibull distribution parameters .................76

Table 4.8. An overview of the results obtained from statistical analysis at different levels .........77

Table 4.9. Sample data for three units’ failure and censoring events ...........................................91

Table 4.10. Summary of the histories for Audible signals ............................................................91

Table 4.11. Estimation and testing results obtained for audible, housing and battery units .........93

Table 4.12. Summary of the chassis/housing failures ...................................................................94

Table 4.13. Summary of the battery failures .................................................................................95

Table 4.14. Estimated standard errors ...........................................................................................96

Table 4.15. Total number of iterations in the complete EM, modified EM and the mid-points

method..........................................................................................................................97

Table 5.1. Parameters of the power law intensity functions and costs ........................................116

Table 5.2. Calculated components of the cost function ..............................................................124

Table 5.3. Parameters of the power law intensity functions, costs and ( )jr x parameters for

different components .................................................................................................134

Table 5.4. Different downtime penalty costs ..............................................................................140

Table 5.5. Parameters of the power law intensity functions, and costs for components with soft

and hard failure ..........................................................................................................151

Table 5.6. Expected cost calculated for 1,2,...,12 in Figure 5.14 .........................................152

Table 5.7. Costs and parameters of the probability function for minimal repair of components

with soft failure ..........................................................................................................158

Table 5.8. Expected costs calculated for 1,2,...,12 in Figures 5.15 and 5.16 .......................158

Table 5.9. of the power law intensity functions, and costs for components with soft

LIST OF TABLES

xvi

and hard failure ..........................................................................................................166

Table 5.10. Parameters ja and jb for different components .....................................................166

Table 5.11. Replacement costs considered in Figure 5.18 ..........................................................167

Table 5.12. The costs for different components with hard failure ..............................................173

Table A.1. Pairwise comparison matrix for the grades of criterion “Function” .........................204

Table A.2. Calculating intensities for the grades of criterion “Function” ...................................205

Table A.3. Function grades and intensities .................................................................................205

Table A.4. Utilization grades and intensities ..............................................................................206

Table A.5. Availability of alternatives grades and intensities .....................................................206

Table A.6. Age grades and intensities .........................................................................................206

Table A.7. Failure frequency grades and intensities ...................................................................206

Table A.8. Failure detectability grades and intensities ...............................................................207

Table A.9. Downtime grades and intensities ...............................................................................207

Table A.10. Cost of repair grades and intensities .......................................................................207

Table A.11. Safety and environment grades and intensities .......................................................207

Table A.12. Recalls and hazards grades and intensities ..............................................................208

Table A.13. Maintenance requirements grades and intensities ...................................................208

Table B.1. Calculating the minimum total score value ....................................................... 209-210

Table C.1. Components/features that can be ignored since they have no failure reported .........211

Table C.2. Components/features that have more failures at non-scheduled work orders (hard

failures) ......................................................................................................................211

Table C.3. Components/features that have more failures at scheduled work orders

(soft failures) ..............................................................................................................212

Table F.1. Estimated parameters of Weibull distribution for times between events – hard failures

LIST OF TABLES

xvii

LA=6.12 (systems are degrading) ..............................................................................216

Table F.2. Estimated parameters of Weibull distribution for times between events – soft failures

LA=11.08 (systems are degrading) ............................................................................217

Table F.3. Estimated parameters of Weibull distribution for times between events – Alarms

failures (hard) LA=4.13 (systems are degrading) ....................................................218

Table F.4. Estimated parameters of Weibull distribution for times between events –

Chassis/Housing failures (soft) LA=11.40 (systems are degrading) .......................218

Table M.1. Estimated expected number of minimal repairs and uptime and their standard errors

calculated for the components with soft failure, for 1,2,...,12

(used in Fig. 5.13) ............................................................................................ 231-232

Table M.2. Expected number of minimal repairs and replacements, and the uptimes and their

standard errors calculated for the components with soft failure, for 1,2,...,12

(used in Figures 5.15 and 5.16) ........................................................................ 234-235

Table P.1. Audible signal dataset for 80 units’ failure and censoring events ..................... 241-242

Table P.2. Housing/Chassis dataset for 38 units’ failure and censoring events .................. 243-244

xviii

LIST OF FIGURES

Figure 2.1. A typical life cycle of medical equipment ..................................................................10

Figure 2.2. Acquisition iceberg (source: Cheng and Dyro, 2004) ................................................11

Figure 2.3. Major tests and actions performed during a device’s life cycle .................................16

Figure 2.4. CMMS overview (source: Cohen, 2008) ....................................................................18

Figure 2.5. A typical work order ...................................................................................................22

Figure 3.1. The RCM process .......................................................................................................25

Figure 3.2. Decision hierarchy for prioritization of medical devices ...........................................29

Figure 3.3. Hierarchy for mission criticality .................................................................................32

Figure 3.4. Hierarchy for risk assessment of a device ..................................................................32

Figure 4.1. General Infusion Pump ...............................................................................................63

Figure 4.2. Pareto analysis for qualitative tests results .................................................................67

Figure 4.3a. The policy for analyzing failures at the system level ...............................................70

Figure 4.3b. The policy for dealing with “no failure” devices at the system level .......................70

Figure 4.4. The policy for dealing with hard failures – component level .....................................71

Figure 4.5. The MCF plot for the failures at the system level (0.5*MTBF) .................................74

LIST OF FIGURES

xix

Figure 4.6. Scheme of hard failures and scheduled inspections in a cycle ...................................81

Figure 4.7. Initial age 0y and failure times of a unit ....................................................................82

Figure 4.8. Lower and upper bounds for failure times and the last censoring time ......................83

Figure 4.9. A cracked door hinge in an Infusion Pump ................................................................93

Figure 4.10. Damaged sealed lead acid battery caused by battery overcharging .........................94

Figure 5.1. Structure of the developed inspection optimization models .....................................107

Figure 5.2. Scheme of scheduled inspections in a cycle .............................................................108

Figure 5.3. Initial age t and failure times of component j .......................................................111

Figure 5.4. Total costs for different inspection frequencies (first model) ..................................117

Figure 5.5. Total costs of scheduled inspections for different inspection frequencies (threshold

model) ........................................................................................................................123


model); For penalty cost $1700Pc , 9n is the optimal number of inspections

when $1500Pc .....................................................................................................125

Figure 5.7. Total costs for different inspection frequencies (combined model) .........................127

Figure 5.8. Initial age and failure times of component j .............................................................130

Figure 5.9. Total costs for different inspection intervals ( ) – finite horizon ...........................135

Figure 5.10. Total costs for different inspection intervals ( ) – infinite horizon ......................140

Figure 5.11. 5 is obtained with downtime penalty costs from the last column

of Table 5.4 ................................................................................................................141

Figure 5.12. Scheme of scheduled and non-scheduled inspections in a cycle ............................142

Figure 5.13. Hard and soft failures in the first inspection interval .............................................146

Figure 5.14. Expected costs for different inspection intervals ( ), assuming soft and hard

failures are minimally repaired ..............................................................................153

LIST OF FIGURES

xx

Figure 5.15. Expected costs for different inspection intervals ( ), assuming soft failures are

either minimally repaired or replaced ...................................................................159


either minimally repaired or replaced with different replacement costs ...............160


either minimally repaired or replaced ...................................................................167

Figure 5.18. Expected costs for inspection intervals ( ), assuming soft and hard failures are

either minimally repaired or replaced with the replacement costs given in Table

5.11 ..........................................................................................................................168

Figure 5.19. Expected costs for different inspection intervals ( ), assuming preventive

replacement of component with hard failure, with the costs given in the third

column of Table 5.12 ................................................................................................174

Figure 5.20. Expected costs for different inspection intervals ( ), assuming preventive

replacement of component with hard failure, with the costs given in the last

column of Table 5.12 ................................................................................................174

Figure F.1. The MCF plot of hard failures .................................................................................216

Figure F.2. The MCF plot of soft failures...................................................................................217

xxi

LIST OF APPENDICES

APPENDIX A- Qualitative grades and intensities for criteria/sub-criteria ..........................204

APPENDIX B- Calculating the lower bound for the total score value .................................209

APPENDIX C- Soft and hard failures ..................................................................................211

APPENDIX D- Rules for analyzing failure data at the system level ....................................213

APPENDIX E- Confidence limits estimation for (the Delta method) .............................215

APPENDIX F- Analysis of failure data ................................................................................216

APPENDIX G - Proof of the recursive equation (4.11) .......................................................219

APPENDIX H- Proof of the recursive equation (4.12) ........................................................221

APPENDIX I- Proof of recursive equations (5.17) and (5.18) .............................................222

APPENDIX J- Proof of recursive equations (5.23) and (5.24) ............................................223

APPENDIX K- Upper bound for 1

( ) ( )k

k

V t V t

...............................................................224

APPENDIX L- Proof of recursive equations (5.31) and (5.32) ............................................228

APPENDIX M- Simulation results for the case of opportunistic maintenance ....................231

APPENDIX N- Proof of recursive equations (5.34) and (5.35) ...........................................236

LIST OF APPENDECES

xxii

APPENDIX O- Proof of recursive equation (5.37) ..............................................................238

APPENDIX P- Audible Signal and Housing/Chassis Datasets ............................................241

APPENDIX Q- Source Codes ..............................................................................................245

xxiii

ABBREVIATIONS

AHP : Analytical Hierarchy Process

CMMS : Computerized Maintenance Management System

CT : CAT Scan

EM : Expectation-Maximization

EM number : Equipment Management Number

FMEA : Failure Mode and Effect Analysis

HPP : Homogeneous Poisson Process

JCAHO : Joint Commission on Accreditation of Healthcare Organizations

MADM : Multi-Attribute Decision Making

MCDM : Multi-Criteria Decision Making

MCF : Mean Cumulative Function

MEMP : Medical Equipment Management Program

MLE : Maximum Likelihood Estimate

MRI : Magnetic Resonance Imaging

NHPP : Non-Homogeneous Poisson Process

ABBREVIATIONS

xxiv

OEM : Original Equipment Manufacturer

PM : Preventive Maintenance

PRN : Probability Risk Number

RCM : Reliability Centered Maintenance

SPI : Safety and Performance Inspection

UPS : Uninterruptible Power Supplies

WHO : World Health Organization

1

1. INTRODUCTION

Nowadays from 5,000 to more than 10,000 different types of medical devices can be

found in an average to large sized hospital. Hospitals and healthcare organizations must ensure

that their critical medical devices are safe, accurate, reliable and operating at the required level of

performance. To achieve these objectives, hospitals must establish and regulate a Medical

Equipment Management Program (MEMP) which describes risk management of medical

equipment. Inspection and preventive maintenance is a fundamental aspect of such a program

and it should be reviewed and improved continuously in order to keep up with the pace of

today’s technological improvement of medical equipment, as well as increasing expectations of

healthcare organizations.

Cost-effective and efficient maintenance decisions could be made after thoroughly

understanding, implementing and leading maintenance excellence in healthcare organizations.

Maintenance excellence is the balance of performance, risk, resource inputs and cost to reach to

an optimal solution (Campbell and Jardine, 2001).

Although maintenance strategies and techniques have been significantly improved in the

last two decades, most of hospitals and healthcare organizations do not benefit from maintenance

1. INTRODUCTION 2

excellence as much as other industries. Unnecessary and excessive preventive maintenance

could be also loss-making likewise inadequate level of maintenance. The time, which is spent

doing the unnecessary preventive maintenance, is robbing an organization of a fraction of one of

its most vital resources (Keil, 2008).

Since 2004, when Joint Commission on Accreditation of Healthcare Organizations

(JCAHO) introduced standard EC.6.10 (JACAHO, 2004), hospitals in US have started adopting

their maintenance programs to put their maintenance resources where most needed. This

standard allows hospitals to not have schedule inspection or maintenance tasks for certain pieces

or types of medical equipment, if these tasks are not needed for safe and reliable operation

(Wang et al., 2006).

However, in Canada, most, if not all healthcare organizations include all their medical

equipment in their maintenance program and just follow manufacturers’ recommendations for

preventative maintenance. Current maintenance strategies employed in hospitals and healthcare

organizations have difficulty in identifying specific risks and applying optimal risk reduction

activities (Rice, 2007).

Moreover, even though the use of reliability engineering tools is well established, their

application to the medical industry is new. Most research in this area merely suggests how to

assess or improve the reliability of devices in their design or manufacturing stages. To this point,

best maintenance strategies for medical equipment in their operating context have not been

considered.

The research reported in this thesis aims to address this gap and proposes methods to

improve current maintenance strategies in the healthcare industry. We first propose a

prioritization model which can be used to decide what medical devices should be included in the

1. INTRODUCTION 3

hospital’s maintenance management program. Then, to develop inspection/maintenance

optimization models, we start with the actual maintenance data received from the Computerized

Maintenance Management System (CMMS) of a hospital. We learn about the nature of the data,

inspection and maintenance actions, and different types of failures. The results of our

observations lead us to find out what methods or models needed to be developed, and what

assumptions should be made based on the evidences in data.

We establish policies to analyze the data statistically due to the presence of censored or

incomplete data, and propose a new method for trend analysis of such data. Then, we make some

assumptions according to the results of statistical analysis, and develop several inspection

optimization models for complex repairable systems such as medical equipment.

The three main contributions of this research can be listed briefly as follows:

- A new model proposed to decide what equipment to include in the MEMP of a hospital.

- Models proposed for reliability and trend analysis of failure data in the presence of censoring.

- Several inspection optimization models developed for a repairable system under periodic

inspections.

The thesis is structured as follows. Chapter 2 presents a general overview of medical

devices, their typical life cycle, and inspection and maintenance management. In Chapter 3, we

develop a model for the prioritization of medical equipment for maintenance decisions. This

model uses a multi-criteria decision making methodology to prioritize medical devices in a

hospital according to their criticality. Devices with lower criticality scores can be given lower

priority in the MEMP. However, for devices with high scores, a root cause analysis should be

performed to find the main causes of having high criticality number. Appropriate actions such as

"preventive maintenance", "user training", "redesign the device", etc. should be taken to reduce

1. INTRODUCTION 4

their risk. Moreover, some guidelines for selecting appropriate maintenance strategies for

different classes of devices are proposed. The information on 26 different medical devices is

extracted from a hospital’s maintenance management system to illustrate an application of the

proposed model.

In Chapter 4, we develop a method to statistically analyze the maintenance data for

complex medical devices with censoring and missing information. We present a classification of

different types of failures and establish policies for analyzing data at system and component

levels taking into account the failure types. As a case study, we conduct the reliability analysis of

a general infusion pump. Moreover, in Chapter 4 a method for trend analysis of failure data

subject to censoring is developed. Trend analysis investigates any decreasing or increasing trend

in the historical failure data for a system or a group of similar systems. Most trend analysis

methods proposed in the literature assume that the failure data is statistically complete and

failure times are known; however, in many situations, such as hidden failures, failure times are

subject to censoring. We assume that the failure process follows a non-homogeneous Poisson

process with a power law intensity function, and the failure data are subject to left, interval and

right censoring. We use the likelihood ratio test to check for the trends in the failure data. More

specifically, we use the EM algorithm to find the parameters that maximize the data likelihood in

the case of no trend and trend assumptions. A recursive procedure is used to calculate the

expected values in the E-step. The proposed method is applied to several components of a

general infusion pump.

In Chapter 5, we use the results of the analysis described in Chapter 4, as basic

assumptions, to develop several periodic inspection optimization models for a complex

repairable system. We assume that the components of the system are subject to soft and hard

1. INTRODUCTION 5

failures. Hard failures are either self-announcing or the system stops when they take place; so

they are fixed instantaneously. Soft failures are unrevealed and can only be detected at

inspections; they do not stop the system functioning, although they reduce the system’s

performance.

First, several models are developed based on the simpler assumption that the system is

periodically inspected and the soft failures are rectified only at the periodic inspection moments.

We start with the assumption of minimal repair of components with soft failure, and construct

two models that respectively take into account the downtime of a component, and the number of

soft failures exceeding a pre-defined threshold. Then, the model with downtime consideration is

extended to two optimization models over finite and infinite time horizons, which assume

possibility of minimal repair or replacement of a component at failure.

In the more complicated models, we assume that the soft failures can also be rectified at

non-scheduled inspections, which are performed due to the occurrence of hard failures. We call

the non-scheduled inspections “opportunistic inspections” for soft failures. We develop several

models based on the assumption of minimal repair or/and replacement of components with soft

and hard failures. Moreover, an optimization model is proposed for the case that components

with hard failure can be preventively replaced at periodic inspections according to some age

dependent probabilities. The main contribution of all these models is in calculating the excepted

cost with delayed replacement or minimal repair of a component.

The conclusions of the research and some guidelines for the future research are given in

Chapter 6.

6

2. INSPECTION AND MAINTENANCE OF MEDICAL DEVICES

Medical devices are fundamental components of modern health services used for

diagnosis, treatment, and monitoring of patients. They are progressively being deployed to

increase the capabilities of health diagnostic and treatment services. On the other hand, the

potential to manage and maintain medical equipment in most developing countries remain rather

weak (World Health Organization, 1998). It is required to have practical methods and powerful

management strategies to meet the challenges of ever increasing number and use of medical

devices.

2.1. Essential Medical Equipment

Basic medical equipment is widely used in the healthcare facilities. This essential

equipment is supportive to provide primary healthcare to the public (Cheng, 2004a). World

Health Organization (WHO) classifies essential medical equipment in four main categories. The

list given in each category includes the devices required for a specified health service delivery.

The type of equipment is significantly dependent on the local health practice, physical

characteristics and culture of the population (World Health Organization, 1998).

2.1. Essential Medical Equipment 7

1. Diagnostic imaging equipment

Diagnostic imaging equipment is used to take pictures, which help physicians to diagnose

a patient’s medical condition (McKay, 1986).

o Diagnostic X-ray equipment

o Ultra-sound equipment

Ultra-sound equipment may be less frequently used, and is usually added if the budget

is available in a hospital.

2. Laboratory equipment

A variety of laboratory equipment is used for analysis or measurement purposes.

o Microscope

o Blood counter

o Analytical balance

o Colorimeter/spectrophotometer

o Centrifuge

o Water bath

o Incubator/oven

o Refrigerator

o Distillation and purification apparatus

3. General electro-medical equipment

o Portable electrocardiograph

o External defibrillator

o Portable anesthesia unit

2.1. Essential Medical Equipment 8

o Respirator

o Dental chair unit

o Suction pump

o Operating theatre lamp

o Diathermy unit

4. Other support equipment

o Operating theatre table

o Delivery table

o Autoclave- for general sterilization.

o Small sterilizer-for specific services (e.g., dentistry)

o Cold chain and other preventive medical equipment

o Electrical generator

o Electrical power regulator

o Air conditioner, dehumidifier

o Refrigerator

o Ambulance-four-cylinder diesel, four-wheel drive vehicle equipped with medical

equipment for emergencies; complete accessories, spare tires and tools

o Gynecological examination table

o Small, inexpensive equipment and instruments

Individual hospital authorities decide which type and what number of these devices are

required for their own health service purposes.

2.2. Medical Equipment Management 9

2.2. Medical Equipment Management

Medical Equipment Management Program (MEMP) is established in hospitals to provide

safe and reliable operation of medical equipment and promote its effective utilization (Stiefel,

2009). This program defines procedures and policies to manage activities related to medical

equipment, from their selection and acquisition to decommission. MEMP ensures that devices

can provide reliable and accurate information to clinicians, operate safely for patients, and are

used to their fullest capacity (University of Michigan Hospitals, 2010).

The life cycle of medical devices should be thoroughly considered for effective

management. Deficiencies in managing each stage of the life cycle, especially in the earlier

phases can cause more problems in the succeeding stages. For example, if the maintenance

capabilities are considered during acquisition stage, it can hinder the challenges that might be

faced during the maintenance stage of the equipment.

2.2.1. Life Cycle of Medical Equipment

A typical life cycle of medical equipment has the stages shown in Figure 2.1 (Cheng and

Dyro, 2004; also World Health Organization, 1998). Proper management of each phase can have

a positive impact on the others.

Planning

In the planning stage, distinct policies on acquisition, utilization and maintenance of

medical equipment are clearly outlined. This can significantly minimize the problems arising

from the contracts, spare parts and maintenance of the equipment (World Health Organization,

1998). For example, considering the skill level of operators, ensure that only appropriate

technology is acquired. Most costs incurred during the life cycle of a device are hidden from


view (Figure 2.2). For proper planning of a large number of devices, a full consideration of all

elements in the equipment’s life cycle is required.

Figure 2.1. A typical life cycle of medical equipment

Planning

Acquisition

Delivery and Incoming Inspection

Inventory and Documentation

Installation, Commissioning

and Acceptance

Monitoring of Use and Performance

Maintenance

Replacement or Disposal

Training of Users and Operators


Figure 2.2. Acquisition iceberg (source: Cheng and Dyro, 2004)

Acquisition

Evaluation and procurement (Harding and Epstein, 2004a; 2004b) are two main aspects

of the acquisition phase. Evaluation process includes safety, performance and maintainability

assessment of devices. Moreover, the models and manufacturers of equipment are standardized.

In the procurement process, it is emphasized that the supplier must supply operating and service

manuals, and must provide operation and service training and essential spare parts.

Delivery and Incoming Inspection

Incoming devices should be checked carefully for possible damages in the shipment

process, conformity with the purchase order, and all required accessories, spares, and documents.

Inventory and Documentation


Inventory and documentation are important aspects of equipment management and

standardization (Barerich, 2004; Cohen and Cram, 2004). Inventory entries should include all

accessories, spares, and manuals of each device.

Installation, Commissioning and Acceptance

In-house technical staff or the suppliers can perform installation and commissioning

stage. In the latter, in-house staff should monitor the process and record it in the equipment

service history.

Training of Users and Operators

Proper training of users and operators assures effectiveness and safety of medical

devices, and decreases maintenance errors.

Monitoring of Use and Performance

In-house technical staff should act as a link between user and supplier and monitor the

supplier’s technical services.

Maintenance

Medical equipment must always be maintained in working condition, and calibrated

periodically for safety and accuracy (McCauley, 2004; Cheng, 2004a).

Replacement and Disposal

When a medical device is old and its spares run out of supplies, it should be replaced and

disposed according to the safety procedures (Cheng, 2004b).


The management of each stage mentioned above can be enriched if the resources are

available. For example, in addition to corrective maintenance, preventive maintenance can be

added to enrich the maintenance element of the management plan (Cheng and Dyro, 2004).

2.2.2. Joint Commission Standards for Medical Equipment

In accordance with the life cycle phases of medical equipment, biomedical/clinical

engineers should comply continuously with two primary Joint Commission medical equipment

standards EC.02.04.01 and EC.02.04.03. Standard EC.02.04.01 must be used by healthcare

organizations to manage safety and security risks. Standard EC.02.04.03 presents guideline to

inspects, tests, and maintains medical equipment. The elements of performance for these two

standards are as follows (JCAHO, 2008):

Standard EC.02.04.01: The organization manages safety and security risks.

1. The organization has a systematic approach to selecting and acquiring medical

equipment.

2. The organization maintains either a written inventory of all medical equipment or a

written inventory of selected equipment categorized by physical risk associated with use

and equipment incident history. The organization evaluates new types of equipment

before initial use to determine whether they should be included in the inventory.

3. The organization identifies the activities for maintaining, inspecting, and testing for all

medical equipment on the inventory. Organizations may use different maintenance

strategies based on the type of equipment. Strategies must include defined intervals for

inspecting, testing, and maintaining equipment on the inventory [bolded by Sharareh

Taghipour]. Defined intervals are based on criteria such as manufacturers'


recommendations, risk levels, and current organization experience. In addition, predictive

maintenance, reliability centered maintenance, interval-based inspections, corrective

maintenance, or metered maintenance [means maintaining according to the working age

of a device] may be selected to ensure reliable performance.

4. The organization identifies frequencies for inspecting, testing, and maintaining medical

equipment on the inventory based on criteria such as manufacturers’ recommendations,

risk levels, or current organization experience.

5. The organization monitors and reports all incidents in which medical equipment is

suspected in or attributed to the death, serious injury, or serious illness of any individual,

as required by the Safe Medical Devices Act of 1990 (Samuel, 1991).

6. The organization has written procedures to follow when medical equipment fails,

including using emergency clinical interventions and backup equipment.

7. For organizations that provide the technical component of advanced diagnostic imaging

and elect to use The Joint Commission CMS imaging supplier accreditation option (Joint

Commission Accreditation Ambulatory Care, 2010): The organization identifies activities

and frequencies to maintain the reliability, clarity, and accuracy of the technical quality

of diagnostic images produced.

Standard EC.02.04.03: The organization inspects, tests, and maintains medical equipment.

1. Before initial use of medical equipment on the medical equipment inventory, the

organization performs safety, operational, and functional checks.

2. The organization inspects, tests, and maintains all life-support equipment. These

activities are documented.

2.3. Preventive Maintenance of Medical Equipment 15

3. The organization inspects, tests, and maintains non–life-support equipment identified on

the medical equipment inventory. These activities are documented.

4. The organization conducts performance testing of and maintains all sterilizers. These

activities are documented.

5. The organization performs equipment maintenance and chemical and biological testing of

water used in hemodialysis. These activities are documented.

2.3. Preventive Maintenance of Medical Equipment

Medical devices are often complex repairable systems consisting of a large number of

interacting components, which perform a system’s required functions. A repairable system,

upon failure, can be restored to satisfactory performance by any method except replacement of

the entire system (Ascher and Feingold, 1984).

Medical devices usually undergo several types of tests/inspections during their life cycles

as described here (Atles, 2008):

Acceptance Test

A series of qualitative and quantitative tasks designed to verify the safety and

performance of newly received equipment, as well as conformity to applicable codes, regulations

and standards.

Operational Check

Visual and operational check of the equipment’s safety and functionality typically

performed at the beginning of the day or work period, or just before using equipment on a

patient.

2.3. Preventive Maintenance of Medical Equipment 16

Safety and Performance Inspection (SPI)

A set of qualitative and quantitative tasks designed to verify the safety and performance

of each piece of equipment by detecting potential and hidden failures and taking appropriate

actions.

After accomplishing the acceptance test for a newly received device, SPIs are scheduled

to be performed periodically. If any problem is found at inspection, corrective actions are taken

to restore the device or its defective parts to an acceptable level. In addition, a set of failure

preventive actions may be taken to prevent future failures and/or restore device function; these

include part replacement, calibration, lubrication, etc. to address age or usage related

deterioration.

When a device fails while it is in use, the operator reports the problem, and again

appropriate actions (corrective maintenance) are taken. When the repair of a device is no longer

technically feasible or cost effective, replacement becomes the best or the only option (Atles,

2008). Figure 2.3 describes major tests and actions performed during a device’s life cycle.

Figure 2.3. Major tests and actions performed during a device’s life cycle

SPI

SPI

SPI

Retirement/

Replacement

SPI

SPI

SPI

Acceptance

Test

SPI (Safety and Performance Inspection)

Corrective Maintenance

2.4. Computerized Maintenance Management Systems 17

2.4. Computerized Maintenance Management Systems

Computerized Maintenance Management Systems (CMMS) are database applications in

an organization that assist in planning and management functions required for effective

maintenance (Gulati and Smith, 2009). A CMMS to a medical device is similar to an electronic

medical record to a patient. It provides the information required for assets management, resource

management, financial management, workload, workflow management and regulatory

compliance (Cohen, 2008). A CMMS is essential in most healthcare organizations due to the

Joint Commission for the Accreditation of Healthcare Organization requirements under the

Environment of Care Standards (Cram, 1998).

One of the most important steps in implementing either a computerized or non-

computerized equipment management system in a hospital, is to have a complete and accurate

inventory of all equipment in the MEMP. The inventory should also include the devices that are

serviced by other organizations, but they still must be tracked. Each device must have an

equipment control number, which is labeled to the device. Equipment control number can be the

hospital asset or property number, or can be an independent number assigned to the device. The

inventory should be frequently updated as new devices are added. Without an accurate inventory

system, tracking equipment maintenance and repair, alerts and recalls is almost impossible

(Cohen and Cram, 2004).

The core functionalities of a CMMS consist of the modules shown in Figure 2.4 (Cohen,

2008; also see Bagadia, 2006). More details of the CMMS modules are explained in Section

2.4.1.


Figure 2.4. CMMS overview (source: Cohen, 2008)

2.4.1. CMMS Core Modules

Inventory control

Inventory control is the core module of a CMMS. It allows an operator to track the

inventory movement, i.e. moving in and out of an item from the inventory or from a location to

another (Sapp, 2010). To proper planning of service and repair/replacement of each device,

healthcare organizations should know the quantity, type, age and the other information related to

the device.

Work order management system

Work order or work request is an electronic document used to schedule routine inspection

and maintenance. Work order management system is the heart of the CMMS. Using this module


a work order can be created, followed up and completed. It stores all corrective and preventive

maintenance requests. It keeps track of the initial customer request, device information, requestor

information, date and time of the request, nature of the problem, its urgency, and a summary of

the assistance provided so far. All activities performed to a work order should be clearly

documented and go into the device’s repair and maintenance history.

If the CMMS is equipped with an invoice system, a bill is issued and documented for the

work order. The work, which is performed, by vendors or external service providers should be

also tracked to have a complete history of all devices.

Scheduling/planning

This module determines the work required to be performed to satisfy a request. It

specifies the most efficient way to perform the work, the schedules and the required resources.

Several scheduling procedures are usually considered in the CMMS. It includes periodic or fixed

scheduling, floating scheduling, and synchronized scheduling by feature such as device type,

location, specialty, and others (Cohen, 2008). Periodic scheduling is scheduling of an activity,

such as preventive maintenance periodically, regardless the time of the last action. Floating is

scheduling of an activity based on the last time the action has been completed for the device, and

the conditions of the device. Synchronized scheduling by feature allows to schedule actions

according to a technician’s expertise, type of the device, department, etc.

Vendor management

In some hospitals, the most sophisticated equipment such as MRI, scanner, analyzers,

etc., are maintained and services by the original equipment manufacturer (OEM). A good CMMS

should integrate the vendor and in-house work to have complete histories of all equipment.


Vendor management system should also include the contracts and purchase management

subsystems to allow for recording the contracts with the external service providers and purchase

transactions made from the vendor.

Parts management

Most organization uses a just-in-time process for ordering repair parts. Parts usually are

divided into three categories: stock parts, contract parts, and noncontracted parts (Cohen, 2008).

Stock parts are those, which are purchased and kept in stock by an organization to be used when

needed. They are not usually immediately assigned to a specific work order. Contract parts are

those included in a prepaid contract. Prepaid contracts are made with external service providers

for a certain period. Noncontracted parts are those, which are purchased usually just in time for a

specific work order or maintenance action.

Preventive maintenance

Preventive maintenance is a fundamental module of a CMMS. It generates PM work

orders, prioritizes them based on some given criteria, and manipulates them until they are

accomplished.

Labor

Tracking labor resources can be performed using this module. It includes the information

of all maintenance personnel and their expertise.

Purchasing

Purchasing module is to initiate the requisition of parts and materials against a work order

and track the delivery and cost data of the item when the part or material arrives (Sapp, 2010).

Budgeting


Budgeting module is integrated with the planned resources (labor hours, parts and

materials) usage on the work orders. It includes the labor, parts and materials rates to calculate

or estimate the costs associated with a work order.

2.4.2. Scheduled and Non-Scheduled Work Orders

The maintenance and inspection data are usually available in the CMMS of a hospital,

stored in either scheduled or non-scheduled work orders. Scheduled work orders are used for

routine tests (SPIs); however, when a device fails or has a defective part, a non-scheduled work

order is requested to fix the problem. Both scheduled and non-scheduled work orders include the

basic information of a device and a test checklist designed for a particular class of device. The

checklist contains qualitative and quantitative tests; technicians or clinical engineers should use

this list to ensure that all necessary tests and checks are accomplished. Figure 2.5 shows a sample

of a work order created for corrective maintenance (non-scheduled work order).

Qualitative tests mainly consist of visual inspection of the main parts/components of a

device. For example, for a general infusion pump, these include testing its chassis/housing, line

cord, battery/charger, etc. Quantitative tests include measuring parameters of a device to check

whether the parameters are in control. Grounding resistance and maximum leakage currents are

among the quantitative tests for a general infusion pump.

A work order presents all PM checks and actions, such as cleaning, lubricating or

replacement of a device or its parts. A work order is also created for an acceptance test of a

newly received device.


Figure 2.5. A typical work order (designed for a general infusion pump)

“P” (pass), “F” (failed), or “N” (not applicable) are the possible results of each qualitative

or quantitative test on the work order. When a test is performed on a component, and it is found

to be non-defective, the result is “P”; however, in the case of failure of a part, the result is shown

as “F”. Since a general test checklist is designed for a class of equipment, some qualitative or

quantitative tests may not be applicable to particular devices in that class; for these devices, the

test result is given as “N”. Quantitative and qualitative tests are specifically performed for

components/features of a device, but PM checks show the actions performed at the device level.

Currently, most hospitals merely follow manufacturers’ recommended intervals for

periodic SPI of devices. SPI intervals differ from 6 to 12 months depending on the device type

and risk level (U.S. FDA/CDRH, 2005). Class III (high risk) devices such as defibrillators should

2.5. Concluding Remarks 23

be inspected every 6 months, and class II (medium risk) devices like ECGs should be inspected

annually. However, the optimality and even the necessity of these recommended intervals are

questionable. It is essential to establish an evidence-based inspection or maintenance regimen

derived from analysis of field data.

2.5. Concluding Remarks

According to standard EC.02.04.01 for Medical Equipment (see Section 2.2.2),

healthcare organizations must identify maintenance activities to maintain, inspect, and test all

medical equipment on the inventory. They must employ appropriate strategies, including the

intervals for maintenance activities. However, not much research has been presented in the

literature to address proper strategies and the methods for implementing them. In the research

reported in this dissertation, we propose several strategies, which can be used to meet some

requirements of standard EC.02.04.01. We propose a model to decide on inclusion of medical

devices in the MEMP, methods for trend analysis of maintenance data, and finally several

models to find the optimal periodic inspection interval for a repairable system such as a medical

device. The proposed models and methods are discussed in the rest of the thesis.

24

3. PRIORITIZATION OF MEDICAL DEVICES

3.1. Literature Review

The ever-increasing number and complexity of medical devices demands that hospitals

establish and regulate a Medical Equipment Management Program (MEMP) to ensure that

critical devices are safe and reliable and that they operate at the required level of performance.

As fundamental aspects of this program (Stiefel, 2009) inspection, preventive maintenance, and

testing of medical equipment should be reviewed continuously to keep up with today’s

technological improvements and the increasing expectations of healthcare organizations.

No longer content to merely follow manufacturers’ recommendations, hospital clinical

engineering departments all around the world including Canada, Australia, and United States

have begun to employ more efficient and cost-effective maintenance strategies. Gentles et al

[http://CESOData.ca accessed 27 April 2010] have begun to develop a unique database to collect

comparative data on inventory and maintenance of the most critical devices used in hospitals

across Canada and the United States. This project will provide a large statistical failure data set

which could be used to establish optimum intervals for routine maintenance scheduling.

Ridgway et al (2009) provide concise guidelines for maintenance management of medical

3.1. Literature Review 25

equipment and address methods, which have been used for a long time in other industry

segments, such as Reliability Centered Maintenance (RCM). RCM is a structured methodology

for determining the maintenance requirement of a physical asset in its operating context through

a thorough and rigorous decision process, as shown in Figure 3.1 (Jardine and Tsang, 2006):

Figure 3.1. The RCM process

Steps 2 to 5 in Figure 3.1 show the process of Failure Mode and Effect Analysis (FMEA).

The results of FMEA are used to select appropriate maintenance tactics using RCM logic for the

various functional failures. Asset criticality analysis is the first step of applying RCM in an

organization, especially when a large number of different devices exist and the worst problems in

terms of failure consequences are not obvious.

Criticality is a relative measure of the importance of an object based on some factors

considered in a particular context. For example, the importance or criticality of a failure mode

depends on the combined influences of several factors such as severity, probability, detectability,

cost and timing, and all these factors play a part in determining the amount of attention that a

failure mode requires (JACAHO, 2005). Asset criticality is a function of the operational impact

to the organization’s mission due to the loss, damage, or destruction of an asset (Vellani, 2006).

Dekker et al (1998) define the equipment criticality as a function of the use of equipment, rather

than of equipment itself and explain how a certain device may be in one case critical and in

another auxiliary.

3. Define

Functional

Failures

4. Identify

Failure

Modes and

Causes

5. Identify

Failure Effects

&

Consequences

6. Select

Tactics Using

RCM Logic

1. Select

Equipment

(Assess

Criticality)

2. Define

Functions

7. Implementation & Refine

the Maintenance Plan


Significant and critical assets should be identified and prioritized, and many techniques

have been developed for criticality assessment of devices. Most use some variation of the

probability risk number or PRN (Moubray, 1997), a product of the probability of failure of an

asset, severity of the consequence of the failure, and detectability of the failure:

Probability of failure Severity DetectabilityPRN (3.1)

In hospitals, risk is a criterion in criticality assessment of medical devices, but the

definition of risk differs from that used in RCM. After running an evaluation on medical devices,

clinical engineers decide which should be included in the MEMP of the hospital based on their

risk scores.

Fennigkoh and Smith (1989) proposed a risk assessment method to group medical

devices on the basis of their Equipment Management (EM) numbers, or the sum of the numbers

assigned to the device’s critical function, physical risk, and required maintenance:

EM= Critical Function + Physical Risk + Required Maintenance. (3.2)

Devices with an EM number above a critical value ( 12) are considered to have critical

risk and thus are included in inspection and maintenance plans. In 1989, the Joint Commission

on Accreditation of Healthcare Organizations (JCAHO) recognized importance of this method

(Fennigkoh and Smith, 1989) and eventually in 2004 approved it as the standard (EC6.10)

(JACAHO, 2004). This standard allows hospitals not to perform scheduled inspection or

maintenance tasks for certain pieces or types of medical equipment, if these tasks are not needed

for safe and reliable operation (Wang, 2006). Since then, Fennigkoh and Smith’s method or its

many variations have been used by clinical engineers (Rice, 2007). Ridgway (2009) in his recent

paper emphasizes that preventive maintenance can provide a benefit for just a relatively few

devices, and a significant number of repair calls are made due to random failures of device’s

3.2. Model for Prioritization of Medical Devices 27

components. Wang and Rice (2003) propose simplified version of gradient risk sampling and

attribute sampling to select a portion of equipment for inclusion.

Clinical engineers believe that risk is not the only inclusion criterion, however, even

though it is the most important one (Hyman, 2003). Other criteria which reflect the needs and

reality of a hospital should be considered, including mission criticality, availability of backup,

hazard notice, and recall history (Wang and Levenson, 2000; Ridgway, 2001). Moreover, current

maintenance strategies employed in hospitals have difficulty identifying specific risks and

applying optimal risk reduction activities (Rice, 2007).

3.2. Model for Prioritization of Medical Devices

We present a multi-criteria decision-making model, which can be used to prioritize

medical devices and establish guidelines for selecting appropriate maintenance strategies.

Multi-Criteria Decision Making (MCDM) is a well-known branch of decision making,

divided into multi-objective and multi-attribute decision making (Triantaphyllou, 2000). A

Multi-Attribute Decision Making (MADM) is making preference decisions such as evaluation,

prioritization, and selection over available alternatives, characterized by multiple attributes

(Yoon and Hwang, 1981). Analytical Hierarchy Process (AHP) (Saaty, 1980, 1990), a MADM

methodology used widely by practitioner and researchers (Leung and Cao, 2001), is a theory of

measurement through pairwise comparisons which relies on the experts judgments to obtain

priority scales (Saaty, 2008). AHP, briefly, is a three-step process. 1- It decomposes a complex

problem into a hierarchy, in which the overall decision objective lies at the top and the criteria,

sub-criteria and decision alternatives are on each descending level of the hierarchy (Partovi et al,

1989) compose of specific factors. 2- Decision makers then compare each factor to all other


factors at the same level of the hierarchy using a pairwise comparison matrix to find its weight or

relative importance. 3- The optimal solution is the alternative with the greatest cumulative

weight (Saaty, 1990).

Two types of comparisons can be employed in the AHP: absolute and relative

measurements. Absolute measurement is applied to rank the alternatives in terms of the criteria

independent of other alternatives; however, in relative measurement the priority of an alternative

depends also on other alternatives.

In absolute comparison, alternatives are compared with a standard in one’s memory that

has been developed through experience. In relative measurement, alternatives are compared in

pairs according to a common attribute. As a result, in absolute measurement, the rank of

alternatives does not reverse when new alternatives are introduced, or the old ones are deleted;

however, the priority of alternatives may change by altering the existing set of alternatives

(Saaty, 1986; 1988).

AHP has been widely used in many applications involving decision making (Vaidya and

Kumar, 2006; Ho, 2008), and is often used for prioritizing alternatives when multiple criteria

must be considered (Modarres, 2006). Fong and Choi (2000) and Mahdi et al (2002) utilize AHP

for selecting contractors. Ramadhan et al (1999) use AHP to determine the rational weights of

pavement priority ranking factors, and Bevilacqua and Barglia (2000) utilize it for maintenance

strategy selection in an Italian Oil refinery. Labib et al (1998) propose a model to help take a

maintenance decision using AHP. Simpson and Cochran (1987) use AHP to prioritize

construction projects to assure that most needed projects receive funding when the budget is

limited. Al Harbi (2001) presents the AHP as a potential decision making method for use in

project management.


3.2.1. Proposed Criticality Assessment Model for Medical Equipment

We consider criticality prioritization of medical devices as a MCDM problem and use

AHP to solve it. The objective is to identify and include the more critical devices in the

equipment management program of a hospital, and investigate in details the reasons of having

high criticality scores to take appropriate actions, such as “preventive maintenance”, “user

training”, “redesigning the device”, etc. when reducing the criticality score is applicable and

manageable.

The first step in applying AHP is to construct the hierarchy structure of the goal, namely,

prioritization of medical devices. All required criteria for assessment of devices must be

identified and placed at the appropriate level of the hierarchy (Saaty, 2008). Figure 3.2 shows a

decision hierarchy for prioritization of medical devices.

Figure 3.2. Decision hierarchy for prioritization of medical devices

The assessment criteria lie at the second level of the hierarchy structure. Relative

measurement method is used for pairwise comparison of the assessment criteria and for

determining their relative importance or weights with respect to the goal. In other words, the

weight of each criterion is determined by comparing its relative contribution to the goal

Prioritization of

Medical Devices

Mission

Criticality Age Risk Recalls and

Hazard Alerts

Maintenance

Requirements

Goal

Criteria

Device 1 Device 2 Device 3 Device m

Alternatives

Function

c1 c2 c3 c4 c5 c6


(prioritization of medical devices) with other assessment criteria. Therefore, if a new criterion is

added or an existing one is deleted from the hierarchy, all criteria should be reassessed to find

their new weights.

The alternatives or medical devices compose the third level of the hierarchy. The

objective is to assign a criticality score for every single device participating in the model.

However, the large number of alternatives (devices) makes their pairwise comparison with

respect to all criteria almost impossible. Moreover, medical devices are dynamic, i.e. devices are

added to or removed from the inventory over time, so to solve this problem, we suggest an

absolute measurement technique for ranking alternatives. By using absolute measurement, each

device is assessed with respect to each criterion and is given the most descriptive grade without

comparing it with other devices. Thus, our proposed model will use both relative and absolute

measurement in the application of AHP.

To be able to assess a device with respect to a criterion, the criterion’s grades and their

associated intensities should be defined in advance. The grades are possible categories or classes

of a criterion. For example, “old”, “average”, and “new” can be considered as three classes of a

device’s age. The definition of each class should be decided and concurred by the decision

makers. The decision makers may consider a device as new when its actual age is 25% of its

expected life span. The grades and their descriptions used in this thesis, are either obtained from

the available standards and literature, or proposed in this research and approved by the clinical

engineers.

Since the grades are subjective, each should be assigned an intensity value indicating its

score or importance with respect to the criterion. Quantifying the grades is a necessary step,

because when a device is assessed with respect to a criterion and is assigned the most descriptive


grade, then it is the assigned grade’s intensity, which participates in the prioritization model. In

order to determine more accurate intensity values for the grades, we propose to use the relative

measurement method to pairwise compare the grades with respect to their criterion. Employing

this method makes it possible to avoid assigning arbitrary intensities for the grades and having

more consistent score values for them.

After defining the grades and intensities for all criteria, the model is ready for use to

assess the devices. Each device is compared with respect to each criterion and is assigned the

most descriptive grade.

The proposed model can be summarized in the following steps:

Identify all sufficient, efficient and independent criteria and sub-criteria for criticality

assessment of devices.

Determine weighting values for all criteria and sub-criteria using relative measurement

method.

Set up grades and determine intensities for each criterion using relative measurement

method.

Evaluate alternatives (devices or failure modes) with respect to each criterion, and assign

the most descriptive grades using absolute measurement method; the assigned grade’s

intensity for an alternative is called its score with respect to a criterion.

Calculate the criticality score for each device i as follows:

, 1

n

j

ijji swCS (3.3)


1,...,i m , where m is the total number of devices, 1,...,j n , where n is the

maximum criteria, jw is the weight of the thj criterion, ijs is the score of the thi device

with respect to the thj criterion,

n

j

jw1

1.

Order devices according to their criticality scores.

In our proposed model (Figure 3.2), six criteria are identified at the top level. Some of

these should be divided into sub-criteria; we divide “Mission criticality” into “Utilization” and

“Availability of alternative devices”. Figures 3.3 and 3.4 show associated “Mission criticality”

and “Risk” sub-criteria.

Figure 3.3. Hierarchy for mission criticality

Figure 3.4. Hierarchy for risk assessment of a device

Risk of failure modes

Failure Frequency Detectability Failure Consequence

Operational Non-Operational Safety and

Environment

Downtime Cost of Repair

c41 c42 c43

c431 c432 c433

c4311 c4321

Failure Mode 1

Alternatives

Failure Mode 2 Failure Mode 3 Failure Mode nf

nf

k 1

k mode failue ofRisk =Risk c4

Utilization

Availability of

Alternative Devices

c2

c21 c22

Mission Criticality


The criteria suggested in this chapter include some proposed criteria in the literature for

MEMP inclusion of medical devices. For example, “Function”, “Physical Risk”, and

“Maintenance requirements” are suggested by Fennigkoh and Smith (1989). To assess the failure

consequences of a device’ failure modes we include in our model “Physical Risk” as “Safety and

environment” criterion. Wang and Levenson (2000) suggest replacing “Function” by “Mission

Critical” in the Fennigkoh and Smith’s model, and also to take into consideration the

“Utilization” rate of each piece of equipment. We consider “Utilization” and “Availability of

alternative devices” as sub-criteria of “Mission criticality”.

3.2.2. Descriptions of Criteria and Sub-criteria

C1. Function

The function of a device is the main purpose for which it is to be used. The Medical

Devices Bureau of Health Canada recognizes four classes of medical devices based on how the

device is represented for use by the manufacturer (Health Canada, 1998). Class I devices present

the lowest potential risk and Class IV present the highest. This classification implicitly represents

the function of a device. For example, a life support device such as a defibrillator is considered

as a class IV device with high risk of failure (death of a patient) if the device fails. However, this

classification does not explicitly describe the function of a device. Moreover, risk or

consequence of a device failure should not be confused with its function, thus we propose in our

model “Life support”, “Therapeutic”, “Patient diagnostic”, “Analytical”, and “Miscellaneous” as

function categories. The proposed categories are an adaptation of Fennigkoh and Smith (1989)

and Dhillion’s (2000) classifications.


C2. Mission criticality

Mission criticality or operational impact describes the extent to which a device is crucial

to the care delivery process of a hospital (Wang, 2006). For example, Magnetic Resonance

Imaging (MRI) equipment might be extremely significant according to the mission of a hospital

but less critical in terms of its function or potential risk through use. Wang (Atles, 2008) suggests

classification of devices in three groups (Critical, Important, and Necessary) according to their

mission criticality. In our model, mission criticality depends on utilization and availability of

similar or alternative devices.

C21. Utilization

Utilization shows the total hours a device is used on average in a hospital (hours per day

or days per week or weeks per year). In this model, we consider the “average hours a device is

used per week” as the utilization criterion. “Utilization” can be defined as a function of average

usage and the number of patients served per unit time. Obviously, incorporating the “Number of

patients” into the model makes calculation of “Utilization” more complicated.

C22. Availability of alternative devices

Although not always true, with decreased backup and fewer similar devices at hand, the

situation of a device in the care delivery process becomes more critical, especially when it is in

high demand and there is only one such device in the hospital.

It should be noted that having several similar devices does not always mean high

availability of alternative devices: the demand per unit time of these devices is also important. If

there are several similar devices in a hospital but all are highly utilized, if either fails, there is

less chance that others can be used as substitutes. Therefore, “Availability of alternative devices”


can be considered as a function of the number of similar or backup devices and their demand per

unit time.

C3. Age

Age score is based on the actual age of a device and its predictable life span. The life

span for a group of similar devices can be obtained from the literature. In general, ten years is the

average life span of a medical device (Taylor, 2005).

C4. Risk

Risk is one of the most important criteria in criticality assessment of a device but cannot

be simply considered as a single number assigned to a device. Rather, the risk of a device should

be an aggregate of all risk values estimated from the actual failures, which have occurred in the

device. All failure modes and their associated frequencies, consequences, and detectabilities

should be extracted or estimated from data history and device maintenance work orders. The risk

value can then be estimated as a function of frequency, consequence, and detectability for each

failure mode. In short, the risk of the device is the total risk of all its failure modes.

C41. Failure frequency

Failure frequency is the likelihood of a failure occurrence. The VA National Center for

Patient Safety has designed “Healthcare Failure Modes and Effects Analysis (HFMEA)”

specifically for healthcare (DeRosier et al, 2002). Their suggested “Failure Frequency” rating

scales are shown in Table A.5.

C42. Detectability


Failure detectability is the ability to detect a failure when it occurs. In the proposed

model, we use the detectability levels described in Table A.6.

C43. Failure consequence

To find the total consequences of each failure mode, its operational, non-operational, and

safety and environment impacts should be assessed. These three categories are the conventional

failures consequences used in the RCM terminology (Moubray, 1997), so we keep this level

although both operational and non-operational criteria have only one related sub-criteria in our

model, as follows.

C431. Operational

Operational consequence of a failure is its impact on the care delivery process of a

hospital.

C4311. Downtime

Downtime is generally the average time that a device is out of service. However, for

medical devices it is also important to consider the number of patients who have to wait while

the device is down. We therefore suggest the average of the total waiting time of all patients as

“Downtime” sub-criteria, considering both the number of patients and the length of time they

wait due to device failure.

C432. Non-operational

Non-operational consequence of a failure is its direct inspection and repair cost, including

“Man power” and “Spare part(s)”.

C4321. Cost of repair


Cost of repair is the sum of labor and spare part(s) costs incurred by fixing a failure or

defect. Grade classification (i.e. high, medium, and low) for cost of repair depends on the budget

of a hospital and the purchase price of devices.

C433. Safety and environment

Safety and environment consequences of a failure are critical impacts that should be

taken into account. These are described in Table A.9 (Fennigkoh and Smith, 1989).

C5. Recalls and hazard alerts

The number and class of recalls and the number of hazard alerts that may occur for a

device are important criteria in prioritization of medical devices. U.S. Food and Drug

Administration (FDA) guidelines categorize recalls into three classes according to the level of

hazard involved (Meados, 2006).

C6. Maintenance requirements

According to Fennigkoh and Smith (1989), equipment that is predominantly mechanical,

pneumatic, or fluidic often requires the most extensive maintenance. A device is considered to

have an average maintenance requirement if it requires only performance verification and safety

testing. Equipment that receives only visual inspection, a basic performance check, and safety

testing is classified as having minimal maintenance requirements.

3.2.3. Determining Weighting Values for Criteria and Sub-criteria

Once all criteria and sub-criteria have been identified, their relative importance can be

determined with respect to their goal or their upper-lever criterion using Saaty’s eigenvector


technique (Saaty, 1980, 1990, and 2008). In other words, relative AHP is employed to determine

criteria and sub-criteria weighting values. Table 3.1 shows the weighting values calculated for all

criteria and sub-criteria in the model. The values given here represent expert opinion; results may

differ with the participation of a different group of experts.

Main Criteria (Weight) Sub-Criteria

(Weight)

Sub-Criteria

(Weight) Sub-Criteria (Weight)

c1-Function (0.45)

C2-Mission criticality (0.10)

c21-Utilization

(0.70)

c22 -Availability of

alternatives devices

(0.30)

C3-Age (0.06)

C4-Total risk (0.16)

c41-Failure frequency

(0.30)

c42-Detectability

(0.24)

c43-Failure

consequence (0.46)

c431-Operational

(0.16)

c4311- Downtime

(1.00)

c432-Non-

operational

(0.08)

c4321-Cost of repair

(1.00)

c433-Safety and

environment

(0.76)

C5-Recalls and hazard alerts

(0.16)

C6-Maintenance requirement

(0.07)

Table 3.1. Criteria/sub-criteria weighting values


3.2.4. Setting up Grades and Intensities for Each Criterion

We suggest using the absolute measurement technique for ranking of medical devices due

to their large number and dynamic nature. To employ absolute measurement, qualitative

(descriptive) grades for each criterion are constructed. Then, to find grade intensities, the grades

are pairwise compared according to their corresponding criterion. If a grade with highest

intensity is assigned to a device with respect to a criterion, the criterion for this device should

contribute with its full capacity (criteria’s weight) to its upper-level criterion or goal. It means

that this intensity should have value of 1. Therefore, the intensities should be divided by the

maximum intensity that can be obtained for each criterion. Finally, using absolute AHP, each

alternative is evaluated with respect to each criterion and is assigned an appropriate describing

grade (Saaty, 1990, 2008).

The grades’ intensities for the assessment criteria proposed by current classification

models in the literature can be criticized, especially with respect to inconsistency in logical

reasoning. For example, in the model proposed by Fennigkoh and Smith (1989), “Death” and

“No significant risk” have score values of 5 and 2 as the major and minor failure consequences,

respectively; however, this does not appropriately reveal their severities ratio. Therefore, in our

model, we propose using AHP to find the intensities of the criteria’s grades.

In the AHP hierarchy, the qualitative grades and intensities should be determined for

criteria/sub-criteria upon which the alternatives are directly evaluated (criteria with no sub-

criteria). Appendix A describes the qualitative grades and intensities of criteria/sub-criteria taken

from the literature or designed for this study. To demonstrate how the intensities can be

calculated for a criterion, the required steps are explained for the criterion “Function” in

Appendix A.


3.2.5. Ranking Medical Devices

Our model is now ready to rank medical devices. Each device should be assessed with

respect to every covering criterion, the lowest level criterion or sub-criterion connected to the

alternatives (Saaty, 2008) and assigned an appropriate grade. The score of a device at a criterion

that has sub-criteria is then the sum product of the sub-criteria’s weights and their grades’

intensities assigned to the device. This part of the AHP process is called “synthesizing” (Saaty,

2008). Therefore, the total score for a device can be obtained as an absolute value from the

weighted sum of the main criteria and their assigned intensities for the device. In order to easily

prioritize and classify devices according to their score values, the absolute total score values

should be normalized by dividing the total score values by the maximum of all devices’ score

values.

To estimate a device’s “Risk”, its failure modes should first be assessed with respect to

the risk criterion. In other words, alternatives in “Risk” hierarchy (Figure 3.4) are the failure

modes of a device. The “Risk” for a device is the sum of risk values of all its failure modes, that

is

1

nf

kk

RFRF (3.4)

The risk of a failure mode can be calculated using weighted sum of risk criteria, that is

434241 FCkGc

FDkGc

FFkGc

kRF (3.5)

where RFk is the risk of failure mode k of the device, 1,2,...,k nf (the total number of

devices’ failure modes). Gk FF

, Gk FD

, and Gk FC

are the intensities assigned to the failure mode

k of the device with respect to “Failure Frequency”, “Failure Detectability”, and “Failure


Consequence”. 41c , 42c , and 43c are the weighting values of Gk FF

, Gk FD

, and Gk FC

,

respectively. The weighted sum allows us to take into account the magnitude of risk factors.

The grades’ descriptions presented in Tables A.7 to A.11 are used to assign appropriate

grades to a failure mode. For example, if a particular failure mode of a device occurs once in 2 to

5 years, “uncommon” grade is assigned to its failure frequency. The failure history of all possible

failure modes of a device should be extracted from the computerized maintenance management

system (CMMS) and analyzed to select appropriate grades. A module can be implemented in the

CMMS to automatically extract and analyze the history of a failure mode. Tables A.7 to A.11

are either adapted from the FMEA proposed for medical devices (DeRosier et al, 2002), or are

designed based on the authors’ experience from the other industries.

In order to compare the risk values of all devices, we need an upper bound for the risk

criterion. After calculating the risk values for all devices, we divide them by the maximum

existing risk value, so the device with the highest risk value will have a risk ratio of 1.

Obviously, when the number of device is just one, the obtained risk value will be 1, but this

method is intended to compare several devices.

Instead of the weighted sum, heuristic reasoning can be used to deduct the grade of a

device/failure mode with respect to a criterion from its sub-criteria’s grades assigned to that

device/failure mode. A deterministic reasoning such as propositional logic or a non-deterministic

reasoning such as fuzzy logic can be employed (Rzevsky, 1995). In this case, the rules of

inference should be established before assessing the alternatives. Rules are knowledge

expressions in the form of conditions and actions. A rule consists of an IF-statement and a

THEN-statement. The IF-statement contains a combination of conditions. If the IF-statement is


satisfied, the THEN-statement can consequently be concluded. For example, the consequence of

a failure mode can be deducted from the grades of its sub-criteria as follows:

IF safety and environment consequence is death AND

operational consequence is low AND

non-operational consequence is high THEN

failure consequence is high.

In general, rules represent the experts’ knowledge, and a rule-based expert system

(Durkin, 1994) is a system in which the knowledge is stored in the condition-action rules.

In our model the criteria may be also considered as either static or dynamic. For example,

function is a static feature of a device, which does not change over time, while other criteria such

as age or risk are dynamic. Therefore, the prioritization model should be applied to devices

periodically to adjust their criticality scores according to changes in the score values of the

dynamic criteria. For example, a recently purchased device may not have enough failures

reported in the CMMS, but gradually some failure records will be added for it to the system.

New failures contribute to more precise estimation of the total risk value of the device and they

may eventually influence the total criticality score of the device.

3.2.6. Numerical Example and Discussion

The CMMS of the pilot hospital has the information of about 11,365 non-imaging and

2,241 imaging physical units (devices) and all of them should be prioritized. Thus, presenting

our complete results would require much more space, which is not possible in this chapter. We

therefore present a simplified example to illustrate the model’s application in the prioritization of

medical devices. We extracted information of 26 different medical devices from a hospital’s


CMMS. The selected devices are similar to the devices listed and discussed in (Fennigkoh and

Smith, 1989) and (Wang and Levenson, 2000). We used the assessment given in these papers for

the selected devices with respect to some of our proposed criteria such as “Function”, “Mission”,

and “Maintenance requirement”.

We extracted multiple failure modes for each device selected for the study, and assessed

them with respect to the risk criteria. Tables 3.2 (a, b)-3.5 demonstrate the approach. As shown

in Tables 3.2 (a, b), Infant Incubator, Defibrillator, and Intra_aortic Balloon Pump have the

highest normalized criticality score of 1.000, 0.964, and 0.943, respectively. For these devices,

the main contribution to the critically score comes from their high score values with respect to

“Function”, “Mission”, and “Risk”. Infant Incubator is also assigned a “Low” grade for

“Recalls” which is an important criterion with a weight of 0.16 in the prioritization hierarchy

structure. Surgical and Exam Lights have the lowest criticality scores, so they can be excluded

from the maintenance management program of the hospital. Additional comments on the results

of the example are given in the following sections.

No Device Name Function Mission Age Risk

1 Infant Incubator Life support (1.00) 0.760 Average (0.43) 1.000

2 Defibrillator Life support (1.00) 1.000 Average (0.43) 0.790

3 Intra-aortic Balloon Pump Life support (1.00) 1.000

Almost New

(0.17) 0.784

4 External Pacemaker Life support (1.00) 0.802

Almost New

(0.17) 0.854

5 Hematology Slide Stainer Analytical (0.13) 0.802 New (0.12) 0.550

6 Treadmill Therapeutic (0.21) 0.538 Old (1.00) 0.739

7 Gamma Camera

Patient Diagnostic

(0.16) 1.000 Average (0.43) 0.775

8 CT Scanner

Patient Diagnostic

(0.16) 0.802

Almost Old

(0.67) 0.614

9 Infusion Pump (B) Therapeutic (0.21) 1.000 Old (1.00) 0.668

10 ECG Physiological Telemetry

Unit

Patient Diagnostic

(0.16) 1.000

Almost Old

(0.67) 0.760


11 Automatic X-Ray Processor

Patient Diagnostic

(0.16) 1.000 Old (1.00) 0.269

12 Ultrasound Machine

Patient Diagnostic

(0.16) 0.802 Average (0.43) 0.576

13 Mobile Hypo/Hyperthermia

Unit Therapeutic (0.21) 0.340 Average (0.43) 0.718

14 Infusion Pump (A) Therapeutic (0.21) 0.760

Almost New

(0.17) 0.886

15 Multi-channel

Electrocardiograph

Patient Diagnostic

(0.16) 0.538

Almost New

(0.17) 0.695

16 Fetal Monitor

Patient Diagnostic

(0.16) 0.340 Old (1.00) 0.610

17 Centrifuge Analytical (0.13) 0.340

Almost Old

(0.67) 0.598

18 Cardiac Cath Harp Therapeutic (0.21) 0.538

Almost New

(0.17) 0.432

19 Scale for patient care Miscellaneous (0.11) 0.165 Old (1.00) 0.817

20 Blood Pressure Modules

Patient Diagnostic

(0.16) 0.340 Old (1.00) 0.328

21 Computer Terminal Analytical (0.13) 0.298

Almost Old

(0.67) 0.737

22 Water Bath Circulator Analytical (0.13) 0.207 Old (1.00) 0.524

23 Sterilizer Miscellaneous (0.11) 0.760 New (0.12) 0.399

24 Ultrasound Doppler

Patient Diagnostic

(0.16) 0.298

Almost New

(0.17) 0.792

25 Surgical Lights Therapeutic (0.21) 0.340

Almost Old

(0.67) 0.432

26 Exam lights Miscellaneous (0.11) 0.207 Average (0.43) 0.677

Table 3.2a. Assessment of the devices with respect to “Function”, “Mission”, “Age” and “Risk”

No Device Name Recalls Maintenance

Requirement

Total

Score

Normalized

Score

Transformed

Score (%)

1 Infant Incubator Low (0.12) High (1.00) 0.801 1.000 77.77

2 Defibrillator Null (0.00) High (1.00) 0.772 0.964 74.55

3 Intra-aortic Balloon Pump Null (0.00) High (1.00) 0.756 0.943 72.71

4 External pacemaker Null (0.00) High (1.00) 0.747 0.933 71.74

5 Hematology Slide Stainer High (1.00) High (1.00) 0.464 0.579 40.10

6 Treadmill Null (0.00) High (1.00) 0.397 0.495 32.58

7 Gamma Camera Null (0.00) High (1.00) 0.392 0.489 32.04

8 CT Scanner Low (0.12) High (1.00) 0.380 0.474 30.72

9 Infusion Pump (B) Null (0.00) Low (0.17) 0.373 0.466 29.97



Unit Null (0.00) Medium (0.50) 0.369 0.460

29.47

11 Automatic X-Ray Processor Null (0.00) High (1.00) 0.345 0.431 26.82

12 Ultrasound Machine Null (0.00) High (1.00) 0.340 0.425 26.27

13 Mobile Hypo/Hyperthermia Unit Null (0.00) High (1.00) 0.339 0.423 26.17

14 Infusion Pump (A) Null (0.00) Low (0.17) 0.334 0.417 25.63

15 Multi-channel

Electrocardiograph Null (0.00) High (1.00) 0.317 0.396

23.71

16 Fetal Monitor Null (0.00) Medium (0.50) 0.299 0.373 21.63

17 Centrifuge Null (0.00) High (1.00) 0.298 0.372 21.60

18 Cardiac Cath Harp Null (0.00) High (1.00) 0.298 0.371 21.51

19 Scale for patient care Low (0.12) Low (0.17) 0.288 0.359 20.42

20 Blood Pressure Modules Medium

(0.21) Medium (0.50) 0.287 0.358

20.35

21 Computer Terminal Null (0.00) Low (0.17) 0.258 0.322 17.12

22 Water Bath Circulator Null (0.00) Medium (0.50) 0.258 0.322 17.11

23 Sterilizer Low (0.12) Medium (0.50) 0.251 0.313 16.29

24 Ultrasound Doppler Null (0.00) Low (0.17) 0.251 0.313 16.26

25 Surgical Lights Null (0.00) Low (0.17) 0.250 0.312 16.18

26 Exam Lights Null (0.00) Low (0.17) 0.216 0.270 12.44

Table 3.2b. Assessment of the devices with respect to “Recalls” and “Maintenance

requirement”, and total, normalized and transformed criticality scores

No Device Name Utilization Alternative

Devices Mission

1 Automatic X-Ray Processor High (1.00) Low (1.00) 1

2 Blood Pressure Modules Medium (0.34) Medium (0.34) 0.34

3 Cardiac Cath Harp Medium (0.34) Low (1.00) 0.538

4 Centrifuge Medium (0.34) Medium (0.34) 0.34

5 Computer Terminal Medium (0.34) High (0.20) 0.298

6 CT Scanner High (1.00) Medium (0.34) 0.802

7 Defibrillator High (1.00) Low (1.00) 1


Unit High (1.00) Low (1.00) 1

9 Exam Lights Low (0.15) Medium (0.34) 0.207

10 External Pacemaker High (1.00) Medium (0.34) 0.802


11 Fetal Monitor Medium (0.34) Medium (0.34) 0.34

12 Gamma Camera High (1.00) Low (1.00) 1

13 Hematology Slide Stainer High (1.00) Medium (0.34) 0.802

14 Infant Incubator High (1.00) High (0.20) 0.76

15 Infusion Pump (A) High (1.00) High (0.20) 0.76

16 Infusion Pump (B) High (1.00) Low (1.00) 1

17 Intra-aortic Balloon Pump High (1.00) Low (1.00) 1

18 Mobile Hypo/Hyperthermia

Unit Medium (0.34) Medium (0.34) 0.34

19 Multi-channel

Electrocardiograph Medium (0.34) Low (1.00) 0.538

20 Scale for patient care Low (0.15) High (0.20) 0.165

21 Sterilizer High (1.00) High (0.20) 0.76

22 Surgical Lights Medium (0.34) Medium (0.34) 0.34

23 Treadmill Medium (0.34) Low (1.00) 0.538

24 Ultrasound Doppler Medium (0.34) High (0.20) 0.298

25 Ultrasound Machine High (1.00) Medium (0.34) 0.802

26 Water Bath Circulator Low (0.15) Medium (0.34) 0.207

Table 3.3. Assessment of the devices with respect to “Mission criticality” sub-criteria

Device Name

Fre

qu

ency

Det

ecta

bil

ity

Con

seq

uen

c

e

Fa

ilu

re

mod

e R

isk

score

Tota

l ri

sk

Dev

ice

risk

=

Tota

l

risk

/max

Automatic X-Ray Processor

- Unit does not power up Frequent High

0.16

0.4048

0.4048 0.2688

Blood pressure modules

- Module receptacle was damaged Occasional Very Low

0.3372 0.4941

0.4941 0.3281

Cardiac Cath Harp

- CRA/CAU bearing noise

- Low contrast resolution loss after previous

HARP test

Frequent

Occasional

High

High

0.284

0.126

0.4618

0.1882

0.6500 0.4316

Centrifuge

- Broken speed control knob

- Centrifuge intermittently drops in speed while

Frequent

Occasional

Moderate

Moderate

0.2172

0.2172

0.4479

0.2469

0.9005 0.5979


running

- Machine turns on but motor does not run

Occasional High

0.164

0.2056

Computer Terminal

- Mouse locked up

- Overflow errors caused data to be missed in

analysis

- Tape speed errors

Frequent

Occasional

Frequent

High

Low

Moderate

0.1044

0.2796

0.164

0.3792

0.3068

0.4234

1.1095 0.7367

CT Scanner

- CT cable loose

- System could not recognize a scan as raw data

- Wire harness for CT dislodged

Occasional

Occasional

Frequent

Moderate

Moderate

Low

0.1044

0.284

0.16

0.1950

0.2776

0.4528

0.9255 0.6145

Defibrillator

- Cable constructed

- Cable tie on paddle handle

- Intermittent break in power cord

Occasional

Frequent

Occasional

Moderate

Moderate

Low

0.1424

0.164

0.8176

0.2125

0.4234

0.5543

1.1902 0.7903

ECG Physiological Telemetry Unit

- Broken terminal connection

- Soot water damage from fire

- Telemetry does not detect lead off

Occasional

Uncommon

Frequent

Low

High

Very Low

0.164

0.284

0.2796

0.2536

0.2218

0.6686

1.1441 0.7597

Exam lights

- Bulb broken inside socket

- Lamp holder broken

- Loosed swivel head

Frequent

Frequent

Occasional

High

Moderate

Moderate

0.1044

0.1044

0.2132

0.3792

0.3960

0.2451

1.0203 0.6775

External Pacemaker

- Broken clip

- Pace lights visible only when pacemaker is

lying down

- Pulse generator failure

Frequent

Occasional

Uncommon

Moderate

High

Low

0.1424

0.312

1.000

0.4135

0.2737

0.5992

1.2864 0.8542

Fetal Monitor

- Monitor not powering-up

- Paper was not moving

- The unit kept losing its configuration

Occasional

Frequent

Occasional

High

High

Low

0.164

0.1044

0.3372

0.2056

0.3792

0.3333

0.9182 0.6097

Gamma Camera

- Fatal timeout problems

- Peak shift

- System froze

Frequent

Uncommon

Occasional

Low

Very Low

High

0.2796

0.3372

0.16

0.5078

0.4551

0.2038

1.1667 0.7747


Hematology Slide Stainer

- Power light did not light up after switching on

- Unit is not auto indexing for different staining

stations

- Vertical movement cam loose on shaft

Occasional

Occasional

Frequent

High

Low

Moderate

0.1044

0.164

0.1044

0.1782

0.2536

0.3960

0.8279 0.5497

Infant Incubator

- Audio alarms are not working

- Missing access grommets

- Motor is stuck

Frequent

Frequent

Uncommon

Moderate

Low

Moderate

0.316

0.1424

1.00

0.4934

0.4447

0.5680

1.5061 1.0000

Infusion Pump (A)

- Dead battery

- Problem with door clip

- Screw missing on bottom left

Uncommon

Frequent

Frequent

Low

Moderate

Moderate

0.8176

0.164

0.1044

0.5153

0.4234

0.3960

1.3348 0.8863

Infusion Pump (B)

- Dead battery


Uncommon

Frequent

Low

Low

0.9376

0.122

0.5705

0.4353

1.0058 0.6678

Intra-aortic balloon pump

- Broken ECG leads

- Vacuumed performance failed

- Worn out ECG connector

Occasional

Frequent

Frequent

Moderate

Low

Moderate

0.164

0.3996

0.1044

0.2224

0.5630

0.3960

1.1815 0.7845

Mobile Hypo/Hyperthermia Unit

- Bad input jack

- Cool warning not working

- Pump had a noisy bearing

Frequent

Frequent

Occasional

Moderate

Moderate

High

0.1044

0.1044

0.3464

0.3960

0.3960

0.2895

1.0816 0.7182

Multi-channel Electrocardiograph

- Computer locked up

- Intermittent noise on all leads

- LCD display is intermittent

Frequent

Frequent

Occasional

High

High

High

0.1044

0.284

0.164

0.3792

0.4618

0.2056

1.0467 0.6950

Scale for patient care

- Garbled display

- Scale will not zero

- System does not go ready mode

Frequent

Frequent

Frequent

Moderate

High

High

0.2132

0.1044

0.16

0.4461

0.3792

0.4048

1.2301 0.8168

Sterilizer

- Cassette jammed and punctured on plastic

housing

- Plaster is cracked

Frequent

Occasional

Moderate

High

0.164

0.1044

0.4234

0.1782

0.6017 0.3995


Surgical Lights

- Broken cover cat

- Light drifting

Frequent

Occasional

High

High

0.2172

0.1956

0.4311

0.2202

0.6513 0.4324

Treadmill

- CRT display tube goes blank

- Does not indicate speed

- Poor contact in ECG lead connector block

Frequent

Frequent

Occasional

Moderate

Moderate

Low

0.16

0.2172

0.1424

0.4216

0.4479

0.2437

1.1132 0.7392

Ultrasound Doppler

- Damaged LCD leads

- Loose battery contact

- Not picking up the signal

Frequent

Frequent

Occasional

High

Moderate

Low

0.2264

0.164

0.3372

0.4353

0.4234

0.3333

1.1921 0.7915

Ultrasound Machine

- Cable holder fell off

- Diagnostic software not working

- system locks up

Frequent

Occasional

Uncommon

Moderate

High

Moderate

0.1044

0.16

0.3464

0.3960

0.2038

0.2673

0.8672 0.5758

Ventilator

- Damaged power supply

- Going to standby during use

Occasional

Uncommon

Moderate

Low

0.16

0.4984

0.2206

0.3685

0.5891 0.3911

Water Bath Circulator

- Pump motor bearing failed

- Relay no longer available

- Water bath not regulating temperature

Occasional

Occasional

Uncommon

High

Moderate

Moderate

0.3464

0.2172

0.316

0.2895

0.2469

0.2534

0.7898 0.5244

Table 3.4. Risk assessment of the failure modes

Device Name

Op

era

tio

na

l

(D

ow

nti

me)

No

n-o

per

ati

on

al

(R

epa

ir c

ost

)

Sa

fety

an

d

en

vir

on

men

t

Co

nse

qu

ence

Sco

re

Automatic X-Ray Processor

- Unit does not power up Medium Low Delayed Treatment 0.1600

Blood pressure modules

- Module receptacle was damaged

High Medium Inappropriate therapy 0.3372

Cardiac Cath Harp

- CRA/CAU bearing noise

High

Medium

Medium

Medium

Delayed Treatment

No consequence

0.2840


- Low contrast resolution loss after previous HARP test 0.1260

Centrifuge

- Broken speed control knob

- Centrifuge intermittently drops in speed while running

- Machine turns on but motor does not run

Medium

Medium

Medium

Medium

Medium

Medium

Inappropriate therapy


Delayed Treatment

0.2172

0.2172

0.1640

Computer Terminal

- Mouse locked up

- Overflow errors caused data to be missed in analysis

- Tape speed errors

Low

Medium

Medium

Low

High

Medium

No consequence


Delayed Treatment

0.1044

0.2796

0.1640

CT Scanner

- CT cable loose

- System could not recognize a scan as raw data

- Wire harness for CT dislodged

Low

High

Medium

Low

Medium

Low

No consequence

Delayed Treatment

Delayed Treatment

0.1044

0.2840

0.1600

Defibrillator

- Cable constructed

- Cable tie on paddle handle

- Intermittent break in power cord

Low

Medium

Medium

Low

Medium

Medium

Delayed Treatment

Delayed Treatment

Death

0.1424

0.1640

0.8176

ECG Physiological Telemetry Unit

- Broken terminal connection

- Soot water damage from fire

- Telemetry does not detect lead off

Medium

High

Medium

Medium

Medium

High

Delayed Treatment

Delayed Treatment


0.1640

0.2840

0.2796

Exam lights

- Bulb broken inside socket

- Lamp holder broken

- Loosed swivel head

Low

Low

Medium

Low

Low

Low

No consequence

No consequence


0.1044

0.1044

0.2132

External Pacemaker

- Broken clip

- Pace lights visible only when pacemaker is lying down

- Pulse generator failure

Low

Medium

High

Low

Low

High

Delayed Treatment

Injury

Death

0.1424

0.3120

1.0000

Fetal Monitor

- Monitor not powering-up

- Paper was not moving

- The unit kept losing its configuration

Medium

Low

High

Medium

Low

Medium

Delayed Treatment

No consequence


0.1640

0.1044

0.3372

Gamma Camera

- Fatal timeout problems

Medium

High


0.2796

0.3372


- Peak shift

- System froze

High

Medium

Medium

Low


Delayed Treatment

0.1600

Hematology Slide Stainer

- Power light did not light up after switching on

- Unit is not auto indexing for different staining stations

- Vertical movement cam loose on shaft

Low

Medium

Low

Low

Medium

Low

No consequence

Delayed Treatment

No consequence

0.1044

0.1640

0.1044

Infant Incubator

- Audio alarms are not working

- Missing access grommets

- Motor is stuck

Medium

Low

High

Medium

Low

High

Injury

Delayed Treatment

Death

0.3160

0.1424

1.0000

Infusion Pump (A)

- Dead battery

- Problem with door clip


Medium

Medium

Low

Medium

Medium

Low

Death

Delayed Treatment

No consequence

0.8176

0.1640

0.1044

Infusion Pump (B)

- Dead battery


High

Medium

Medium

Low

Death

No consequence

0.9376

0.1220

Intra-aortic balloon pump

- Broken ECG leads

- Vacuumed performance failed

- Worn out ECG connector

Medium

High

Low

Medium

High

Low

Delayed Treatment


No consequence

0.1640

0.3996

0.1044

Mobile Hypo/Hyperthermia Unit

- Bad input jack

- Cool warning not working

- Pump had a noisy bearing

Low

Low

High

Low

Low

High

No consequence

No consequence

Delayed Treatment

0.1044

0.1044

0.3464

Multi-channel Electrocardiograph

- Computer locked up

- Intermittent noise on all leads

- LCD display is intermittent

Low

High

Medium

Low

Medium

Medium

No consequence

Delayed Treatment

Delayed Treatment

0.1044

0.2840

0.1640

Scale for patient care

- Garbled display

- Scale will not zero

- System does not go ready mode

Medium

Low

Medium

Low

Low

Low


No consequence

Delayed Treatment

0.2132

0.1044

0.1600

Sterilizer

- Cassette jammed and punctured on plastic housing

- Plaster is cracked

Medium

Low

Medium

Low

Delayed Treatment

No consequence

0.1640

0.1044


Surgical Lights

- Broken cover cat

- Light drifting

Medium

Low

Medium

Low



0.2172

0.1956

Treadmill

- CRT display tube goes blank

- Does not indicate speed

- Poor contact in ECG lead connector block

Medium

Medium

Low

Low

Medium

Low

Delayed Treatment


Delayed Treatment

0.1600

0.2172

0.1424

Ultrasound Doppler

- Damaged LCD leads

- Loose battery contact

- Not picking up the signal

Medium

Medium

High

High

Medium

Medium

Delayed Treatment

Delayed Treatment


0.2264

0.1640

0.3372

Ultrasound Machine

- Cable holder fell off

- Diagnostic software not working

- system locks up

Low

Medium

High

Low

Low

High

No consequence

Delayed Treatment

Delayed Treatment

0.1044

0.1600

0.3464

Ventilator

- Damaged power supply

- Going to standby during use

Medium

High

Low

High

Delayed Treatment

Injury

0.1600

0.4984

Water Bath Circulator

- Pump motor bearing failed

- Relay no longer available

- Water bath not regulating temperature

High

Medium

Medium

High

Medium

Medium

Delayed Treatment


Injury

0.3464

0.2172

0.3160

Table 3.5. Consequence assessment of the failure modes

The proposed model in this chapter incorporates all criteria suggested by clinical

engineers (Fennigkoh and Smith, 1989; Wang and Levenson, 2000; Ridgway, 2001; Hyman,

2003; Wang, 2006) to assess the criticality of medical equipment. Moreover, the model gives a

realistic estimate of the total risk of a device by taking into account its different failure modes

and assessing their frequency, detectability and consequences. The failure modes are extracted

from the device’s failure history available in the CMMS of the hospital and are analyzed with

respect to the risk’s sub-criteria. AHP enables the model to accommodate multiple criteria and


integrates scientific judgments with personal opinion in the evaluation of the alternatives (Herath

and Prato, 2006). Calculating the consistency ratio (Saaty, 1990) in pairwise comparison of the

criteria makes the model able to produce more precise and consistent criteria’s weights compared

to direct assignment of the weights. Furthermore, the intensities, which are obtained for the

criteria’s grades, are also more consistent due to applying the relative measurement method. One

of the limitations of the proposed model is that it requires experts to be involved in the process of

applying the model to medical devices and this process may be expert intensive. Moreover, the

prioritization results may not always be accepted by clinical engineers and requires reassigning

the criteria’s weights and/or grades’ intensities. To apply the proposed model to the CMMS of a

hospital and obtain accurate results, the CMMS should be up-to-date and should include the

information required for assessment of devices.

3.2.7. Classification and Maintenance Strategies

As has been noted, the proposed model prioritizes devices according to their criticality.

The normalized score value indicates the relative criticality of a device compared to other

devices. Given such a model, hospitals could focus their maintenance efforts on more critical

devices. Devices with lower criticality scores could be discarded from a maintenance

management program, while devices with high score values could be monitored and supervised.

Normalized scores (e.g. the values in the last column of Table 3.2b) can be used for

prioritizing or ranking of devices. The normalized scores depend on the total number of devices

involved in the model. However, total scores of devices (e.g. the values in the 5th

column of

Table 3.2b) can be used as absolute measurements for classification. The total score is a metric

that can be compared with predefined thresholds to decide to which category the device belongs.


In our proposed model, devices can have a total score between (0.1050, 1.0). Score 1.0 is for a

device which gets the highest intensity when assessed against every single criterion, and 0.1050

is obtained when the device gets the lowest intensity from all criteria (see Appendix B for this

calculation). Therefore, the score in the proposed model is always between 0.1050 and 1. The

total score can then be mapped to (0, 100%) using the following equation:

score value-min score value-0.1050

Transfomed score value % % max min 0.8950

TSV

(3.6)

We suggest the following classification of thresholds and maintenance strategies:

Criticality Class Transformed score

value

Maintenance strategy

High 40% 100%TSV Proactive, predictive, or time-based maintenance

Medium 20% 40%TSV Proactive or time-based maintenance

Low 0% 20%TSV Corrective maintenance

Table 3.6. Proposed classes and the thresholds

The thresholds in Table 3.6 are based on the score values obtained for the devices given

in the numerical example. In general, the thresholds can be adjusted after applying the model to

the inventory of a hospital and investigating the obtained transformed score values. The

thresholds can be adjusted, depending on the minimum and maximum of these values. Different

type of devices with different transformed score values should participate in the model to decide

how the budget should be allocated for management of devices and which maintenance strategies

should be applied. Participation of all devices is normally expected when the prioritization

module is integrated into the CMMS of a hospital. Otherwise, the output decisions might be

inaccurate. For example, if the prioritization model is just applied to a group of devices in a


hospital and not all devices, the managers may allocate the budget only to this group and decide

to apply predictive maintenance for all of them, which is not necessarily required.

Therefore, thresholds should always be established according to both the characteristics

of participant devices and their estimated transformed score values. Moreover, the number of

classes and established maintenance strategies depend on available resources (budget, personnel,

etc.) in the hospital.

In general, maintenance strategies can be classified according to their required resources

(cost, labor, and special equipment) and their impact on maintaining a piece of equipment

(Mobley, 2002). In corrective maintenance, a device is just run until it breaks. This strategy does

not require any staff or money to be applied; however, unplanned downtime and inefficient use

of the staff to repair the device can even be more costly. In time-based maintenance, the device is

periodically checked and preventive maintenance is performed if necessary. This strategy is easy

to implement and can reduce the failure rate of a device. However, it requires some resources

such as budget and labor to be implemented. Time-based maintenance may not always be

optimal and a device may be over or under maintained when the preventive maintenance is

carried out on a calendar-based scheme. In predictive maintenance, some variables such as the

device vibrations are measured to assess the device’s physical status. The measured variables are

used to predict failure of the device. This maintenance strategy requires some sensors or special

equipment to measure the prediction variables. However, it can increase the operational life of

the device and its availability.

Devices with low critically score can be excluded from the maintenance management

program of a hospital and just get fixed whenever they break down (corrective maintenance).

This group of medical devices more likely have a low score value with respect to the criteria


possessing higher weights such as function, risk, and recalls and hazard alerts; so the asset

management resources are better not to be spent on this group. Table 3.2b shows that corrective

maintenance can be used for the last eight devices on the list starting with Scale for patient case

and ending with Surgical and Exam lights. In other words, these devices are run to failure and

just get fixed when they fail.

Proactive maintenance could potentially be applied to high critical devices. This type of

maintenance employs monitoring and correction of failing root causes (Swanson, 2001) or in

general, detection and elimination of root causes of such a high criticality score when it is

applicable and manageable. If physical conditions such as vibration, noise, etc. of a device with

high criticality score can be measured, condition-based or predictive maintenance is

recommended to avoid or reduce failure consequences; if not, time-based maintenance could be

applied here as well. Time-based maintenance can be considered for devices with average

criticality score. Unlike predictive maintenance, time-based maintenance is easier to implement,

with no need to have any special sensor equipment. In Table 3.2b, devices such as Infant

Incubator, Defibrillator, and Balloon Pump with transformed score of greater than 40% belong to

the high criticality class.

To facilitate a risk reduction strategy, devices should be investigated periodically with

respect to their dynamic criteria, and their current maintenance policies should be adjusted

accordingly. For example, when the total risk score of a device is high, its root causes should be

investigated and appropriate actions should be taken to reduce the risk. If risk is high due to the

high frequency of failures, the root causes might be “use error” or imperfect design. In this case,

user training or redesigning the device if possible, could be effective risk reduction actions.


We also can use the score value obtained for a device with respect to each individual

criterion to determine appropriate maintenance strategies for the device. Here, the decision factor

is the score value of a device with respect to a criterion or a combination of criteria (antecedent

or IF-statement) and the consequence (THEN-statement) is the most recommended maintenance

strategy. This multi-criteria methodology introduced here is a method that requires more research

and will not be further discussed, but some initial heuristic guidelines for selecting appropriate

maintenance strategies are suggested in Table 3.7. For example, Table 3.7 recommends “design-

out maintenance” when a device has a high score value for either “recalls” or “risk”, since the

device more likely has an engineering design problem. When the score value of the device for

both “recalls” and “risk” is low, and the device is function-critical with significant failure

consequence with low possibility of failure detection, failure-finding strategy is recommended to

minimize the consequence of hidden failures.

Antecedent Consequent

(Recalls ↑) or (Risk ↑) Design-out maintenance

(Recalls ↓) and (Risk ↓)

(Function ↑)

and

(Failure consequence ↑)

(Utilization ↑) Condition monitoring

(Utilization ↓) Inspections

(Detectability ↑) Failure-finding interval

(Function ↓) (Mission ↓) Operate to failure

(Function ↕) (Utilization ↑) Calendar-based maintenance

(Utilization ↓) Utilization-based maintenance

Symbols: ↑ high/critical ↓ low/non-critical ↕ medium/mid-critical

Table 3.7. Heuristic reasoning for maintenance decisions



This chapter presents a multi-criteria decision-making model to prioritize medical devices

according to their criticality. The model uses AHP to identify and include the more critical

devices in the equipment management program of a hospital. The proposed hierarchy structure

contains six criteria to assess criticality of the devices, “Function”, “Mission criticality”, “Age”,

“Risk”, “Recalls and hazard alerts”, and “Maintenance requirements”. The model gives a

realistic estimate of the total risk of a device by taking into account its different failure modes

and assessing their frequency, detectability and consequences. The proposed model uses both

relative and absolute measurement in the application of AHP to determine weighting values for

the criteria and their grades’ intensities, and to evaluate the alternatives (devices). Calculating the

consistency ratio in pairwise comparison of the criteria makes the model able to produce more

precise and consistent criteria’s weights compared to direct assignment of the weights.

Furthermore, the intensities that are obtained for the criteria’s grades are also more consistent

due to applying the relative measurement method.

This model can be integrated as a module into the CMMS of a hospital to prioritize

medical devices. Devices with lower criticality scores can be assigned a lower priority in a

maintenance management program. However, those with higher scores should be investigated in

detail to find the reasons of having such high criticality scores to take appropriate actions, such

as “preventive maintenance”, “user training”, “redesigning the device”, etc. when reducing the

criticality score is applicable and manageable. In addition, individual score values for each

device with respect to a criterion or a combination of several criteria can be used to establish

guidelines to select an appropriate maintenance strategy.


In our proposed model, the pairwise comparison matrices are constructed using Saaty’s

crisp 1-9 scales. Alternatively, fuzzy analytic hierarchy process (Kwong and Bai, 2002) can be

used; in this case, scales are fuzzy numbers instead of crisp values to handle uncertainty in

assigning the scale values.

60

4. RELIABILITY AND TREND ANALYSIS OF FAILURE DATA


A common belief of a healthcare society is that medical devices fail independent of age,

following exponential distribution. This belief originates from the general conviction that

electronic equipment has a constant failure rate. Even the most widely known and used

reliability prediction handbook, MIL-HDBK-217 (United States Department of Defense, 1995),

proposes reliability models whose construction is based on the constant failure rate assumption.

This assumption has been criticized as inaccurate, however, and its use may result in erroneous

decisions (Choi and Seong, 2008; Puran, 1990; Mortin et al., 1995). Reliability and failure

patterns of a device may be affected by external factors such as operating conditions,

environmental stress, expertise level of operators, etc. Therefore, to realistically determine their

reliability, devices should be studied in their operating context. To this end, learning about

existing maintenance procedures of equipment and statistical analysis of field data are essential

steps towards developing an optimal evidence-based maintenance/inspection plan.

Trend analysis is a common statistical method used to investigate the operation and

changes in a repairable system over time. This method takes historical failure data of a system or


a group of similar systems and determines whether the recurrent failures exhibit an increasing or

decreasing trend. Both graphical methods and trend tests are used for trend analysis. The latter

are statistical tests for the null hypothesis that the failure process follows a homogeneous Poisson

process (HPP) (Lindqvist, 2006).

Crow/AMSAA test (Crow, 1974; 1982) assumes that the failure process of a repairable

system has a Weibull intensity function and finds the maximum likelihood estimates of the

parameters. The shape parameter is then used as an indicator for a growth or deterioration in the

system reliability. The Laplace and the Military Handbook tests (Ascher and Feingold, 1984) are

used to test whether data follow a HPP. Kvaløy and Lindqvist (1998) propose the Anderson-

Darling trend test for a NHPP (Ross, 2007) with a bathtub shaped intensity function based on the

Anderson-Darling statistic test (Anderson and Darling, 1952). The Lewis-Robinson (Lewis and

Robinson, 1974) is a modification of the Laplace test and is used as a general test to detect trend

departures from a general renewal process (Lawless and Thiagarajah, 1996). The Mann test

(Mann, 1945) corresponds to a renewal process null hypothesis and the monotonic trend as the

alternative. Kvaløy et al. (2002) declare that the Mann test is a more powerful test against

decreasing trend. The MIL-HDBK-189 (1981; Crowder et al., 1991) is used for testing a NHPP

with a power law intensity function.

Most trend tests (Ascher and Feingold, 1984; Wang and Coit, 2005) assume that failure

times are known, so the failure data are complete. Currently in the literature, except for right

censoring, there is no available method for estimating the parameters of a non-homogeneous

Poisson process (NHPP) which incorporates left and interval censored failure data if repairs are

not instantaneous or not performed immediately. However, it is expected that the data received

from industry include missing and incomplete information. Sometimes a particular type of data


of interest is not measured at all, or if measured may be incomplete or unreliable. For example,

hidden failures, which make up to 40% of all failure modes of a complex industrial system

(Moubray, 1997), are not evident to operators and remain dormant until they are rectified at

scheduled inspections. In this case, the times of hidden failures are either left or interval

censored, and the censoring interval is the interval between two consecutive inspections.

Weibull or power law (Crow, 1974) and log linear (Cox and Lewis, 1996) intensity

functions are two common models used to describe recurrent even data underlying NHPP. The

maximum likelihood estimates of the parameters are obtained for two intensity parameters

(Crowder et al., 1991; Meeker and Escobar, 1998).

Although a number of researchers have considered the reliability prediction of medical

systems at the design and development stage over the last two decades (Fries, 2005; Dhillon,

2000), less literature deals with the reliability prediction and trend analysis of medical devices

while they are in use in hospitals. Ion et al. (2006) analyze the field data for medical imaging

systems during the warranty period. Roelfsema (2004) presents the results of early reliability

prediction of Philips medical systems based on field data. Baker (2001) analyses a large database

for failures of many types of repairable medical devices including ventilators, infusion pumps,

and ECG monitors. She assesses the validity of some well-known failure rate models such as

power law and log linear and linear Poisson processes. The results reveal an increasing rate of

failures with equipment age. Moreover, the power law process model is found to fit better to the

data. In addition, she considers the imperfect repair models, and finds out that for some

equipment, such as ventilator and ECG monitors, the repair effect models are better fits.

4.2. A Model Proposed for Reliability Analysis of Medical Devices’ Failure Data 63

4.2. A Model Proposed for Reliability Analysis of Medical Devices’ Failure Data

In this study, we analyzed maintenance data (collected from 2000 to 2007) from a

Canadian General hospital. We conducted statistical analysis to learn about the failure types and

trends of a particular general infusion pump used to deliver liquids for therapeutic and/or

diagnostic purposes. However, since all medical devices undergo periodical safety and

performance inspections, the general policies derived from this study can be applied to other

medical devices. Although dealing with a specific pump, this chapter describes how available

maintenance data in a hospital can be statistically analyzed.

4.2.1. Preliminary Analysis of Failure Data

For the purposes of this study, we selected a general infusion pump. General-purpose

infusion pumps (Figure 4.1) are used to accurately deliver liquids through intravenous (IV) or

epidural routes for therapeutic and/or diagnostic purposes (ECRI Institute, 2009). The reason for

selecting this model is its significant number of non-scheduled work orders.

Figure 4.1. General Infusion Pump

The pilot hospital has 681 pieces of this model used in three departments. All scheduled

and non-scheduled work orders were extracted from the CMMS, and we performed a visual


examination of the data looking for possible errors. A typical work order is shown in Figure 2.4

of Chapter 2.

Finding at the inspection Non-scheduled work orders Scheduled work orders

Device Contaminated 1 0

Component Failure 910 1182

User Input/Setup Error 3 1

Problem Not Found 275 316

Not Applicable 0 2

Device over used 1 0

NULL (the cause was not specified) 6 18

Total 1196 1519

Table 4.1. The total number of scheduled and non-scheduled

work orders and findings at inspection

Table 4.1 shows the total number of scheduled and non-scheduled work orders and the

associated findings at inspection. Minor defects are not usually reported by the operators and are

rectified only at SPIs. Moreover, clinical engineers perform opportunistic maintenance of some

components like batteries, which may be replaced before complete depletion. Therefore, as Table

4.1 makes clear, more failed components are identified in scheduled work orders than in non-

scheduled work orders. The average time between non-scheduled work orders is 452.8 days, with

a standard deviation of 423.1 days. However, 661.6 days is the average time between scheduled

work orders with relatively small variation (a standard deviation of 175.3 days).

We counted the percentage of times that a test resulted in “P” (pass), “F” (failed) and

“N/A” (not applicable) for different components/features of the devices (see Section 2.4.2 in

Chapter 2). Tables 4.2-4.4 summarize the results. Components/features with zero or few failures

or with mostly “N/A” tests were excluded from the study.


The Pareto analysis of the qualitative test results of non-scheduled work orders identified

the most problematic components (Figure 4.2). “Chassis/Housing”, “Alarms”, and

“Fittings/Connectors” are the most problematic components/features with the total number of

failures being 502, 321, and 240, respectively.

Non-scheduled work orders

Component P

(%)

F

(%)

N/A

(%)

AC Plug/Receptacles 98.92 1.08 0.00

Alarms 73.29 26.71 0.00

Audible Signals 91.93 8.07 0.00

Battery/Charger 97.92 2.08 0.00

Chassis/Housing 58.24 41.76 0.00

Controls/Switches 91.26 8.74 0.00

Fittings/Connectors 80.03 19.97 0.00

Indicators/Displays 91.01 8.99 0.00

Labeling 90.68 9.32 0.00

Line Cord 88.77 11.23 0.00

Mount 83.94 15.89 0.17

Circuit Breaker/Fuse 3.99 1.08 94.93

Strain Reliefs 98.34 1.50 0.17

Scheduled work orders

Component P

(%)

F

(%)

N/A

(%)

AC Plug/Receptacles 90.75 0.54 8.71

Alarms 76.20 15.08 8.71

Audible Signals 75.12 16.17 8.71

Battery/Charger 79.39 11.90 8.71

Chassis/Housing 37.26 54.03 8.71

Controls/Switches 86.12 5.17 8.71

Fittings/Connectors 64.48 26.80 8.71

Indicators/Displays 83.11 8.17 8.71

Labeling 63.88 27.34 8.77

Line Cord 67.13 24.16 8.71

Mount 79.27 12.02 8.71

Circuit Breaker/Fuse 1.98 0.18 97.84

Strain Reliefs 88.52 2.46 9.01

Table 4.2. Qualitative tests results extracted from scheduled and non-scheduled work orders



Component P

(%)

F

(%)

N/A

(%)

Flow Rate Accuracy

(%) 12.90 0.00 87.10

Grounding Resistance

(mohm) 99.75 0.17 0.08

Maximum Leakage

Currents 100.00 0.00 0.00

Occlusion Alarm 99.08 0.75 0.17


Component P

(%)

F

(%)

N/A

(%)

Flow Rate Accuracy

(%) 90.50 0.00 9.50

Grounding Resistance

(mohm) 91.17 0.12 8.71

Maximum Leakage

Currents 91.29 0.00 8.71

Occlusion Alarm 90.56 0.72 8.71

Table 4.3. Quantitative tests results extracted from scheduled and non-scheduled work orders


PM

Check

P

(%)

F

(%)

N/A

(%)

Clean 99.75 0.25 0.00

Lubricate 99.67 0.33 0.00

Replace 29.37 70.63 0.00


PM

Check

P

(%) F (%)

N/A

(%)

Clean 91.29 0.00 8.71

Lubricate 91.11 0.18 8.71

Replace 13.04 78.19 8.77

Table 4.4. PM checks results extracted from scheduled and non-scheduled work orders


0

100

200

300

400

500

600

Chas

sis/Hous

ing

Alarm

s

Fittings

/Con

necto

rs

Mou

nt

Line

Cor

d

Labe

ling

Indica

tors/D

isplay

s

Cont

rols/S

witc

hes

Aud

ible S

igna

ls

Bat

tery

/Char

ger

Stra

in R

eliefs

Circ

uit B

reak

er/Fus

e

AC P

lug/Rec

epta

cles

0

10

20

30

40

50

60

70

80

90

100

Figure 4.2. Pareto analysis for qualitative tests results

4.2.2. Hard and Soft Failures

Failures can be classified into two broad categories: soft and hard failures. Soft failures

are the gradual loss of performance of a product, while hard failures cause it to stop working

(Meeker and Escobar, 1998). However, in this chapter, “hard” failures signify those failures

where the user is notified as soon as they occur. Hard failures either directly influence

functioning of a device or are self-announcing. An example is a faulty alarm; as soon as the

problem occurs, users are notified that something is wrong with the device, and it is sent for

repair. We can assume that a hard failure occurs just before a SPI or corrective maintenance;

therefore, the work order’s starting time can be considered as the failure time (complete data).

Arguably, soft failures are identified at SPI when operators have postponed reporting them.


“Soft” failures are minor defects that have no or little influence on the device’s

functioning. These defects such as “chassis/housing” defects may occur a long time before a

scheduled or non-scheduled work order’s starting time, but the device still can function properly.

The problem is not reported by the user and is detected only at the next SPI or during corrective

maintenance triggered by a hard failure. Therefore, there is a time delay between the real

occurrence of a soft failure and its detection. Ignoring this time is misleading; it should be

considered as left or interval censoring time.

While we can interpret soft failures as the result of system deterioration over time, hard

failures are mostly unpredictable and random in nature.

Interval censoring takes place when a minor failure or defect happens between a “no

failure found” inspection and a “failure detected” inspection. If a defect occurs between the

acceptance date and the first corrective maintenance or SPI, interval censoring is reduced to left

censoring in which the starting time for the interval is zero.

Although there is not always a clear distinction between soft and hard failures, we used

them to classify the components/features of a system. We categorized components/features in

two groups depending on whether their associated failures were soft or hard failures and

analyzed the failure data separately for each group. We classified components/features according

to their failure percentage at scheduled and non-scheduled work orders. Classification of soft and

hard failures can be found in Appendix C. This classification has been approved from a practical

point of view by clinical engineers involved in this study. In addition, it should be noted that

Failure Mode and Effects Analysis (FMEA) could be also used to decide which components

should be included in each category of components with soft and hard failures.


In order to check the validity of our treatment with soft and hard failures, we calculated

the number of hard and soft failures detected at each single scheduled and non-scheduled work

order. Table 4.5 shows the results. The significant percentage of single hard failure detected in

non-scheduled work orders emphasizes that it is correct to assume all hard failures’ times as

complete data. The close values obtained for one and two detected soft failures at both scheduled

and non-scheduled work orders show that we can treat soft failures as left or interval censored

data.

Failures frequency (%) in work orders

No SPI NS

Soft failures

1 33.92 57.55

2 34.39 31.87

3 22.13 8.24

4 7.79 2.34

5 1.77 ---

Hard failures

1 79.72 83.72

2 18.88 14.88

3 1.22 1.24

4 0.17 0.16

Table 4.5. Soft and hard failure frequencies at scheduled and non-scheduled work orders

4.2.3. Proposed Policy for Analyzing Soft and Hard Failures

Because of the large number of censored data and combination of hard and soft failures,

we propose separate policies for analyzing failure data at different levels, including system level,

components corresponding to hard failures, and components corresponding to soft failures.


System level

All non-scheduled work orders’ starting times should be considered as a system’s exact

failure times. The time between SPIs having at least one failure identified should be considered

as either left or interval censored. The right censored data is specified according to the result of

the last inspection (failure or pass), and its distance to the end of the test. Figures 4.3a and 4.3b

illustrate the proposed policy, and the policy rules are explained in detail in Appendix D.

Figure 4.3a. The policy for analyzing failures at the system level

Figure 4.3b. The policy for dealing with “no failure” devices at the system level

TEnd

Event Times WO Type Data Type

x Right Censored (if x <= 0.5*MTBF)

0.5*MTBF Right Censored (if x > 0.5*MTBF)

x

TStart

Non-scheduled

work order time

with Failed status

Scheduled

work order time

with Failed status

TEnd


x1 NS Complete

x2 PM Left Censored

x3 PM Left Censored

x4 Pass

x4+ x5 Right Censored (if x5 <= 0. 5*MTBF) x4+0.5*MTBF Right Censored (if x5 > 0.5*MTBF)

x1

x2

x3

x4

x5

TStart

Scheduled

work order time

with Pass status

t4

t3

t1

t2

Non-scheduled

work order time

with Failed status

Scheduled

work order time

with Failed status

TEnd


x1 NS Complete

x2 PM Left Censored

x3 PM Left Censored

x4 Pass

(x4, x4+x5] PM Interval Censored

x6 Right Censored (if x6 <= 0. 5*MTBF)

0. 5*MTBF Right Censored (if x6 > 0. 5*MTBF)

MTBF= Mean time between failures

x1

x2

x3

x4

x5

TStart

x6

Scheduled

work order time

with Pass status

t5

t3

t1

t2

t4


When the time between the last inspection or acceptance test (when there is no failure)

and the end of the test is long, particularly for soft failures, there are two possibilities: 1. a soft

failure may have occurred somewhere between the last inspection and the end of the test; 2. no

failure has occurred in the meantime. Since there is no information available regarding these two

possible scenarios, and as it is expected that a failure occurs sometime between the average (or

proportion of) times between failures and the end of the test, we consider a proportion of that

time as the right censoring time. In order to be consistent, we consider (0.5*mean time between

failures) as right censored data. We conducted different analyses with different values (0.5, 0.75

and 1) as the ratio to decide which proportion of mean time between failures should be

considered as the right censoring rule. Even though we may have several failures in that interval,

we only use an approximation to the first event.

Component level – hard and soft failures

For the components corresponding to hard failures, work orders’ starting times are

considered as a component’s exact failure times if they report a failure of that component. The

time interval between the last failure of the component and the end of the test is considered as

right censored data.

Figure 4.4. The policy for dealing with hard failures – component level

Non-scheduled

work order time

Scheduled

work order time

TEnd

Failure Times WO Type Data Type

x1 NS Complete

x2 PM Complete

x3 PM Complete

x4 NS Complete

x5 Right Censored

x1

x2

x3

x4

x5

TStart


For soft failures, all rules are similar to the system level’s rules, except that we never

have complete data. If there is no inspection with “no problem found” between two consecutive

failures, the time interval between these two failures is considered as left censored data.

4.2.4. Trend Analysis – The Laplace Trend Test

The “Laplace trend test”, the “Reverse Arrangement test” or the “Military Handbook

test” (NIST/SEMATECH, 2006) can be used to quantitatively determine whether the failure

times exhibit a significant improvement or degradation trend.

The Laplace trend test can be used for multiple repairable systems to check between no

trend and non-homogeneous Poisson process (NHPP) exponential models. Assume that there are

m similar and independent systems. Let ijT represent the time of the thi failure of the thj system,

mjni j ,...,2,1,,...,2,1 , observed over the period ],[endjstartj TT , where

jn jendT T . Let:

if the process is time truncated

1 if the process is failure truncated

j

j

j

nn

n

The trend test can then be conducted using the following formula:

m

jstartjendjj

endjstartjj

m

j

n

i

m

j

ij

TTn

TTnT

LA

j

1

2

1 1 1

)(12

1

)(2

1

. (4.1)

The value of LA should be compared to high (for improvement) or low (for degradation)

percentiles of the standard normal distribution. The Laplace trend test compares the average of

failure arrival times to the midpoint of the observed time interval. There is a trend when the

average of the failure arrival times deviates from the midpoint of the observation interval. In the


Laplace trend test, when nj 3, the approximation with the Normal distribution generally

provides good results.

For a trend test at a significant level of , we have: If / 2LA z or / 2LA z , then

there is a trend, with direction of the trend (Ansell and Phillips, 1994) indicated by the following:

If / 2LA z , the process is improving, so the times between failures are increasing.

If / 2LA z , the process is deteriorating, so the times between failures are decreasing.

4.2.5. Trend Analysis and Distribution Fitting of Failure Data

We analyze the failures data based on the proposed policy described in Section 4.2.3. We

use the Laplace trend test for trend analysis of failure data, taking the midpoint of censoring

intervals as an approximation of the actual failure times. Even though this is an approximation

and therefore not precise, we opt to use it because no available trend test analysis in the literature

incorporates interval censored data (except right censored). Trend analysis using the midpoints is

simple and does not require any complex calculation. Moreover, when the censoring intervals are

short, reliable results can be obtained. In Section 4.4 we propose using the likelihood ratio test to

check for trends in the failure data with censoring. We use the EM algorithm to find the

parameters, which maximize the data likelihood in the case of no trend and trend assumptions. A

recursive procedure is proposed to calculate the expected values in the E-step of EM.

The mean cumulative function (MCF) and confidence limits are plotted versus system

age in days using the RELIABILITY procedure in SAS. The MCF plot describes average

behavior of a system or multiple systems under study. It is constructed incrementally at each

failure event by considering the number of systems at risk at that point in time. The number of


systems at risk depends on how many systems are contributing information at that particular

point in time. Information can be obscured by the presence of censoring and truncation (Trindade

and Nathan, 2008). The RELIABILITY procedure in SAS provides a nonparametric estimate and

plot of the MCF for the number or cost of repairs for a population of repairable systems.

The nonparametric estimate of the MCF, the variance of the MCF estimate, and

confidence limits for the MCF estimate are based on Nelson (Nelson, 1998). MCF, usually

denoted by )(tM , is defined as the mean of the cumulative number of events up to time t . The

method does not assume any underlying structure for the repair process (Johnston,

http://support.sas.com/rnd/app/papers/reliability.pdf, accessed in Feb 18, 2011).

Analysis of failures data at the system level

As shown in Figure 4.5, the MCF plot at the system level is not a straight line. Its initial

concave appearance is followed by a stable period, and a convex curve appears at the end.

Therefore, the assumption of HPP for failure data is not valid. The end-of-history or right-

censoring ages are plotted in an area at the top.

days

S

a

m

p

l

e

M

C

F

0 500 1000 1500 2000 2500

- 2. 5

0

2. 5

5. 0

7. 5

10. 0

No. of Uni t s 674

No. of Event s 2358

Conf . Coef f . 95%

Figure 4.5. The MCF plot for the failures at the system level (0.5*MTBF)


Using equation 4.1, the Laplace trend test value for the failure data is LA=12.59. At the

significant level / 2 0.05, z 1.96 , the systems show a degrading trend.

Since the failure times are not independently and identically distributed, we first classify

the times between two consecutive events (n-1th

, nth

) of each system. An event can be either a

failure or a censoring event. The nth

event of a system is censoring if at inspection no failure is

found; otherwise, it is a failure event. We then pool the times between n-1th

and nth

events of all

systems and conduct the Weibull analysis for each category separately. For example, for

calculating the times to the first event, all devices that have at least one failure during their life

are identified. For these devices, the time to the first failure is extracted with “failure” as an

event. For devices with no failure at all, the first and only event is a censoring event with the

censoring time from 0 to the last updating time of the CMMS database. Finally, the Weibull

analysis is performed for the times between n-1th

and nth

events using LIFEEG procedure in SAS.

Tables 4.6 and 4.7 summarize the Weibull parameters estimated for all times between events.

The results of the Weibull distribution fitted to the times between n-1th

and nth

event

(failure/censoring)

n No of

observations

No of

non-

censored

No of

right

censors

No of

left

censors

No of

interval

censors

)11

(

1 674 237 28 348 61 1.5680 396.9852 356.6204

2 646 215 111 283 37 1.5175 318.9879 287.5696

3 536 177 100 222 37 1.3499 306.6525 281.2007

4 433 143 116 162 12 1.5622 311.9259 280.3152

5 318 101 115 96 6 1.3042 301.7416 278.5013

6 203 48 114 40 1 1.1024 381.6272 367.9733

7,8 129 34 58 35 2 0.9718 247.9506 251.0791

9 89 41 30 15 3 0.8472 156.9852 171.1464

Table 4.6. Estimated parameters of Weibull distribution for times between events – system level


n )11

(

estimate

95% confidence

limits estimate

95% confidence

limits estimate

95% confidence

limits

1 1.5680 1.4414 1.7058 396.9852 372.1665 423.4590 356.6204 333.3963 379.8445

2 1.5175 1.3867 1.6606 318.9879 297.9330 341.5308 287.5696 267.7068 307.4324

3 1.3499 1.2165 1.4868 306.6525 282.5753 332.7812 281.2007 257.6725 304.7289

4 1.5622 1.3977 1.7461 311.9259 288.1422 337.6727 280.3152 257.8563 302.7741

5 1.3042 1.1365 1.4967 301.7416 269.8376 337.4177 278.5013 246.4507 310.5519

6 1.1024 0.8865 1.3708 381.6272 307.6500 473.3927 367.9733 284.4419 451.5047

7,8 0.9718 0.7618 1.2395 247.9506 192.9140 318.6887 251.0791 182.2192 319.9390

9 0.8472 0.6710 1.0695 156.9852 115.1459 214.0272 171.1464 112.4674 229.8254

Table 4.7. Estimated 95% confidence limits for the Weibull distribution parameters

As can be seen in the tables, earlier failures show a degrading trend; however, later

failures appear more random, i.e. follow exponential distribution. The values are slightly

decreasing for the first five events, which have more non-censored data; thus, they constitute a

more reliable result. The results are also consistent with the result of the Laplace trend test.

Since SAS only gives the 95% confidence limits estimation for the estimated parameters

( and ), we calculated the limits for using the Delta method (Meeker and Escobar, 1998).

For details, see Appendix E. The estimated lower and upper limits for using the Delta method

appear in the last two columns of Table 4.7.

Analysis of all hard failures, soft failures, and the failures of individual components

Appendix F describes the results obtained from analysis of all hard and soft failures

separately, and some of the system’s components individually. An overview of the analysis

results appears in Table 4.8.


Level *LA

System level 12.59 (↓) ↓ ↓ ↓

All hard failures 6.12 (↓) ↓ ↓ ↓

AC Plug/Receptacles 0.87 (--) -- -- --

Alarms 4.13 (↓) ↓ ↓ ↓

Circuit Breaker/Fuse 2.39 (↓) N/A N/A N/A

Controls/Switches 1.15 (--) -- -- --

Indicators/Displays -2.72 (↑) ↓ ↑ ↑

Mount 9.31 (↓) ↑ ↓ ↓

Occlusion Alarm -1.11 (--) -- -- --

All soft failures 11.08 (↓) ↕ ↕ ↕

Audible Signals -7.94 (↑) ↓ ↑ ↑

Battery/Charger 5.69 (↓) ↓ ↑ ↑

Chassis/Housing 11.40 (↓) ↕ ↓ ↓

Fittings/Connectors 8.15 (↓) ↕ ↓ ↓

Labeling 4.13 (↓) N/A N/A N/A

Strain Reliefs 3.03 (↓) N/A N/A N/A

Line Cord 10.61 (↓) ↕ ↕ ↕

Grounding Resistance -2.10 (↑) N/A N/A N/A

Symbols: ↑ improving/increasing ↓ degrading/decreasing ↕ no obvious trend -- no trend

*LA (the Laplace test value) N/A (not applicable)

Table 4.8. An overview of the results obtained from statistical analysis at different levels

The total number of failures and their completeness or censoring status determine the

accuracy of the results. With more complete failure data, the results are better and more

consistent. For example, due to the large number of exact hard failure data, the results for this

group exhibit a degrading trend consistent with the results of the Laplace trend test. However, in

the case of only a few failures, nothing can be learned because of the large number of censored

data. “N/A” and “↕” results in Table 4.8 stem from the scarceness of failure data. Nor were the

4.3. Basic Assumptions for the System Under Study 78

exact soft failure times available; thus, all data for this group were either left or interval

censored, and consequently, the results showed a large variation. Overall, the Laplace trend test

appears to yield results that are more reliable: analyzing the data marginally may show large

variations due to the censoring and scarceness of failure data; and when the results are

aggregated, they do not show a clear trend.

4.3. Basic Assumptions for the System Under Study

The results described in Section 4.2 show that components of a general infusion pump

(Figure 4.1) or any other medical device or more general complex repairable system could be

categorized into two groups. Failures of some components such as indicators, switches, and

occlusions make the system stop functioning as soon as they occur, and they are fixed

immediately. Then, these failure times are known. These components have so-called hard

failures. The second group consists of components with hidden or soft failures such as circuit

breakers or informative components such as audible signals. The pump can continue to operate if

one of these components, as for example, an audible signal, fails. Therefore, soft failures are

usually not self-announcing and are rectified only at inspections, and their failure times are either

left or interval censored. In other words, there is a time delay between the real occurrence of a

soft failure and its detection. Although the system still may be able to continue working, but the

presence of soft failures can reduce its performance and eventually require to be fixed.

Once the component experiences a soft failure it stays in the same condition until it is

fixed. Even if the component may deteriorate in some way after the failure the repair at

inspection will return it to the state just before failure (minimal repair), or will renew it.


Soft and hard failures can be also interpreted by their consequence on the system. Hard

failures have more significant influence on the system operation. Soft failures are less critical for

the system. Meeker and Escobar (1998, p. 327) also consider two types of failures: hard and soft.

They suggest that soft failures may be defined as when degradation (gradual loss of

performance) exceeds certain specified level, while hard failures cause the system to stop

working.

There are different categories of soft failure of components in complex devices. One

category includes a wide range of integrated protective components used to protect a system

from unwanted transient incidents such as current and voltage surges. For example, infusion

pumps and medical ultrasounds are equipped with circuit breakers to protect them against

overload and short circuits. If these protective components fail, the system can continue its main

function, although the risk of damage increases if an overload or short circuit takes place. The

failures of protective components are not self-announcing and can only be rectified at inspection.

This group of components does not age further when a failure occurs and until the

component is fixed and put back in service. Standby redundant components are another category

of components with soft failures. They are used in many critical systems to enhance system

safety and reliability. The system, depending on its design, can continue functioning if a required

number of these redundant components work. Uninterruptible Power Supplies (UPS), dual

redundant processors, and redundant fan trays are examples of redundant components used to

increase system safety and availability. A CAT scan (CT) may have several CPUs running

different algorithms for image processing. If one CPU fails, processing is routed to other CPUs

without interrupting or degrading the system’s performance. Again, components in this category

do not age after failure, and usually their failures are detected only at inspection.


In addition to protective and redundant components, there are other components whose

failure does not halt the system, even though failure can have serious consequences, even

catastrophic, if left unattended. With infusion pumps, for example, audible or visual signals are

used to communicate with operators and inform them of the status of the patient to whom the

device is attached. When the level of liquid delivered to a patient reduces to a certain level, the

component responsible for producing signals starts to send a warning alarm. If the component

fails, the pump can still function, but the patient’s health risk increases if the operator does not

take action. Devices in overcrowded hospitals are always in use, and even though some of their

informative features such as audible signals may fail, they remain in operation until the next

scheduled inspection.

The failures of components that do not carry out the main functions of a system can

sometimes be hidden. For example if an external hard drive backup of a computer fails, the

system can still function, so the operator may not be informed about its failure and the chance of

losing data may be increased.

The followings are the basic assumptions, which we make for the system under study in

this and all future chapters. Additional assumptions if required will be made and stated at each

section of the thesis.

Basic Assumptions

A repairable system with components (units) subject to hard or soft failures

Hard failures are detected and fixed immediately.

Soft failures are detected at inspection.

4.4. A Proposed Method for Trend Analysis of Failure Data Subject to Censoring 81

At the beginning, a component may have any initial age, which can be zero if it starts as

good as new.

Failures of the units follow a NHPP with a power law intensity function.

Components are inspected periodically at times k ),...,2,1( nk (Figure 4.6), over a given

life cycle of length T n .

Components with hard failure do not fail simultaneously.

Soft failures or their combination cannot convert to hard failure.

Inspections are perfect and inspection and repair times are negligible.

Figure 4.6. Scheme of hard failures and scheduled inspections in a cycle

4.4. A Proposed Method for Trend Analysis of Failure Data Subject to Censoring

We consider a system with the assumptions stated in Section 4.3. Since soft failures are

subject to censoring, in this section we propose a method for trend analysis of this type of failure.

We assume that the failure process follows an NHPP with a power law intensity function. We

use the EM algorithm to estimate parameters of the power law process and propose a recursive

method to calculate the expected values in the EM expectation step. In the maximization step, we

use the Newton-Raphson method. We then apply the general likelihood ratio test to test HPP

against NHPP alternative (Crowder et al., 1991). In some practical situations, the midpoints of

the censoring intervals may be considered as the failure times and used to estimate the

2 )1( k k

HF HF HF

S S S S

HF: Hard failures S: Scheduled inspection

n T


parameters of the failure process and to perform trend test analysis. If the inspection intervals are

not short, this approximation of actual failure times can be inaccurate.

Additional assumptions for this section are as follows:

Additional Assumptions

The system has a power law intensity function 1( )x e x and ( )x e x .

The unit is minimally repaired at the moment of inspection, if found failed.

It should be noted that a more common form to present a power law is

1

( )x

x

or 0

( ) ( )x x

x t dt

; however we use

1( )x e x for convenience of maximum

likelihood estimation. 0y is the initial age of a unit.

The censoring intervals are the intervals between two consecutive inspections (Figure 4.7).

Figure 4.7. Initial age 0y and failure times of a unit

Our first goal is to estimate parameters of the NHPP from censored data on

1 2, ,...X X using the maximum likelihood method and to test for possible trend in the data using

the likelihood ratio test. Because the random variables 1 2, ,...X X are dependent, our problem is

2 T )1( k k

rX

1st failure rth failure

0 0x y 3

2X1X

2nd failure


complex. Note that in our approach, inspection times need not be periodic; nonequal inspection

intervals can also be used.

Our main objective is to develop a method to perform a trend analysis of censored failure

data for a unit or a group of similar units.

4.4.1. Parameter Estimation and Trend Analysis

We use the likelihood ratio test (Meeker and Escobar, 1998; Engelhardt et al., 1990),

with null ( 0H ) and alternative ( 1H ) hypotheses, as follows:

0H : Homogeneous Poisson process ( 1 )

1H : Non-homogeneous Poisson process ( 1 )

Let 0L and 1L be the maximum likelihoods of the data when 1 , and without

restrictions, respectively. The likelihood ratio test in our case uses the statistic 2

0 12ln( / )L L

with 1 degree of freedom and rejects the null hypothesis if 2 is greater than the appropriate

critical value ( 0L assumes 1 parameter less than 1L ).

The likelihood function incorporates failure events and a possible last censoring event

(right censoring) for each unit (Figure 4.8).

Figure 4.8. Lower and upper bounds for failure times and the last censoring time

1u

2lT

rl

ru

rX1st failure 2nd failure rth failure

1l

2u

2X 1rX

0 0x y

1ru

1rl

1X


In Figure 4.8, il is the last known (inspection) time when the unit was still working before

the thi failure, and iu is the time that the unit was found failed, where 1,2,...,i r , 0r .

When ( 1)thr event is a right censoring event (not a failure), 1rl is the right censoring time, and

1ru . The origin for measuring 1il and 1iu is the previous failure detection time ( iu ).

Without right censoring, 1 2 ... ru u u T , and with right censoring, 1 2 1... r ru u u l T .

Note that this case may be considered as a Type I censoring, with T being a calendar time

interval, but not the operating time interval, as common in other applications.

Let

0

i

i j

j

y x

, 0 0y x .

where jx are the actual survival times measured from 1ju . Therefore, iy is the unobserved

actual age of the unit at the time of iu and 1 1i i i i iy l y y u . In the case with right censoring,

we have the following observations:

1 1{ , 1,..., 1}i i i i iy l y y u i r r +11H .

Let 1 1{ ( , ]}i i i i i iA y y l y u . Thus,

1 2 1... rA A A r +11H .

In the case with r failures and no right censoring, we have 1 2... rA A Ar1H .

Let ( , ) . Let 1n r in the case with right censoring, and let n r in the case

without right censoring. The common case when a failure is observed as soon as it occurs and the

system is repaired instantly (that is, when all inspection intervals are of length zero, Crowder et


al., 1991)) reduces to our case when 1n r . If all iy were known, the complete data likelihood

would be

1 0

1

1

[ ( ) ( )] ( ( ) ( ))

1 1

( | ) ( , ) ( | )

( ) ( ( )) ,

i

i i n

n

Y Y i i

i

n ny y y y

i i

i i

L y f y f y y

y e y e

(4.2)

where ( ) ( , )i i iy y e y .

The log-likelihood is then

01

0

1

ln ln ( , ) ( ) ( )

ln ( 1) ln( ) ( )

nyn

Y i

i y

n

i n

i

L f y y s ds

n n y e y y

(4.3)

The following system of equations should be solved to obtain parameters ( , ) ,

which maximize the log-likelihood:

0

0 0

1

ln ( ) 0

ln ln( ) ( ln ln ) 0

n

n

i n n

i

L n e y y

nL y e y y y y

(4.4)

The actual likelihood of observed data 1H n is

1 1 0( | ) ( , ,..., | , )n nL P A A A y n1H

1 1

1 0 1

1,...,

... ( ,..., | , ) ...i i i i i

Y n ny l y y ui n

f y y y dy dy

11 1

1 1 0 1

1,...,

... ( | , )... ( | , ) ...n

i i i i i

Y n n Y ny l y y ui n

f y y f y y dy dy

(4.5)


We use the Expectation-Maximization (EM) algorithm (McLachlan and Krishnan, 1996)

to find the maximum likelihood estimates (MLE) of the parameters ( , ) given the

observed data. The EM finds the MLE iteratively using the two following steps:

Expectation Step: In this step, the expected value of the log-likelihood function is calculated:

( )

( )( , ) [ln ( , ) | ]t

t

YQ E f Y

n1H

Maximization Step: In this step, ( 1)t is found as to maximize the expected log-likelihood:

( 1) ( ) ( )( , ) max ( , )t t tQ Q

The iteration continues until the convergence of ( )t .

Therefore, in our problem the ( )( , )tQ function is calculated as follows:

( )

( )

0 0 0

1

( , | , ) [ ln ( ) ( ( ) ( )) | , ]t

nt

i n

i

Q y E Y Y y y

n n1 1H H

lnn n ( ) ( )0 0 0

1

( 1) [ ln | , ] ( [ | , ] )t t

n

i n

i

E Y y e E Y y y

n n1 1 H H (4.6)

For simplicity, we will assume that 0 0y , i.e., the initial age of a unit is zero in the rest

of the section.

To maximize ( )( , )tQ in ( , ) , we need the first and second derivatives of ( )( , | )tQ n

1H

( )

( ) ( )

1

[ | ]

[ ln | ] [ ln | ]

t

t t

n

n

i n n

i

Qn e E Y

Q nE Y e E Y Y

n1

n n1 1

H

H H , (4.7)


( )

( )

( )

2

2

2 2

22

2 2

[ | ]

[ ln | ]

[ (ln ) | ]

t

t

t

n

n n

n n

Qe E Y

Q Qe E Y Y

Q ne E Y Y

n1

n1

n1

H

H

H

. (4.8)

We use both the “complete” EM algorithm and a modification of the EM - the EM

gradient algorithm (Lange, 1995) which solves the M-step of the algorithm using one iteration of

the Newton-Raphson method (Kelley, 2003) as follows:

( )

2 ( ) ( )( 1) ( )

2

( , ) ( , )( ) t

t tt t Q Q

, (4.9)

where 2 ( )

2

( , )tQ

is the Hessian matrix of

( )( , )tQ .

The parameters in the complete EM algorithm are obtained by applying the Newton-

Raphson method to find ( 1)t . The ( 1)t is the limit of the sequence 0

( , ){ }k

t k

where

( ,0) ( )t t for {1,2,...}t (Booth, 2001),

( , )

2 ( ) ( )( , 1) ( , )

2

( , ) ( , )( ) t k

t tt k t k Q Q

. (4.10)

In equations (4.6 - 4.8) we need to calculate the following expected values:

( )

1

[ ln | ]t

n

i

i

E Y

n1H , ( ) [ | ]t nE Y

n1H , ( ) [ ln | ]t n nE Y Y

n1H , ( )

2[ (ln ) | ]t n nE Y Y

n1H .

Calculation of these expected values involves intractable integrals hard to perform

analytically. For example,


1 1

( )

1 1

1 1

1,...,

1 1

1,...,

... ( ,..., ) ...

[ | ]... ( ,..., ) ...

i i i i i

t

i i i i i

n Y n ny l y y ui n

n

Y n ny l y y ui n

y f y y dy dy

E Yf y y dy dy

n

1H .

Therefore, we will derive a general recursive procedure to calculate the required expected

values.

Let ( ) ( )n n nB y g y , where ( )ng y is a function of nY , such as ( )n ng y y

or

( ) lnn n ng y y y . Then,

1 1 1 1( ) ( ) ( | )i

i

i

u

i i i i X i

l

B y B y x f x y dx , (4.11)

where 1 1

1 1

i i

j i j

j j

l y u

and

1

1

( )

1 1( | ) ( )

y xi

yi

i

s ds

X i if x y y x e

, and , 1,...,1i n n .

As a special case of nB , let ( ) ( ) 1n n nC y g y .

Then,

( ) 0 0[ ( ) | ] (0) / (0)t nE g Y B C

n1H .

For a proof, see Appendix G. A similar recursive equation can be derived to calculate the

expected value of 1

lnn

i

i

Y

, as follows:

Let ( ) 0n nD y , then

1 1 1 1 1 1( ) [ln( ) ( ) ( )] ( | )i

i

i

u

i i i i i i i X i

l

D y y x C y x D y x f x y dx , (4.12)

, 1,...,1i n n . Then

( ) 0 0

1

[ ln | ] (0) / (0)t

n

i

i

E Y D C

n1H .


For a proof of this equation, see Appendix H.

To obtain 0L , or the maximum likelihood of the data when 1 , first note that because

1 2, ,...X X are then independent random variables, we have

0 1 1 1 2 2 2

e e

1

( | ) ( , ,..., | )

( ) ,i i

n n n

nl u

i

L P l X u l X u l X u

e e

n1H

and

( ) ( ) 1

1 1

e

e e1

[ | ] [ ... | ]

[ | ] [ | ]

e

( )

t t

i

i

i

i i

n n

n n

i i i i i

i i

u

x

nl

l ui

E Y E X X

E X E X l X u

x e dx

e e

n n1 1

n1

H H

H

e e e e

e e1

e e

e e1

( ) e ( )

( )

e .

i i i i

i i

i i

i i

l u l uni i

l ui

l uni i

l ui

l e u e e e

e e

l e u en

e e

In the case of right censoring, i.e. when 1n r and 1ru , the last element in the

sum is just 1rl .

Parameter ( 1)t can be obtained directly from (4.7) as follows:

( )

( 1) ln( )[ | ]t

t

n

n

E Y

n1H

. (4.13)

When there is more than one unit, equation (4.6) should be changed appropriately.

Assume there are m similar independent units with jn histories ( 1,...,j m ), and that each unit

may have failures and a right censoring event. Then, equation (4.6) is revised as follows:


( )

10 0( , | ,..., , ,..., )t

mQ y y 1 mn n1 1H H

( ) 0

1 1 1 1

ln ( 1) { [ ln | , ]}j

t

nm m m

j j ji j

j j j i

n n E Y y

jn1H

( ) 0 0

1

{ [ | , ] }tn

m

j j j

j

e E Y y y

jn1H (4.14)

where 0jy is the initial value for unit j , 1,2,...j m .

4.4.2. Case Studies

In Section 4.2.5, for trend analysis of components of an infusion pump, we considered the

mid points of the censoring intervals as the failure times and applied the Laplace trend test.

We apply the method proposed in this section to subsets of failure data for several

components of the general infusion pump with hidden failures and compare the results with those

obtained by using the “mid-point” method. To obtain the maximum likelihood estimates of the

parameters we use both the EM algorithm (equation (4.10)) and the modification of the EM

(equation (4.9)) with one Newton-Raphson iteration, and compare their results and the

algorithms’ performances.

The first component considered is an audible signal. We use the failure data of 80

randomly selected units out of 674 units. The dataset contains 125 records or histories collected

over intervals of between one and seven years with inspection intervals of approximately 1.6

years. Each unit has up to 3 histories including failure censoring intervals and the last right

censoring interval. Table 4.9 shows sample data for three units. The third and fourth columns

show the lower and upper bounds of the censoring interval for a failure. The symbol in the

jiu column denotes right censoring. See Appendix P for complete dataset.


Unit No. j History No. jil (years) jiu (years)

1

1 0.000 1.300

2 2.886 3.597

3 1.408

2 1 1.289

3 1 1.906 3.419

2 2.886

Table 4.9. Sample data for three units’ failure and censoring events

In total, we found 41 units with right censoring only; a summary of the data appears in

Table 4.10.

No of units with right censoring only 41

No of units with one failure and one censoring 33

No of units with two failures and one censoring 6

Total No of units 80

Table 4.10. Summary of the histories for Audible signals

To apply the proposed method, we used the initial parameters

(0) (0) (0)( , ) ( 0.3,0.5) . We obtained functions iB , iC and iD for 0,1,..., ji n

recursively, using equations (4.11) and (4.12), for every j , 1,...,80j . We applied the

Composite Simpson’s rule for numerical integration, with step length h depending on the

required accuracy; after some experimentation, we used 310h . We then used equation (4.14)

and the first and second derivatives and applied equations (4.9) and (4.10) to obtain

( 1) ( 1) ( 1)( , )t t t from ( )t . In the modified EM algorithm, ( )t converged to


ˆ ˆˆ( , ) ( 1.672,0.921) after 27 iterations. In this case, we obtained 1ln 120.332L . We

also obtained an estimate ˆ 1.781 when 1 , with 0ln 120.396L . Therefore, the

likelihood ratio test statistic 2

0 12ln( / ) 0.128L L , which is less than the critical value

2

1,0.05 3.841 . The results are summarized in Table 4.11. The parameter /e is also

calculated for comparison. Accordingly, we do not reject the null hypothesis and conclude there

is no visible trend in the pattern of given failures. However, this may also be due to the small

number of histories per unit (up to three), not enough to show the trend clearly. Applying the

complete EM resulted in an estimate ˆ ˆˆ( , ) ( 1.672,0.920) after 24 iteration for t and 52

iteration in total for both t and k . The parameters obtained are very close to the parameters

obtained by applying the modified EM. The required process time for the complete EM

algorithm was about 2 times longer than the time of the modified EM. The same conclusion

about trend as above is made when the parameters obtained from the complete EM are used in

the likelihood ratio test (Table 4.11).

Next, we estimated the parameter using the mid-point method. We obtained

ˆ ˆˆ( , ) ( 1.858,1.050) , slightly different from the above results using the complete EM and

its modified version. Although in this example, there is no significant difference between the

results, a larger dataset and more failure histories may yield different results, and the results

obtained when considering mid points might be misleading. On the other hand, using mid-points

is much simpler (as it does not need numerical integration), and the obtained values may be used

as initial values in the EM algorithm. This may greatly reduce the number of iterations.

We used another failure dataset of 38 housing/chassis units of general infusion pumps.

Figure 4.9 shows an infusion pump with a cracked door hinge. A summary of the number of


failures for the units are given in Table 4.12. See Appendix P for complete dataset. All units have

a right censoring record. We applied the proposed method with the same initial parameters

(0) (0) (0)( , ) ( 0.3,0.5) . The results are summarized in the third column of Table 4.11.

The chi-square test for this case study also shows no trend in the failures. However, the value

obtained from the mid-points is slightly greater than 1 that would indicates the units are

degrading over time. Considering that the likelihood ratio test is more accurate, this would be

likely a wrong conclusion.

Component Name Audible Signal Housing/Chassis Battery

No of records 125 164 897

No of units 80 38 674

Maximum Likelihood ( 1 )

Modified EM 1

ˆˆ 1.672, 0.921

ˆ 6.145

ln( ) 120.332L

1

ˆˆ 0.144, 0.918

ˆ 1.17

ln( ) 153.425L

1

ˆˆ 2.908, 1.778

ˆ 5.131

ln( ) 698.311L

Maximum Likelihood ( 1 )

Complete EM 1

ˆˆ 1.672, 0.920

ˆ 6.149

ln( ) 120.332L

1

ˆˆ 0.141, 0.916

ˆ 1.166

ln( ) 153.425L

1

ˆˆ 2.908, 1.779

ˆ 5.130

ln( ) 698.310L

Maximum Likelihood ( 1 ) 0

ˆˆ 1.781, 1.000

ln( ) 120.396L

0

ˆˆ 0.280, 1.000

ln( ) 153.536L

0

ˆˆ 1.926, 1.000

ln( ) 751.228L

2 0.128 0.222 108.835

Conclusion no trend no trend trend

Parameters obtained from

mid-points ˆˆ 1.858, 1.050 ˆˆ 0.719, 1.215 ˆˆ 2.875, 1.763

Table 4.11. Estimation and testing results obtained for audible, housing and battery units

Figure 4.9. A cracked door hinge in an Infusion Pump


Number of failures Number of units

0 0

1 2

2 9

3 13

4 6

5 6

6 1

7 1

Total 38

Table 4.12. Summary of the chassis/housing failures

As the last case study, we consider the battery of a general infusion pump. An infusion

pump can be powered either from an AC line or from a rechargeable or disposable battery. The

infusion pump in our case study is equipped with a rechargeable battery. Battery depletion,

battery overcharged, battery low voltage, are common failure modes of a battery. Some battery

failures, which damage the battery, require it to be replaced. For example, Figure 4.10 shows a

damaged sealed lead acid battery, which is damaged by battery overcharging. A depleted battery

is charged if the depletion is not due to physical damage of the battery.

Figure 4.10. Damaged sealed lead acid

Battery caused by battery overcharging


The battery dataset consists of 897 records describing failures of 674 batteries. Table 4.13

gives a summary of the number of failures. All units have a right censoring record. Due to the

large dataset, it is not reported in the thesis.

Number of failures Number of units

0 507

1 129

2 27

3 6

4 4

5 0

6 1

Total 674

Table 4.13. Summary of the battery failures

The results of applying the proposed model for batteries are shown in the last column of

Table 4.11. High value of 108.835 for the Chi-square test indicates a low p-value. According to

the results, there is an obvious increasing trend in the failure of batteries. In other words,

batteries exhibit more failures over time.

We compared the proposed method of using likelihood test with the standard errors of the

estimation (Meeker and Escobar, 1998). We calculated the standard errors of the parameter

estimators using the modified EM from the observed information matrix. Calculation of the

observed information matrix requires additional effort and more complex numerical integration

and will not be discussed here. The standard errors of the parameter estimators for all three

components are summarized in Table 4.14. As an example, the following variance-covariance

matrix of the parameters and is obtained for the first case study (Audible Signals):


12 2

2

2 2

2

ˆˆ ,

ln ln

ˆ ˆˆ ˆ ˆ( ) ( , ) 0.0622 0.0281

ˆ ˆ ˆˆ 0.0281 0.0198ln lnˆ( , ) ( )

L L

Var Cov

L LCov Var

.

The estimated standard errors of and are the square roots of the diagonal elements

of the variance-covariance matrix.

Case Study Standard Error

Audible Signal 0.2494 0.1407

Housing/Chassis 0.2133 0.1176

Battery 0.1387 0.0851

Table 4.14. Estimated standard errors

The standard errors obtained are in accordance with the likelihood ratio test. For

example, in the case of batteries, the 95% confidence interval for is

ˆ ˆˆ1.96 ( ) 1.779 1.96 0.0851 [1.612,1.946] ,

well above 1. Or, for audible signals, the 95% confidence interval is [0.645,1.197], covering

1.

The total numbers of iterations required to obtain the parameters in the complete EM, the

modified EM and Mid-Point method are given in Table 4.15 for all three cases. In the complete

EM we have iterations over both t and k . In the modified EM, iterations are over t only. In the

midpoints method they are also over t (the number of iterations required for Newton-Raphson

method until convergence). In all methods, we considered a difference less than 0.001 as the

condition for convergence of parameters.


Case Study Complete EM

t k ( t )

Modified EM

t Midpoints

t

Audible Signal 52 (24) 27 6

Housing/Chassis 42 (17) 19 6

Battery 98 (42) 43 7

Table 4.15. Total number of iterations in the complete EM,

modified EM and the mid-points method

The total number of iterations in the complete EM is 2~2.5 times larger than in the

modified EM, and the required times for executing these two methods follow about the same

proportion.


The analysis of data from complex medical devices is not simple, largely because of the

amount of censored and missing information. In this chapter, we propose a solution to this

problem. Even with scarce failure data and a large number of censoring events, we are able to

conclude that the reliability of systems degrade over time. This result runs counter to the

common belief that failures of electronic equipment, including medical devices, are

exponentially distributed and the times between failures of the same device are independent.

Practicing statisticians make this erroneous assumption when they fit a distribution to the failure

data without checking whether the data exhibit a trend. When there is a trend, models such as

NHPP can be used to describe the failure process, and the time to each event can be analyzed

separately.


Current trend analysis methods in the literature mention only right censoring and do not

address recurrent failure data with left or interval censoring and periodic or nonperiodic

inspections if repairs are not instantaneous or not performed immediately.

In this chapter, we first propose a method to analyze statistically maintenance data for

complex medical devices with censoring and missing information. We present a classification of

different types of failures and establish policies for analyzing data at system and component

levels taking into account the failure types. We use the midpoints of the censoring intervals as

the failure times to apply the Laplace trend test at several levels of medical equipment (infusion

pump): the system level, hard failures, soft failures, and individual components. Trend analysis

using the midpoints is simple with no need of complex calculation. Reliable results are obtained

when the censoring intervals are short.

Obviously, precise and reliable results demand precise failure data. Since medical devices

tend to be highly reliable, scarce failure data is always a problem in statistical analysis. A

solution could be to aggregate the CMMS data from hospitals with similar equipment. This

aggregation should be performed cautiously, however, since the same device may exhibit

different failure patterns depending on operating and environmental conditions.

Then we assume a system whose failure process follows a non-homogeneous Poisson

process with a power law intensity function. We propose using the likelihood ratio test to check

for trends in the failure data with censoring. We use the EM algorithm and a modification of the

EM to find the parameters which maximize the data likelihood in the case of null and alternative

hypotheses (no trend and trend assumptions). A recursive procedure is used to solve the main

technical problem of calculating the expected values in the E-step requiring numerical

integration. The proposed method is used to perform a trend analysis of some components of a


general infusion pump such as audible signal, chassis/housing, and battery using a hospital’s

maintenance data. The results are compared to an ad-hoc method of replacing unknown failure

times with the mid points of censoring intervals.

The results reveal that the parameters estimates which are obtained from the complete

EM and its modified version are very close. Moreover, although the parameters obtained from

the EM may be close to what is obtained from the mid-points method, the latter may not always

give a precise approximation of the parameters and the conclusion made for trend in failure data

may not be correct. At least, the values obtained from the mid-points method may be used as

initial values in the EM algorithm.

It should be also noted that other methods, such as Monte-Carlo simulation can be also

used to estimate the expected values required in the E-step of the EM (Levine and Casella,

2001). In this case, no recursive procedure needs to be used, which is not discussed here.

The results of the statistical analysis can be used to develop an inspection/optimization

model. To establish an optimal inspection policy at each level of the system, a model should take

into account the failure trend and life pattern at that level. This problem will be discussed in the

following chapters.

100

5. INSPECTION AND MAINTENANCE OPTIMIZATION MODELS

In this chapter we propose several inspection optimization models under different

assumptions for a multi-component repairable system with soft (hidden) and hard failures.

Although the inspection optimization models are developed particularly for finding the optimal

periodic inspection interval for medical devices, they can still be applied more generally to any

kind of repairable system. The assumptions used to construct the optimization models are made

based on the results of Chapter 4.


Complex repairable systems such as medical devices, telecommunication systems, and

electronic instruments consist of a large number of interacting components, which perform the

system’s required functions. A system is called repairable, if after failure can be returned to

satisfactory functioning by any method other than replacement of the entire system (Ascher and

Feingold, 1984). A repairable system is usually subject to periodically or non-periodically

planned inspections during its life cycle. Periodic preventive maintenance (PM) due to its

relatively


simple implementation is perhaps the most common maintenance policy applied to repairable

systems in practice. Planned periodic inspections are performed to verify the system safety and

performance by detecting potential and hidden failures and taking appropriate actions. The

actions can be to fix the potential failure if the device is found defective or perform preventive

maintenance if no failure is detected to avoid or reduce future failures.

In many real-world situations where the safety and reliability of devices is vital, devices

must be inspected periodically. For example, according to regulations, all major components and

functions of clinical medical devices in a hospital must be checked at inspection to assure that

the devices are safe for use on patients. If a device fails between two consecutive scheduled

inspections and operators discover the failure, it is checked and repaired immediately regardless

of scheduled plans. Often hospitals take non-scheduled maintenance as an opportunity to check

the device’s failed component and to inspect all other major components/functions. In fact,

clinical engineers working in a hospital are supposed to go through a predesigned checklist when

inspecting major components/functions of a device at both scheduled and non-scheduled

inspections.

Numerous models have been developed for inspection and maintenance optimization of

repairable systems. These models can be classified according to the degree to which the

operating condition of the system is restored by maintenance (Nakagawa and Mizutani, 2009;

Wang and Pham, 2006; Wang, 2002; Pham, 2003, Pham and Wang, 1996; Kijima, 1989; Pham,

2008; Abdel-Hameed, 1995). Perfect PM models assume that the system is replaced or is as good

as new after PM (Barlow et al., 1965; Nakagawa, 2005; Dohi et al., 2003). In models with

assumption of minimal repair, the hazard rate of the system remains undisturbed by PM (Barlow

and Hunter, 1960; Dohi et al., 2003; Berg and Cleroux, 1982). Imperfect repair restores the


system to somewhere between as good as new and as bad as old. Worse and worst repairs make

the failure rate increase; however, the system does not break down in the worse repair, while the

worst repair unintentionally makes the system fail.

Some models assume that at failure a system is minimally repaired with probability

p and replaced with probability 1 p (Brown and Proschan, 1983; Nakagawa and Yasui, 1987;

Murthy and Nguyen, 1981). This assumption is extended in other models in which the

probability of a repair type (replacement, minimal repair, or imperfect repair) is age dependent

(Block et al., 1985; Sheu and Griffith, 2001).

Nakagawa and Mizutani (2009) in their recent paper give an overview of replacement

policies with minimal repair, block replacement and simple replacement for an operating unit in

a finite time horizon. In all three policies, they assume that the unit is replaced at periodic

replacement times and is as good as new. In the policy with minimal repair, the unit is minimally

repaired if it fails between periodic replacements. In the block policy, the unit is always replaced

at failures between replacements, and in the simple policy, there is an elapsed time between a

failure and its detection at the next inspection time after failure. Moreover, the paper presents

optimal periodic and sequential policies for an imperfect preventive maintenance and an

inspection model for a finite time span. In the inspection model, it is assumed that the unit is

replaced if a failure is detected at inspection; so, this model describes mainly a failure-finding

optimization problem.

Sometimes the probability of a repair type depends on both age and other time-dependent

variables. In the so-called “repair number counting” and “reference time” policies, the

probability of perfect maintenance depends on time and number of imperfect maintenance

actions in a cycle (Makis and Jardine, 1992; Sheu and Chang, 2009; Sheu et al., 2006; Sheu et


al., 2005). In some shock models the probability of replacement depends on system age and the

number of shocks that the system tolerates since the last replacement (Chien and Sheu, 2006;

Lam, 2009). Some shock models assume that the system is replaced depending on age, number

of imperfect repairs, and the cumulative damage exceeding a specific threshold (Qian et al.,

2003; Li and Pham, 2005). In repair limit policies, replacement of the system at failure is decided

according to the cost of repair (Chien et al., 2009; Lai, 2007) or the required repair time (Dohi et

al., 2001a). If the cost or repair time exceeds a certain limit, the system is replaced.

Further inspection and maintenance models for single-unit and multi-unit systems are

discussed in detail in Nakagawa (2003, 2005, 2008), Dohi et al. (2003),Wang and Pham (2006),

and Wang (2002) including age-dependent, periodic PM, failure limit, sequential PM, and repair

limit policies in an infinite time span.

The majority of maintenance optimization models assume that the system is not operative after a

failure, so the failure is rectified immediately and the failure time is known. However, up to 40% of all

failure modes of a complex industrial system fall into the category of hidden failures (Moubray,

1997). Hidden failures are not evident to operators; they remain dormant, do not stop the

system’s operation, and are only rectified at inspections. Hidden or soft failures may not halt the

system, but can have serious consequences, even catastrophic, if left unattended. These types of

failures can reduce the system’s performance or cause production loss and require to be fixed

eventually.

There are some models developed for failures with delayed detection when the failure

can only be rectified at inspection. Optimization models developed for failure finding intervals

are among this category. The delay-time model (Christer, 2002; Wang and Christer, 2003)

regards the failure process as a two-stage process. First, a defect is initiated and if unattended, it


will lead to a failure. The delay time between the initiation of the defect and time of failure

represents a window of opportunity for preventing the failure. Wang in his recent paper (2009)

presents an inspection optimization model for minor and major inspections, which are carried out

for a production process subject to two types of deterioration. Minor and major defects are

identified and repaired respectively at routine and major inspections. Taghipour and Banjevic

(2011a) present two inspection optimization models over finite and infinite time horizon for a

multi-component repairable system subject to hidden failures.

A great number of models assume that the system is replaced when a failure is rectified at

inspection, so it is as good as new after failure (Cui, 2005; Lienhard et al., 2008; Okumura,

2006). Available models in the literature mainly consider a general cumulative distribution

function for the system’s time to failure (Okumura, 2006; Jiang and Jardine, 2005). Lienhard et

al. (2008) assume that the failure rate and occurrence rates of operational demand and corrective

maintenance actions are given, and propose a model to determine the optimal length of failure-

finding task interval. Taghipour et al. (2010a) consider a repairable system with components

subject to hard and soft failures; soft failures are only rectified at periodic inspections and are

fixed with minimal repairs. They propose a model to find the optimal periodic inspection interval

on a finite time horizon.

In the literature, the opportunistic inspection and maintenance is also discussed. When

individual components are not stochastically and economically independent, an opportunistic

maintenance policy is more effective. This policy is beneficial when the cost of joint

maintenance actions of several components is less than the cost of maintenance action for

individual components (Sherif and Smith, 1981). Nowakowski and Werbińka (2009) present an

overview of some recent maintenance optimization models, including opportunistic maintenance


for multi-unit systems. Gertsbakh (2000) formulates a policy for opportunistic replacement of a

two-component system. Laggoune et al. (2009) propose a maintenance plan based on

opportunistic multi-grouping replacement optimization for a multi-component system subjected

to random failures. They assume that the components are configured in series, and propose a

model to find the replacement time at which a decision should be made regarding the preventive

replacement of each component. Moreover, at a system failure, it is decided for unfailed

components whether an opportunistic replacement should be performed or the components

should not be replaced until the next scheduled inspection. A general failure distribution for each

component is considered and the total cost is formulated for a cycle. The cycle is the minimal

time of simultaneous replacement of all components.

Jia (2010) focuses on the opportunistic maintenance of an asset, which is composed of

multiple non-identical life-limited components with both economic and structural dependence.

Zhou et al. (2006) develop a dynamic opportunistic maintenance policy for a continuously

monitored multi-unit series system with integrating imperfect effect into maintenance activities.

They assume that a scheduled or unscheduled maintenance action has to be carried out for a unit,

whenever it fails or reaches a certain reliability threshold. At a maintenance action, the whole

system is shut down and this provides a maintenance opportunity for all other units. They

propose a dynamic decision process, which includes calculating the reliability threshold for each

unit, determining opportunistic maintenance cost saving, and maintenance activities grouping

and decision making.

Total expected cost and expected downtime per unit time are usually considered for the

inspection/replacement decision problems. Cui et al. (2004) present several inspection models

with the emphasis on meeting the availability requirement. Barlow and Hunter (1960) define an


optimal maintenance policy as a policy which maximizes the fractional amount of uptime over

long intervals. Boland and Proschan (1982) present two models, which minimize the expected

cost for a system under minimal repair-periodic replacement policy. Boland (1982) assumes that

the cost of minimal repair of a system is a nondecreasing function of its age and derives the

expected minimal repair cost in an interval in terms of the cost function and the failure rate of the

system. Hartman and Murphy (2006) present a dynamic programming approach to the finite-

horizon equipment replacement problem with stationary costs. Dohi et al. (2001b) propose a

model to find the optimal preventive maintenance schedule, which minimizes the relevant

expected cost criterion for a system under an intermittently used environment.

In this chapter, we present several periodic inspection optimization models for a

repairable system subject to hard and soft failures. For practical purposes and simpler

implementation, most of organizations especially dealing with a large number of devices use a

periodic inspection policy. A non-periodic inspection policy can be also considered using the

same mathematical models but with more time consuming calculation and more demand on

practical implementation.

We consider two main cases with the following assumptions and construct the

optimization models for each case:

Case (1). A system under periodic inspections

Case (2). A system under periodic and opportunistic inspections

The structure of the inspection optimization models developed in this research is shown

in Figure 5.1. The models are classified according to the assumptions introduced.

5.2. Case 1: A System Under Periodic Inspections 107

Figure 5.1. Structure of the developed inspection optimization models

5.2. Case 1: A System Under Periodic Inspections

We consider a repairable system with the assumptions described in Section 4.3, and

develop a model to find the optimal periodic inspection interval for the system. In general, the

expected cost incurred from both soft and hard failures that include their inspection and failure

costs should be incorporated in inspection optimization model. However, in this section we

consider a non-opportunistic policy in which at hard failures only the failed component is


inspected and fixed and at periodic scheduled inspections only all components with soft failures

are checked and fixed if found failed. In this case, because of instantaneous repair or replacement

of hard components, they do not have impact on the inspection policy, and will be ignored in

calculation in this section. Therefore, the model reduces to a model of a system consisting of

several units/components with different soft failure rates.

5.2.1. Minimal Repair of Soft Failures

We assume that the components with soft failures are repaired if found failed at

inspections to the same condition as just before failure (minimal repair), even if they age in

certain way while in the failed state. It should be noted that at inspection only the states of

components with soft failures are found, but not their age at failure (i.e., the times of soft failures

are censored), which makes the problem of estimating the failure rate complicated (see Section

4.2. in Chapter 4). A summary of additional assumptions in this section are given as follows:


Periodic inspection/maintenance is performed.

Component with soft failures are minimally repaired.

Periodic inspections are taken at times k ),...,2,1( nk (Figure 5.2), where T n is

given. Our objective is to find the optimal scheduled inspection interval, which minimizes the

expected cost incurred over the cycle of length T .

Figure 5.2. Scheme of scheduled inspections in a cycle

2 )1( k k

S S S S

n T

S: Scheduled inspection


We assume that soft failures have influence on the system’s performance. For example,

any changes in the shape of a measuring head in a thickness gauge used in steel industry can

produce less accurate measurement of a steel strip, although the whole gauge is still working.

Therefore, the components with soft failures affect performance and maintenance cost, but not

basic operation of the system.

Several methods can be employed to incorporate soft failures into the model. In fact, the

method depends on the actual application for which the optimization model is developed. The

importance and consequences of soft failures on the system’s performance can be used as criteria

by managers to decide about the form of including this type of failure in the model.

5.2.1.1. Model Considering Downtime and Repair of Components

When the downtime of a component with soft failure is important, a method considering

downtime should be employed. At times, the longer a component is in a failed state, the more

significant is its influence on system performance or safety. For example, a defective head in a

thickness gauge can produce inaccurate measurements starting from the moment of its

deformation and lasting until the defect is rectified at inspection. Therefore, a penalty should be

imposed for the period of less accurate measurements. Another method of incorporating soft

failures in the cost model is discussed in Section 5.2.1.2. This mehtod considers the number of

soft failures which exceeds a pre-defined threshold. The method of considering downtime of soft

failures can be combined with the total number of failed components (Section 5.2.1.3) to

emphasize both total downtime and the number of failures exceeding the threshold.


Notation

m is the number of components with soft failure.

( )j x intensity function of the non-homogeneous Poisson process associated with the

number of failures of component j over period [0, ]x .

T system cycle length, which is known and fixed.

n the number of inspection intervals.

length of periodic scheduled inspection interval; n T , or /T n .

jc cost of minimal repair of component j .

[ ]T

SE C expected cost incurred from scheduled inspections in a cycle.

Ic cost of one inspection.

(( 1) , ][ ]k k

SE C expected cost incurred from a scheduled inspection at k over time period

],)1(( kk .

For each inspection interval (( 1) , ] , 1 2k k k , ,...,n , and for all components with soft

failure a penalty cost is incurred proportional to the elapsed time of the failure. The objective is

to find the optimal inspection interval, which minimizes the total penalty cost. We assume that

inspection and possible repairs are also done at the end of the cycle, T , that is, for k n . This

action may be considered as a sort of minimal preparation for the next cycle, but also makes the

model slightly simpler. This assumption can be easily removed, if appropriate.

The expected cost incurred from scheduled inspections in a cycle:

(( 1) , ]

1

[ ] [ ]n

T k k

S S

k

E C E C

. (5.1)

The costs incurred at a scheduled inspection are:


- Cost of inspection ( Ic ).

- Cost of minimal repair of component j if found failed ( jc ).

- Penalty cost for the elapsed time of failure for component j ( jc ).

When a component fails, it remains failed until the next inspection, when it is minimally

repaired. So, the expected cost incurred from the thk scheduled inspection in a cycle is:

(( 1) , ]

1

1

[ ] (component fails in (( 1) , ])

(downtime of component in (( 1) , ])

mk k

S I j

j

m

j

j

E C c c P j k k

c E j k k

(5.2)

From additivity of the cost function, it is clear that any structure of the system can be

ignored because only the marginal distributions of failure times are needed. Still the components

are economically dependent, by sharing the same inspection interval. Let kX be the survival

time of the component j in the thk interval, calculated from ( 1)k (Figure 5.3), 1,2,...,k n

(index j is suppressed for simplicity). Let 0X is the initial age of the component at the

beginning of the cycle. Using initial age is convenient for recursive calculation, but in real

application, it is also possible that some components are not as good as new. In the numerical

example 0t will be used. Let

0 0( ) (component does not fail in (( 1) , ] | ) ( | )j j

k kP t P j k k X t P X X t

We will derive a recursive equation for ( )j

kP t using conditioning over the first failure time 1X .

Figure 5.3. Initial age t and failure times of component j

2

T

)1( k k

2XkX

3

3X1X

1st failure 2nd failure rth failure

tX 0


Let 1 min{ , }X X , where X is the time of the first failure. Therefore, 1X , and the CDF

of 1X is

( )

1 1 01 ,0( | ) ( | )

1 ,

t x

jt

s dsj j e xF x t P X x X t

x

(5.3)

and pdf of 1X at its continuous part

( )

1 1( | ) ( | ) ( )

t x

jt

s dsj j

jxf x t F x t t x e

, 0 x . (5.4)

Notice that ( )

1 1 0 1 1( ) ( | ) ( | ) ( 0 | )

t

jt

s dsj jj jP t P X X t F t F t e

.

From the assumptions of NHPP and minimal repair, we have

0 0 1

0 1 1 0 1 1 0

0

1 1 1 1

0

1 1

0

( ) ( | ) [ ( | , )]

( | , ) ( | ) ( | , ) ( | )

( ) ( | ) ( ) ( )

( ) ( | ) , for 2 .

j j j

k k k

jj j j

k k

jj j j

k k

jj

k

P t P X X t E P X X t X

P X X t X x f x t dx P X X t X P X X t

P t x dF x t P t P t

P t x dF x t k ,...,n

So, the recursive equation for ( )j

kP t , 0t , is

( )

1

1 1

0

( ) ,

( ) ( ) ( | ) , 2

t

jt

s dsj

jj j

k k

P t e

P t P t x dF x t k ,...,n

. (5.5)

Note that if we want to calculate ( )j

kP t for 0 t A , we have to calculate 1( )j

kP t for

0 t A . For example, when we want to calculate (0)j

kP , 1,2k ,...,n , we have to calculate

( )j

kP t for 0 ( )t n k , 1,2 1k ,...,n . Actual numerical calculation of ( )j

kP t will be


mentioned later in the numerical example. Direct calculation of ( )j

kP t would be tedious, except

for small n . For example, from equation (5.5),

2

2

2 1 1 1 1

0

( ) ( ) ( ) ( )

0

( ) ( )

0

( ) ( ) ( | ) ( ) ( )

( )

( ) ,

t x t x t t

j j j jt x t t t

t x t

j jt t

jj j j j

s ds s ds s ds s ds

j

s ds s ds

j

P t P t x dF x t P t P t

e t x e dx e e

t x e dx e

and

2

2 3

3 2 1 2 1

0

( ) ( ) ( )

0 0

( ) ( ) ( )

0

( ) ( ) ( | ) ( ) ( )

[ ( ) ] ( )

[ ( ) ]

t x y t x t x

j j jt x t x t

t x t t

j j jt t t

j j j j j

s ds s ds s ds

j j

s ds s ds s ds

j

P t P t x dF x t P t P t

t x y e dy e t x e dx

t x e dx e e

2

( ) ( )

0 0 0

( ) ( ) ( )

t x y t x

j jt t

s ds s ds

j j jt x t x y e dxdy t x e dx

2 3

( ) ( )

0

( ) .

t x t

j jt t

s ds s ds

j t x e dx e

Let

0( ) [ | ] 1 2 .j j

k ke t E X X t k , ,...,n

Then for 2k , similar to derivation of (5.5),

0 1 0 1 1

0

1 0 1 1 1

0 0

( ) [ [ | , ]] [ | , ] ( | )

[ | ] ( | ) ( ) ( | ) .

jj j j j

k k k

j jj j

k k

e t E E X X t X E X X t X x dF x t

E X X t x dF x t e t x dF x t

So, the recursive equation for ( )j

ke t is


1 1 1 1

0

( ) ( ) ( | ) ( ) ( ) , 2j j jj j

k k ke t e t x f x t dx e t P t k ,...,n

. (5.6)

For 1k ,

( )

1 1 0 1 1 1 1

0 0 0 0

( ) [ | ] ( | ) ( | ) ( ) ( | ) .

t x

jt

s dsj j j j j je t E X X t xdF x t xf x t dx P t R x t dx e dx

If we consider a penalty cost of jc for the elapsed time of failure for component j , we will have

1 1 1 1

1 1 1 1

[ ] ( ( )) ( ( ))

( ) ( ( ) ( ))

m n n mT j j

S I j k j k

j k k j

m m m nj j

j I j j k j k

j j j k

E C nc c n P t c e t

T c n c c c P t c e t

. (5.7)

Let 1 1 1( ) ( ) ( )j j j

j jg t c P t c e t . Then for 2k ,

1 1 1

0

1 1

0

( ) ( ) ( ) ( ( ) ( )) ( | ),

( ) ( ) ( | ).

j

k

jj j j j

k j k j k j j k

jj j

k k

g t c P t c e t c P t x c e t x dF x t

g t g t x dF x t

So, )(tg j

k can be calculated recursively using the single recursive equation

1 1 1

1 1

0

( ) ( ) ( ) ,

( ) ( )( ) ( | ) , 2,3,...,

j j j

j j

jj j

k k

g t c P t c e t

g t g t t x dF x t k n

,

instead of separately calculating ( )j

kP t and ( )j

ke t .

Finally, if 1

( ) ( )n

j j

n k

k

g t g t

, the expected cost per cycle is given as follows:

1 1 1

[ ] ( ) ( ).m m m

T j

S j I j n

j j j

E C T c n c c g t


Our objective is to find the optimal inspection frequency n , which minimizes the

expected cost per cycle [ ]T

SE C incurred from scheduled inspections. For that purpose, we

calculate [ ]T

SE C for different values of n and find the value of n . The procedure is numerically

intensive. It should be noted that n is limited by an upper bound Un . This upper bound may

result from a lower bound 0L on , where L is the minimum feasible inspection

interval, e.g., one week, or one month. Then Ln T , or / L Un T n . Note that [ ]T

S IE C nc

for all n . Let (0, ][ ]T

SE C be the expected cost incurred from a plan with only one inspection

( 1n ). Then if (0, ][ ]T

I Snc E C , n cannot be the optimal frequency, that is (0, ][ ] /T

U S In E C c .

Numerical example

Consider a complex repairable system with five components. We assume that the number

of failures for component j follows an NHPP with power law intensity

function

1

( )

j

j

j

j j

xx

. The parameters of the power law processes and the minimal repair

costs of the components are given in Table 5.1. The system is currently inspected periodically

every 4 months. The number of components and the parameters are adapted from a case study on

medical equipment described in Section 4.2 in Chapter 4. We are interested in finding the optimal

inspection interval, which minimizes the total cost incurred from scheduled inspections over a

12-month cycle length. For simplicity, we will assume that the system starts as good as new,

which is not a limitation in our model, as we may assume any initial age, t , of the system (or

even different initial ages, jt t , of different components j ). We may assume that after 12


months a major overhaul of the system will be performed, e.g., some components are replaced

with new or used spare components, and the system will start again. Then, a new optimal

inspection frequency should be found. We also assume that the inspection interval cannot be

shorter than one month, that is, 12n .

Component i i (months) Repair cost ic Downtime cost/month ic

1 1.3 3.5 $70 $100

2 1.1 4.6 $45 $25

3 2.1 6 $100 $200

4 1.8 10 $75 $50

5 1.7 3.6 $150 $150

Table 5.1. Parameters of the power law intensity functions and costs

Therefore, we have 5m , and 1T year = 12 months in our model. In the hospital’s

current policy = 4 months, or 12/4 3n . The inspection cost is $200Ic . We use equation

(5.7) for the optimization purpose.

We obtain ( )j

kP t , ( )j

ke t , 1,...,k n , recursively using equations (5.5) and (5.6), and

numerical integration (Composite Simpson’s rule was applied for numerical integration with step

length h depending on n and required accuracy; after some experimentation, 410h was

used). The value 0t is used in the calculation, that is, we assume the equipment is as good as

new at the beginning of the cycle.

The total cost [ ]T

SE C for different values of n is shown in Figure 5.4. As it can be seen

on the Figure, the minimal cost is obtained for 4n with the total cost [ ]T

SE C = $3,932.88. So,


if the real costs were as given, the current inspection policy should be changed from inspecting

the system every 4 months to 3 months.

Total cost for n = 1,2,3,...,12 in $1000

3.00

3.50

4.00

4.50

5.00

5.50

Cost 4.55 4.08 3.94 3.93 3.99 4.08 4.19 4.33 4.47 4.63 4.79 4.96

1 2 3 4 5 6 7 8 9 10 11 12

Figure 5.4. Total costs for different inspection frequencies (first model)

This model is sensitive to the penalty costs incurred for components’ downtimes. If the

downtime penalty costs jc are not small, the optimal frequency will likely not be 1n , as it

may happen in the next proposed model (model considering the number of failures). In other

words, the longer interval between inspections the longer the elapsed time of failures, and

consequently the larger downtime penalty cost.

5.2.1.2. Model Considering the Number of Failures

In this variation of the model, we introduce a threshold trhN for the acceptable number of

failures observed at a scheduled inspection. According to the total number of failures that

exceeds the threshold, a penalty cost is incurred. This method can be used when only the total

number of failures is important regardless of which combination of failures has occurred. An

example of an application of this method can be a computer system with several external hard


drive backups; even if all hard drive backups fail, the system can function, but a penalty should

be incurred since the chance of losing data is increased by increasing the number of failed

backups. Thresholds can be set by managers responsible for the safety and reliability of the

system. A threshold may also be related to the number of redundant components in the system

when a minimum required number of components must be operational, or to the additional costs

of labor and downtime required to fix accumulated problems. We also assume that a penalty cost

is incurred for exceeding the threshold.

Additional Notation

trhN acceptable threshold for the number of failures detected at a scheduled inspection.

Pc penalty cost incurred for exceeding the acceptable number of failures at a scheduled

inspection.

Pc penalty cost incurred per number of failures exceeding the threshold trhN .

The cost incurred at a scheduled inspection includes:

- Cost of inspection ( Ic ).

- Cost of minimal repair of a component if found failed ( jc ).

- Penalty cost incurred for exceeding the acceptable number of failures ( Pc ).

- Penalty cost which takes into account the number of failures exceeding trhN ( Pc ).

The expected cost incurred from the thk scheduled inspection in a cycle is:

(( 1) , ]

{ }

1

[ ] (1 ( )) E[ ] E[ max{0, }]k trh

mk k j

S I j k P S N P k trh

j

E C c c P t c I c S N

, (5.8)


where kS is the total number of failures in the thk interval, and { } k trhS NI

is the indicator function.

Let ( ) ( | initial age of all components )k kp l P S l t , 0,1,2,...,l m ( n and t are suppressed

for simplicity). Then the expected penalty cost of exceeding the acceptable number of failures,

trhN , can be calculated as follows:

{ } { }

- 0

[ ] [ max{0, - }] [ ] [max{0, - }]

( ) ( - ) ( ) ( ) ( ( ) ( ))

( ) (

k trh k trh

trh trh trh trh trh

P S N P k trh P S N P k trh

P k P trh k P k P k trh k

l N l N l N l N l N

P trh P k

E c I E c S N c E I c E S N

c p l c l N p l c p l c lp l N p l

c N c p l

) ( ) ( ) ( ) ( ).trh trh trh

P k P trh P k trh P k

l N l N l N

c lp l c N c P S N c lp l

In our proposed model the more the number of failures exceeds trhN , the larger the

penalty cost will be incurred.

We need to find the probability of having 1trhN or more of failures, which may be

considered as a ( 1) :trhN out of m F system problem. However, since the probabilities of

failure of components need not be equal, the number of failed components in the system does not

follow the binomial distribution. So, in order to find ( )kp l , one can use the following equation

(Levitin, 2005):

1 2

1 2 1 11 2

1 2

1 1 11

( ) [ (1 )][ ... ]1 1 1

l

l l l

m m l m l mjj j

k j

j j j j jj j j j

qq qp l q

q q q

(5.9)

where 1 ( )j

j kq P t is the probability of failure for component , 1j j ,...,m , in the thk

interval. In this calculation, we assume that components are independent. Otherwise, the

calculation would require knowledge of system structure and a joint distribution of failure times,

which is not considered here.


Computation of ( )kp l based on this equation is complicated, so a more efficient

algorithm suggested by Barlow and Heidtmann (1984) based on generating function can be used

to reduce the computational complexity.

The probability mass function of each component can be represented by the generating

function:

0 1

0( ) ( | ) ( ) (1 ( )) ( ) (1 ( ))kXj j j j j j

k k k k ku z E z X t P t z P t z P t P t z .

01

( ) ( ) ( )m m

j l

k k k

lj

U z u z p l z

generates the distribution of kS . We are interested in

finding ( )trh

k

l N

p l

and ( )trh

k

l N

lp l

.

Define

1

( ) ( )trh

ml

k k

l N

U z p l z

(5.10)

Then ( ) (1)trh

kk

l N

p l U

and ( ) (1)trh

k k

l N

lp l U

, respectively. So, we have:

{ }[ ] [ max{0, - }] ( ) (1) (1)k trh

kP S N P k trh P trh P P kE c I E c S N c N c U c U (5.11)

Replacing equation (5.11) in (5.8) gives:

(( 1) , ]

1

[ ] (1 ( )) ( ) (1) (1).m

k k jkS I j k P trh P P k

j

E C c c P t c N c U c U

(5.12)

Therefore, the expected cost incurred from all scheduled inspections over the cycle nT is:


(( 1) , ]

1

1 1 1

1 1 1 1

1 1

[ ] [ ]

(1 ( )) [( ) (1) (1)]

( ( )) ( ) (1) (1)

( ) ( ) (

nT k k

S S

k

n m nj

kI j k P trh P P k

k j k

m n n nj

kI j k P trh P P k

j k k k

m mj

nI j j P

j j

E C E C

nc c P t c N c U c U

nc c n P t c N c U c U

n c c c P t c N

1 1

) (1) (1), n n

ktrh P P k

k k

c U c U

(5.13)

where 1

( ) ( )n

j jn k

k

P t P t

.

This method is particularly useful when the components are similar or make almost the

same contribution to reducing the whole system’s performance. In other case, a weight can be

assigned to each component to describe its contribution to the system performance, system

production, etc. In the latter, different combinations of failures can cause different penalty costs

since both the occurrence of a failure and its individual influence on system performance or

production are incorporated into the model through the weight. The combined influence of

several failures can be considered as a weighted sum of the influence of all individual

components, or it can be a more complicated function, or a value decided by managers. In this

section, we consider additive downtime costs incurred for individual but simultaneous soft

failures as their so-called joint failure consequence; but the model can be revised to intensify the

failure consequence or risk of simultaneous failures. In this case, if for example two soft failures

take place simultaneously, their joint downtime consequence should be greater than the sum of

their individual downtime consequences. For this variation of the model, possessing a good


knowledge of the system, its structure, components dependency, and their joint failure

consequences is required.

Moreover, we assume that soft failures cannot convert into hard failures, and hard and

soft failures have independent consequences on the system. In other words, we do not consider

joint risk or failure consequence of soft and hard failures even though they may be related

somehow. For example, if both protective component (component with soft failure) and

protected component (component with hard failure) fail simultaneously, our proposed model

considers only the consequence of the soft failure, since it assumes that a hard failure is fixed

instantaneously; however, the model can be extended to consider the joint failure consequence of

soft and hard failures when they have some sort of dependency. Such failure dependency and

joint risk and cost consequences have been extensively discussed in the risk assessment literature

(Vaurio, 1995; 2003; 2005; Yu et al., 2007).

Numerical example

We use the same data given in the numerical example in Section 5.2.1.1 to demonstrate

the model suggested in this section. The other parameters of the model are: threshold 3trhN ,

$200, $1500, $1000I P Pc c c .

We will use equation (5.13) in the following form for convenience of comparing the

values associated with different inspection frequencies n .

, ,,

1 1 1 1

[ ] ( ( )) (1) ( (1) (1))m n n n

jTn n k n kS I j P P n k trh

j k k k

E C nc c n P t c U c U N U

.


To calculate ,

1

( ) ( )n

j jn n k

k

P t P t

for 12,...,3,2,1n we use , ( )j

n kP t , 1,...,k n . Dependence

of , ( )j

n kP t and )(, zU kn on n is emphasized in the notation. , ( )j

n kP t values are also used to find

, (1)n kU and , (1)n kU . We first extract the terms with 4z and 5z from

5

, , ,

1

( ) [(1 ( )) ( )]j j

n k n k n k

j

U z P t z P t

and use it to create )(, zU kn . Then , (1)n kU and , (1)n kU can be

easily obtained. The value 0t is used in the calculation, that is, we assume the equipment is as

good as new at the beginning of the cycle.

The cost curve is shown in Figure 5.5. 1n is the optimal inspection frequency, or to

inspect once a year, with the minimum cost of $3,759.82.


3.00

3.50

4.00

4.50

5.00

5.50

Cost 3.76 4.93 4.92 4.73 4.53 4.40 4.32 4.31 4.35 4.42 4.52 4.64

1 2 3 4 5 6 7 8 9 10 11 12

Figure 5.5. Total costs of scheduled inspections for different inspection frequencies

(threshold model)

Obviously, the optimal n depends on the input parameters of the model. With the cost

parameters given in this example, 1n is the optimal policy. However, we would like to find

conditions for costs under which some other value of n , such as 9n is the optimal solution.


Let , ,1 2 3 ,

1 1 1 1

( ) ( ( )), ( ) (1), ( ) (1) (1)m n n n

j

n n k n kj n k trh

j k k k

Q n c n P t Q n U Q n U N U

. Then the

expected cost for n is 1 11 2 3[ ] ( ) ( ) ( )T

S I P PE C nc Q n c Q n c Q n , if we assume that jc costs are

fixed. Table 5.2 shows Inc , 1( )Q n , 2 ( )Q n , and )(3 nQ calculated for 1,2,3,...,12n .

n Inc )(1 nQ )(2 nQ )(3 nQ

1 200 416.8154 0.9800 1.6730

2 400 699.7925 1.3340 1.8328

3 600 888.5510 1.2423 1.5699

4 800 1029.8770 1.0732 1.2886

5 10000 1141.4125 0.8986 1.0440

6 12000 1232.2306 0.7453 0.8458

7 14000 1307.6402 0.6180 0.6895

8 16000 1371.9082 0.5154 0.5676

9 18000 1426.6128 0.4320 0.4711

10 20000 1474.8199 0.3660 0.3940

11 22000 1516.8528 0.3110 0.3340

12 24000 1553.7449 0.2660 0.2850

Table 5.2. Calculated components of the cost function

For a particular n to be better than n the following relationship of Pc and Pc should

hold:

1 2 3 1 2 3( ) ( ) ( ) ( ) ( ) ( )I P P I P Pnc Q n c Q n c Q n n c Q n c Q n c Q n ,

For example if 2171.3141-0.456P Pc c , then 9n is better solution than 1n . Similar

inequalities can be obtained for other values of n . For $1500Pc , the previous inequality gives


the lower limit $1490Pc . It appears that for, e.g., $1700Pc , 9n is also the optimal

inspection frequency. The cost function for that case is shown in Figure 5.6. Obviously, when the

cost curve is flat near the optimal point, i.e. when there is no a significant difference between the

total cost of the optimal frequency and the other frequencies close to the optimal, we can select

the frequency n close to the optimal that is easier to implement practically. For instance, the

costs for 12n may be considered close enough to the minimal cost, so 12n can be selected

to be implemented in practice, or to inspect every month. If 12n may be inconvenient for

implementation, then 1n , or only one inspection at the end of the cycle may be the next

choice. A compromise would be 6n , or to inspect every two months.


3.00

3.50

4.00

4.50

5.00

5.50

6.00

6.50

Cost 4.93 6.22 6.02 5.63 5.26 4.99 4.81 4.71 4.68 4.69 4.75 4.84

1 2 3 4 5 6 7 8 9 10 11 12


model); For penalty cost $1700Pc , 9n is the optimal number of inspections when

$1500Pc

Comparing the values of Inc , 1( )Q n , 2 ( )Q n , and )(3 nQ given in Table 5.2 for different n

indicates that for some n the values are always greater than for some other n and consequently

no value for Pc and Pc can make them optimal. For instance, since all values for 2n are

greater than the values for 1n , 2n can never be optimal. Therefore, some kind of “peculiar”


behavior of the cost function for 1n in both Figure 5.5 and 5.6 should not be a surprise in this

example. In practical terms, it means that to decrease expected penalty costs it is better to leave

soft failures unattended ( 1n ), than to inspect them after 6 months ( 2n ) and minimally fix

them, because it will then only create another a not small chance for them to fail again and pay a

penalty due to exceeding the threshold. But, using 2n may significantly reduce the chance of

exceeding the threshold, and then may reduce the expected penalty cost.

In the threshold model (equation (5.13)) we need to calculate 1( ) |k zU z to obtain the

expected penalty cost. However, this calculation can be very complicated when the number of

components increases. Also, using a threshold may produce the situation discussed above that it

is sometimes better not to fix failures. The model discussed in Section 5.2.1.1 suggests another

version of the cost model whose optimization requires simpler calculation while it takes into

account downtime of components.

5.2.1.3. The Combined Model

Equations (5.7) and (5.13) can also be combined to take into account both the exceeding

number and elapsed times of failures, thus making the cost model more flexible. It should be

noted that the “threshold” part of the model requires calculation of the distribution of the total

number of failures, whereas the “downtime” part of the model requires only the marginal

distributions of times to failure. We also assume that components are independent.

In the combined model we have:

, ,,

1 1 1 1 1 1

[ ] ( ( )) (1) [ (1) (1)] ( ( )).m n n n n m

jT jn n k n kS I j P P n k trh j k

j k k k k j

E C nc c n P t c U c U N U c e t


The results of applying the combined model on the data given in the examples in Sections

5.2.1.1 and 5.2.1.2 and the models considering threshold and elapsed times are shown in Figure

5.7. As it can be seen in Figure 5.7 10n with the total cost [ ]T

SE C = $5,571.84 is the optimal

solution for the combined model. Moreover, obviously, there is a significant difference between

costs for smaller n and larger n . The cost for 1n is still smaller than the cost for 2n , but

not significantly, due to significant downtime cost for 1n . For practical purposes, it may be

simpler to use 12n , i.e. to inspect every month, which makes only a minor difference from the

optimal solution.


2.50

3.50

4.50

5.50

6.50

7.50

8.50

with threshold 3.76 4.93 4.92 4.73 4.53 4.40 4.32 4.31 4.35 4.42 4.52 4.64

with elapsed time 4.55 4.08 3.94 3.93 3.99 4.08 4.19 4.33 4.47 4.63 4.79 4.96

combined model 7.70 7.91 7.38 6.83 6.38 6.04 5.81 5.67 5.59 5.57 5.59 5.64

1 2 3 4 5 6 7 8 9 10 11 12

Figure 5.7. Total costs for different inspection frequencies (combined model)

In general, the methodology for considering the influence of different failures can be set

up by managers responsible for maintaining devices in a safe and reliable state. When

information about the influence of individual failures and different combinations of failures is

available, a threshold model with a weighted sum of the performance reduction or loss

production can be employed. However, in many real situations, such as hospitals dealing with

many different devices, the availability of this type of information is problematic. In this case,


simpler and more practical methods, such as considering the total number of failures or the sum

of the individual downtimes as the penalty, can be employed.

5.2.2. Minimal Repair and Replacement of Soft Failures

In this section, we have the same assumptions as Section 5.2.1, but in addition we assume

that the components with soft failures can be either minimally repaired or replaced if they fail

with some age dependent probabilities. So, in this section more general assumptions are made

compared to Section 5.2.1. We propose models to find an optimal periodic inspection interval for

the system over finite and infinite time horizons. We consider a penalty cost for the downtime of

the failed component. Additional assumptions of this section are summarized as follows:


Periodic inspection/maintenance is performed.

Failed components may be minimally repaired or replaced with age dependent probabilities.

A penalty cost is incurred for the elapsed time from a failure to its detection at the next

inspection. The longer the elapsed time, the larger penalty is incurred.

5.2.2.1. Model Over a Finite Time Horizon

We assume that periodic inspections are taken at times k ),...,2,1( nk . Our objective is

to find the optimal scheduled inspection interval, which minimizes the expected cost incurred

over the cycle of fixed length T .

The costs incurred at scheduled inspections are:


- Cost of inspection ( Ic )

- Cost of minimal repair of a component if found failed and needing minimal repair ( M

jc )

- Cost of replacement of a component if found failed and needing to be replaced ( R

jc )

- Penalty cost for the elapsed time of failure for component j ( D

jc )

We assume that component j is minimally repaired at age x with probability ( )jr x , or is

replaced with probability ( ) 1 ( )j jr x r x . Thus, ( )jr x is a function of the component’s age. For

all failed components, a penalty cost is incurred proportional to the elapsed time of the failure.

We assume that inspection and possible repairs are also done at the end of the cycle, T , that is,

for k n .

The total expected cost incurred from inspections in a cycle is

(( 1) , ]

1

[ ] [ ]n

T k k

S S

k

E C E C

. (5.14)

The expected cost incurred at the ( 1,2,..., )thk k n scheduled inspection in a cycle is:

(( 1) , ]

1

1

[ ] Pr{component fails in (( 1) , ] and is minimally repaired}

Pr{component fails in (( 1) , ] and is replaced}

mk k M

S I j

j

mR

j

j

E C c c j k k

c j k k

1

[downtime of component in (( 1) , ]] . m

D

j

j

c E j k k

.

(5.15)

Let 0

j

jX t is the initial age of the component at the beginning of the cycle. The initial

age 0jt when the device starts as good as new.


Let j

kX be the survival time of component j in the thk interval, calculated from ( 1)k

(Figure 5.8), where 1,2,...,k n and 1,2,...,j m .

Figure 5.8. Initial age t and failure times of component j

Let j

kI denotes the status of component j at thk inspection:

0, minimal repair at inspection

1, replacement at inspection

2, no failure in interval (( -1) , ]

th

j th

k

k

I k

k k

.

Let

0 0( ) Pr{ 2 | } { | }j j j j j

k k kP t I X t P X X t ,

0( ) Pr{ , 0 | }j j j j

k k kM t X I X t ,

0( ) Pr{ , 1| }j j j j

k k kR t X I X t ,

0( ) [ | ] j j j

k ke t E X X t .

Then from (5.14) and (5.15) the total expected cost incurred over [0, ]T can be written as follows:

1 1 1 1

1 1 1 1 1

[ ] ( ( ) ( ) ) ( ( ))

[ ( ) ( ) ( )].

m n m nT M j j R D j

S I j k k j j k

j k j k

m m n n nD M j R j D j

j I j k j k j k

j j k k k

E C nc c M t R t c c e t

T c nc c M t c R t c e t

(5.16)

We can ignore any structure of the system since the cost function is additive and only

marginal distributions of failure times are required. However, the components still are

economically dependent, by sharing the same inspection interval.

2

T

)1( k k

2

jX j

kX

3

3

jX1

jX

1st failure 2nd failure rth failure

0

jX t


We will derive recursive equations to find ( )j

kP t , ( )j

kM t , ( )j

kR t , and ( )j

ke t

( 1,2,..., )k n , which applies for every component j . For simplicity, we will suppress index j

in the following, unless it is needed for clarity. The recursive equations are identical aside from a

placeholder function ( , )k kX I , which differ according to the quantity we wish to calculate.

The CDF and pdf of 1X are the same as what were introduced in Section 5.2.1.1

(equations 5.3 and 5.4 ) and 1( )P t is with the same definition as follows:

( )

1 1 0 1 1( ) Pr{ | } ( | ) ( 0 | )

t

t

s ds

P t X X t F t F t e

.

We will now derive recursive equations to find the conditional expectation of a

placeholder function ( , )k kX I given initial age of the component 0X t . Let

1 1 1 0( ) [ ( , ) | ]t E X I X t and 0( ) [ ( , ) | ]k k kt E X I X t .

Then from the assumptions of NHPP and minimal repair or replacement, we get the

recursive equations:

1 1 1 0 1 1

0

( ) [ ( , ) | ] [ ( ,0) ( ) ( ,1) ( )] ( | ) ( ,2) ( ),t E X I X t x r t x x r t x f x t dx P t

(5.17)

0 1 1 1 1 1 1

0

( ) [ ( , ) | ] ( ) ( ) ( | ) (0) ( ) ( ) ( ),k k k k k kt E X I X t t x r t x f x t dx g t t P t

(5.18)

for 2k ,...,n , where 1 1

0

( ) ( ) ( | )g t r t x f x t dx

and ( ) 1 ( )r x r x . For a proof, see Appendix

I. For example, when ( )r x r we have

1 1 1 1 1 1

0

( ) ( ) ( | ) (0)(1 ( )) ( ) ( )k k k kt r t x f x t dx r P t t P t

.


Using initial age is necessary for recursive calculation. In our numerical examples, initial

age 0t will be used. In real applications, it is also possible that some components are not as

good as new, so that our method can be applied as well. We now can apply the general recursive

equation (5.17) for some special cases needed for calculating expected cost in equation (5.16).

The only differences between these cases are in their placeholder functions ( , )k kX I and first

functions 1( )t .

(a) For the probability of survival to the end of the thk interval:

( )

1 1 0 0

( , ) ( , 2) ( ),

( ) [ ( ) | ] Pr{ | } .

t

t

k k k k k

s ds

k

X I X I X

P t E X X t X X t e

where denotes the indicator function.

(b) For the probability of failure with minimal repair in the thk interval:

1 1 1 0 0 1

0

( , ) ( , 0),

( ) [ ( , 0) | ] Pr{ , 0 | } ( ) ( | ) .

k k k k

k k

X I X I

M t E X I X t X I X t r t x f x t dx

(c) For the probability of failure with replacement in the thk interval:

1 1 1 0 0 1

0

( , ) ( , 1),

( ) [ ( , 1) | ] Pr{ , 1| } ( ) ( | ) .

k k k k

k k

X I X I

R t E X I X t X I X t r t x f x t dx

(d) For the expected survival time of the component in the thk interval:

1 1 0 1 1

0 0

( , ) ,

( ) ( | ) ( | ) ( | ) ,

k k kX I X

e t E X X t xdF x t F x t dx

where 1 1( | ) 1 ( | )F x t F x t .


In real applications, the last inspection interval does not need to be of the same length as

the planned periodic inspection interval . The device is checked periodically at every

interval, but the time between the two last inspections, i.e. ( 1)T n , need not necessarily be

equal to , if is fixed in advance. We extend the equation (5.18) to include the case when the

last inspection interval has a length of ( 1)T n , when ( 1)n T n . If T n

then and the equation (5.18) can be used without any changes; however, in general we

have:

1 1 1

0

( , ; ) [ ( ,0) ( ) ( ,1) ( )] ( | ) ( ,2) ( ; ), t x r t x x r t x f x t dx P t

1 1 1 1 1 1

0

( , ; ) ( , ; ) ( ) ( | ) ( , ;0) ( ) ( , ; ) ( ; ), k k k kt t x r t x f x t dx g t t P t

(5.19)

for 2,3,...,k n , where ( )

1( ; )

t x

t

s ds

P x t e

.

Then the total expected cost incurred over T is:

1

1 1 1 1 1

1 1 1 1 1

[ ] ( ( ) ( ) ) ( ( )) ( ( ))

[ ( ) ( ) ( )] .

j j j j

j j j j

m n m n mT j j j j

S I M k k R D k D n

j k j k j

m m n n nj j j

D I M k R k D k

j j k k k

E C nc c M t R t c c e t c e t

T c nc c M t c R t c e t

(5.20)

Our objective is to find the optimal inspection frequency n which minimizes the

expected cost per cycle [ ]T

SE C if /T n . For that purpose, we calculate [ ]T

SE C for different

values of n and find the optimal value n. If we use the model with a possibly different last

interval, we first choose , then calculate n and , and then calculate [ ]T

SE C . We usually

assume that is a multiple of L , that is, Lk , {1,2,...}k .


Numerical example

We use the same data given in the numerical example in Section 5.2.1.1 to demonstrate

the model suggested in this section. Assume that ( ) , 0jb x

j jr x a e x

. The parameters of the

power law processes, minimal repair and replacement costs, and parameters ja and jb of ( )jr x

are given in Table 5.3. The system is currently inspected periodically every 4 months. We are

interested in finding the optimal inspection interval, which minimizes the total cost incurred from

scheduled inspections over a 12-month cycle length. The value 0t is used in the calculation

since we assume the equipment is as good as new at the beginning of the cycle. We assume that

the inspection interval cannot be shorter than one month, that is, 12n , and that is a multiple

of one month time unit. The last inspection interval does not need to have the same length as the

preceding inspection intervals.

Component j j (months) Minimal

repair cost M

jc

Downtime penalty

cost/month D

jc

Replacement

cost R

jc

ja jb

1 1.3 3.5 $70 $100 $700 0.9 0.2317

2 1.1 4.6 $45 $250 $450 0.9 0.1763

3 2.1 6 $100 $220 $1000 0.9 0.1352

4 1.8 10 $75 $170 $750 0.9 0.0811

5 1.7 3.6 $150 $260 $1500 0.9 0.2253

Table 5.3. Parameters of the power law intensity functions, costs and ( )jr x

parameters for different components

Using equation (5.20) we calculate the total cost for different inspection intervals over

12 monthsT . We obtain ( )j

kM t , ( )j

kP t , and ( )j

ke t , 1,...,k n , recursively using equations


(5.17) and (5.18) and known functions 1 ( )jM t , 1 ( )jP t , and 1 ( )je t . ( )j

kR t is calculated from

1 [ ( ) ( )]j j

k kM t P t . Composite Simpson’s rule was applied for numerical integration with step

length 410h .

Note that if we want to calculate ( )j

k t for 0 t A , we have to calculate 1( )j

k t for

0 t A . For example, when we want to calculate (0)j

k , 1,2k ,...,n , we have to calculate

( )j

k t for 0 ( )t n k , 1,2 1k ,...,n . Direct calculation of ( )j

k t would be tedious, except

for small k .

1 2 3 4 5 6 7 8 9 10 11 12

Cost 10.33 9.37 9.20 9.20 9.11 9.36 9.25 9.16 9.15 9.24 9.51 9.80

8.50

9.00

9.50

10.00

10.50

11.00

Total cost for Tau = 1,2,3,...,12 in $1000

Figure 5.9. Total costs for different inspection intervals ( ) – finite horizon

As shown, 5 is the optimal inspection interval. This means that the system should be

inspected three times in its life cycle of length 12T months. The first inspection should be

performed after 5 months, the second inspection after 10 months, and the third two months later,

at the end of the 12 months interval. Some kind of peculiar behavior of the cost curve is due to

the combination of cost function for different components with different failure patterns.


5.2.2.2. Model Over Infinite Time Horizon

The inspection optimization model over an infinite time horizon uses the renewal-reward

theorem (Tijms, 2003). With this theorem, the objective function can be formulated as the

expected cost per unit time in long run and calculated as the sum of expected costs per unit time

for each component.

We first formulate the expected cost per cycle for component j . The cycle length for

component j is time of its replacement.

Let u k . The cost incurred for component j over u is

f i f ( ) ( ) ( ) ( ) ( )j j j j j Ic

C u C k C u C u C u km

,

where Ic

m is the inspections cost shared by component j and f

( )jC u expected cost due to

failures and replacements of component j over interval [0, ]u .

Then f ( )( )

jj

IC kC u c

u k m

, and

u ( )( )

lim limjj j

I I

ju k

C kC u c EC c

u k m ET m

, (5.21)

from the renewal reward theorem, where jT is the cycle length for component j , and jC is the

cycle cost for component j . jT can take values , 2 , … so that jT , because the first

replacement can be performed at earliest at the end of the first inspection interval.

We then have

1

( ),j j

k

T T k


and 0 0

1

[ | ] Pr{ | }j j j j

k

E T X t T k X t

where 0Pr{ | } 1j jT X t . We will consider later

when 0[ | ]j jE T X t .

Equation (5.21) can be applied to all components j , 1,2,...,j m , to find the expected

cost for the whole system:

1

1 1 1

( )( ) ( )

lim lim lim

mj

j j jm m mj I

ju l kj j j

C uC u C k EC c

u u k ET

. (5.22)

It should be noted that there is no renewal cycle for the system, but separate renewal

cycles for each component.

We give a general recursive equation for the infinite case similar to equation (5.18). For

simplicity, we will suppress index j in the following.

1 1 1 0 1 1

0

( ) [ ( , ) | ] [ ( ,0) ( ) ( ,1) ( )] ( | ) ( ,2) ( )t E X I X t x r t x x r t x f x t dx P t

, (5.23)

0 1 1 1 1

0

( ) [ ( , ) ( )| ] ( ) ( ) ( | ) ( ) ( ), k k k k kt E X I T k X t t x r t x f x t dx t P t

(5.24)

for 2 3,k , ... . For a proof, see Appendix J.

In the following special cases, we apply equations (5.23) and (5.24). The differences

between different cases are in their placeholder functions ( , )k kX I and 1( )t .

(a) For the number of cycles:

( , ) 1k kX I , 1 1 1

0

( ) ( | ) ( ) 1V t f x t dx P t

.

(b) For the number of minimal repairs:


1, 0( , )

0, 1,2

k

k k

k

IX I

I

,

1 1

0

( ) ( ) ( | )n t r t x f x t dx

.

(c) For downtime:

, 0,1( , )

0, 2

k k

k k

k

X IX I

I

,

1 1 1

0 0

( ) [( ) ( ) ( ) ( )] ( | ) ( ) ( | )d t x r t x x r t x f x t dx x f x t dx

.

Let

0

1 1

( ) [ ( , ) ( ) | ] ( )k k k

k k

G t E X I T k X t t

converges for all t .

Depending on the placeholder function, ( )G t represents a quantity such as expected

downtime, expected number of minimal repairs, etc. over one cycle.

Then

1 1 1 1 1

2 0

1 1 1 1 1

2 20

1 1 1

0

( ) ( ) [ ( ) ( ) ( | ) ( ) ( )]

( ) ( ) ( | ) ( ) ( ) ( )

( ) ( ) ( | ) ( ) ( ) ( ),

i i

i

i i

i i

G t t t x r t x f x t dx t P t

t r t x f x t t x dx t P t

t r t x f x t G t x dx G t P t

or

1 1 1

0

( ) ( ) ( ) ( | ) ( ) ( ) ( )G t t r t x f x t G t x dx G t P t

. (5.25)

Equation (5.25) implies that 1

( ) ( )k

k

G t t

can also be calculated recursively from

( )G t x .


To calculate 1

( ) ( )k

k

n t n t

, 1

( ) ( )k

k

d t d t

, and 1

( ) ( )k

k

V t V t

, we have to check

whether they are finite. The convergence of ( )V t is considered in Lemma 1 and Lemma 2 in

Appendix K, and the convergence of ( )G t in Lemma 3 in Appendix K.

The results of Lemma 1, 2, and 3 can be used to calculate 1

( )k

k

V t

, 1

( )k

k

n t

, and

1

( )k

k

d t

numerically. Details are given in Appendix K.

Therefore, in equation (5.21) we have for given j :

( ) ( )

( )

R M j D jjj j j

j j

c c n t c d tEC

ET V t

. (5.26)

From (5.22) and (5.26) the expected cost for the whole system is

1 1 1

( ) ( ) ( ) ( )1[ ]

( ) ( )

R M j D j R M j D jjm m mj j j j j jI I

Ij j jj j j

c c n t c d t c c n t c d tEC c cc

ET V t V t

. (5.27)

Numerical example

We use the same data given in the numerical example of Section 5.2.2.1. The downtime

costs are given in the second column of Table 5.4. With restriction 12 , 12 months is the

optimal inspection interval, that is, to inspect every 12 months.


1 2 3 4 5 6 7 8 9 10 11 12

Cost $948. $825. $775. $746. $726. $711. $699. $690. $683. $678. $674. $671.

$650.00

$700.00

$750.00

$800.00

$850.00

$900.00

$950.00

$1,000.00

Total cost for Tau = 1,2,3,...,12

Figure 5.10. Total costs for different inspection intervals ( ) – infinite horizon

Obviously, the optimum result depends on input parameters of the model. If we try the

calculation with higher penalty costs for downtime given in the last column of Table 5.4, the

optimal interval 5 months is obtained (Figure 5.11).

Component Downtime penalty cost/month D

jc

used in Figure 5.10

Downtime penalty cost/month D

jc

used in Figure 5.11

1 $100 $150

2 $200 $250

3 $220 $300

4 $70 $100

5 $60 $150

Table 5.4. Different downtime penalty costs

5.3. Case 2: A System Under Periodic and Opportunistic Inspections 141

1 2 3 4 5 6 7 8 9 10 11 12

Cost $986 $894 $869 $861 $858 $859 $860 $863 $866 $870 $874 $878

$840.00

$860.00

$880.00

$900.00

$920.00

$940.00

$960.00

$980.00

$1,000.00

Total cost for Tau = 1,2,3,...,12

Figure 5.11. 5 is obtained with downtime penalty costs from the last column of Table 5.4

5.3. Case 2: A System Under Periodic and Opportunistic Inspections

In this section, we assume that at a hard failure, in addition to the failed component, all

other components are also inspected and fixed if found to have failed. This model is called

opportunistic maintenance since a hard failure provides an opportunity to inspect the other

components as well. We call the opportunistic inspections at hard failures “non-scheduled

inspections”. In real situations such as medical devices in a hospital, clinical engineers have to

go through a check list at both scheduled and non-scheduled inspections to test and verify all

major components and features of a device to assure their safety and reliability.

In this section, we want to find the optimal scheduled inspection interval in presence of

opportunistic inspections/maintenance. We consider a system consisting of components subject

to either hard or soft failures with the assumptions stated in Section 4.3. The system is inspected

at scheduled, or at non-scheduled moments. Scheduled (periodical) inspections are performed at


times k ( 1,2,..., )k n , where T n is given, and non-scheduled inspections are performed at

hard failures (Figure 5.12).

Figure 5.12. Scheme of scheduled and non-scheduled inspections in a cycle

It should be noted that some components may have several failure modes including soft

and hard, but in the analysis they may be considered separately. For simplicity, we will identify

components with their dominant failure modes and assume that each component has only one

failure mode either soft or hard.

Additional Assumption

At a non-scheduled inspection, all components are inspected and fixed if found to have failed.

At scheduled inspections, only components with soft failures are inspected.

Notation

1m , 2m and m are the numbers of components with hard and soft failure, and the total

number of components, respectively; 21 mmm

( )j x intensity function of the non-homogeneous Poisson process associated with the

number of failures of component j over time period [0, ]x .

T system cycle length which is known and fixed.

2 )1( k k

NS NS NS

S S S S

NS: Non-scheduled inspection S: Scheduled inspection HF: Hard failure

HF HF HF

n T


n the number of scheduled inspection intervals.

length of periodic scheduled inspection interval; n T , or /T n .

[ ]TE C expected cost incurred in a cycle.

[ ]T

SE C expected cost incurred from soft failures in a cycle.

[ ]T

HE C expected cost incurred from hard failures in a cycle

Ic cost of scheduled inspection.

NIc cost of non-scheduled inspection.

The cost incurred over the cycle T consists of the costs arising from occurrences of soft and

hard failures, including the cost of inspections:

[ ] [ ] [ ]T T T

S HE C E C E C . (5.28)

5.3.1. Minimal Repair of Soft and Hard Failures

We consider a system with the assumptions stated in Section 4.3 and 5.3. Additional

assumptions are assumed as follows:


Opportunistic maintenance is applied.

Component with soft and hard failures are minimally repaired.

Additional Notation

Cost of minimal repair of a component if found failed ( M

jc )


Penalty cost for the elapsed time of soft failure for component j ( D

jc )

We assume that right after the occurrence of a hard failure, the failed component and all

components with soft failure are inspected (non-scheduled) with a cost greater than of a planned

inspection. The failed component with hard failure is minimally repaired. Due to the NHPP

assumption, the expected number of hard failures over the cycle T for component i is

0( )

T

i x dx , 11,2,...,i m . Therefore, the expected cost of unplanned inspections and minimal

repairs for 1m components with hard failure over cycle length T is:

1

01

[ ] [( ) ( ) ]m

TT M

H NI i i

i

E C c c x dx

. (5.29)

Equation (5.29) implies that [ ]T

HE C does not depend on the periodic inspection interval .

Therefore, this cost does not affect the optimal inspection policy and can be excluded from the

optimization model. It should be noted that the cost of non-scheduled inspections is accounted in

[ ]T

HE C , and will not be included in [ ]T

SE C . The optimal inspection interval is a value of

which minimizes the expected cost incurred from soft failures in a cycle, i.e. [ ]T

SE C . We include

in [ ]T

SE C the cost of periodic inspections since the components are periodically inspected to

rectify their soft failures.

If a component with soft failure fails before a hard failure, the hard failure time is when

the soft failure is rectified. If no hard failure takes place between the soft failure and the next

scheduled inspection, the scheduled inspection is when the soft failure is rectified and fixed. In

both cases, in addition to the cost of minimal repair, a penalty cost is incurred for the downtime

of the component, i.e. the time between the occurrence of a failure and when it is rectified.


Therefore, in the opportunistic case, hard failures contribute to the expected number of minimal

repairs and downtime of components with soft failures.

The total expected cost incurred from 2m components with soft failure over the cycle

T with n scheduled inspections is

1 2

1 1

[ ] cost of repairs cost of downtime

[ ( , , ) ( ( , , ))]

T

S I

m mM j D j

I j n j n

j m

E C nc

nc c M t s c T e t s

We will derive recursive equations to calculate the expected number of minimal repairs

and downtime over T for a component ( j ) with soft failure.

For the convenience of the presentation, we consider now a subsystem of the whole

system consisting of all components with hard failures combined in a serial configuration. Some

of these components may already have some initial ages when they are configured in the

subsystem. So, when the subsystem starts operating, not all its components are new. Let ( )i x is

the intensity function of the thi component with hard failure, and i is its initial age,

11,2,...,i m . Then, following from the serial configuration, the intensity function of the “hard”

subsystem at time z is

1

1

( ) ( ) ( )m

H H i i

i

z z z

, 11 2( , ,..., )m .

Then, if Z is the time to failure of the “hard” subsystem,

0

( )

( ) ( ) 1

z

HZ

x dx

F z P Z z e

,

For simplicity we suppress index .

When the “hard” subsystem has an initial age s , then


( )

( | ) 1

s z

H

s

x dxZF z s e

,

and ( )

( | ) ( )

s z

H

s

x dxZ

Hf z s z e

.

To develop the model, we first assume that there is only one inspection interval [0, ] ,

i.e. one scheduled inspection is performed at the end of the interval (time ), and all other

inspections in [0, ] , if there are any, are opportunistic inspections.

Let 0

jX t is the age of component j with soft failure, and 0Z s is the age of the

“hard” subsystem at the beginning of the cycle. If the first hard failure takes place in the

inspection interval , and the first soft failure occurs before the hard failure, the soft failure is

rectified at the hard failure time and the time between the occurrence of the soft failure and hard

failure is considered as the downtime of the component with soft failure (Figure 5.13).

Figure 5.13. Hard and soft failures in the first inspection interval

Let for a given , 1 min{ , }Z Z , where Z is the time of a hard failure. Therefore,

1Z , and

CDF of 1Z is:

1

( | ), ( | ) ,

1,

Z

Z F z s zF z s

z

1

jX

1st hard failure

0

jX t

0Z s

2

jX

2st hard failure

1st soft failure

1Z2Z


and ( )

1 1 0( | ) ( | )

s

H

s

y dyZP s P Z Z s e

.

For simplicity, we suppress index j in the following.

Let 1 1min{ , }X X Z , where X is the time of the soft failure. Therefore, 1 1X Z .

The joint CDF of 1Z and 1X is,1 1

1 1 1|X Z Z X ZF F F , where 1 1|X Z

F is the conditional CDF

of 1X given 1Z . Assuming that X and Z are independent, 1 1|X ZF is

1 1 1|

1

( ),( | ) ( | ) ,

1,

XX Z X F x x z

F x z F x Z zx z

( )

1 1 1 0 0 1 1 1 0 1( | , , ) ( | , ) ( | )

t z

t

y dyXP X Z Z z Z s X t P X Z Z z X t P z t e

.

To develop a general recursive formula to calculate the expected value of different

random variables such as the total uptime of the component over the inspection interval, we

calculate the conditional expectation of the random variable given the times of the first hard

failure and the first soft failure.

Let V is a random variable which is additive over disjoint intervals and is

calculated/observed over an interval [0, ] . V can be the total uptime, number of minimal

repairs, etc. To calculate the expected value of V , we start with interval [0, ] , but after

conditioning on the first hard failure’ time , e.g. 1Z z , the interval for which the expected value

of V needs to be calculated is reduced to [ , ]z . Thus, the initial interval becomes shorter at the

next step of the recursive calculations.

1 1[0, ] [ , ][0, ]

0 0 0 0[ | , ] [ | , ]Z Z

E V X t Z s E V V X t Z s

1 1[0, ] [ , ]

1 1 0 0[ [ | ] [ | ] | , ]Z Z

E E V Z E V Z X t Z s

. (5.30)


The recursive equations for the total uptime, number of minimal repairs, etc. are identical

aside from a placeholder function ( , )X Z , which differ according to the quantity we wish to

calculate. We derive recursive equations to find the conditional expectation of a placeholder

function ( , )X Z given that the initial age of the component with soft failure is 0X t , and the

initial age of “hard” subsystem is 0Z s .

Let, 1 0 0( , , ) [ | , ]G t s E V X t Z s

It can be shown, following equation (5.30), that (see Appendix L for the details)

1 1 1 1

0 0

( , , ) { ( , ) ( , , )} ( | ) ( | )

z

X ZG t s x z G z t x s z f x t f z s dxdz

1 1 1

0

1 1 1 1

0

{ ( , ) ( , , )} ( | ) ( | )

( | ) ( , ) ( | ) ( | ) ( | ) ( , )

Z X

Z X Z X

z z G z t z s z f z s P z t dz

P s x f x t dx P s P t

. (5.31)

To calculate 1( , , )G t s when there is only one inspection interval of length , i.e. T ,

1( , , )G t s must be calculated for all , starting with 1(0, , ) 0G t s .

When there are n inspection intervals of length over T , i.e. T n , we first calculate

1( , , )G t s using equation (5.31) for 0 ( 1)t n , 0 ( 1)s n which is the expected value

of variable V in the last inspection interval ([( 1) , ]n T ) given t and s . In this case also

1( , , )G t s must be calculated for all . We then proceed to the second to the last inspection

interval (i.e. time ( 2)n ), and we need to find 2 ( , , )G t s for 0 ( 2)t n , 0 ( 2)s n

and . 2 ( , , )G t s gives the expected value of V over [( 2) , ]n T , so it includes also the

expected value calculated from 1G . In other words, the calculations are performed backward.


To generalize the equation for the interval [0, ]k , ( 1,2,..., )k n , we need to find

[0, ]

0 0( , , ) [ | , ]U

kG t s E V X t Z s , where ( 1)U k is the length of the interval, for

0 ( )t n k , 0 ( )s n k , and for all .

By taking similar steps as for 1( , , )G t s , the following recursive equation can be derived

(see Appendix L):

0 0( , , ) [ | , , ( 1) ]kG t s E V X t Z s U k

1 1

0 0

{ ( , ) ( , , )} ( | ) ( | )

z

X Z

kx z G z t x s z f x t f z s dxdz

1 1

0

{ ( , ) ( , , )} ( | ) ( | )Z X

kz z G z t z s z f z s P z t dz

1 1 1

0

( | ) { ( , ) ( , , )} ( | )Z X

kP s x G t x s f x t dx

1 1 1( | ) ( | ){ ( , ) ( , , )}Z X

kP s P t G t s .

(5.32)

It should be noted that in practical applications, the last inspection interval may have a

length 0 shorter than , or 0( 1)T n . In this case, we first calculate 1( , , )G t s for

0 ( 1)t n , 0 ( 1)s n and all 0 , which calculates [ ]E V for the last inspection

interval , i.e. 0[( 1) , ( 1) ]n n , and eventually we calculate ( , , )nG t s for all , which

gives the [ ]E V over 0[0, ( 1) ]n .

The general equation ( , , )kG t s can be considered for some special cases as follows. The

difference is in their ( , )X Z function:

(a) For the expected number of minimal repairs of the component:


1, ( , )

0,

x zx z

x z

,

1 1 1 1

0 0

1 1 1 1 1

0 0

( , , ) {1 ( , , )} ( | ) ( | )

( , , ) ( | ) ( | ) ( | ) ( | )

z

X Z

Z X Z X

M t s M z t x s z f x t f z s dxdz

M z t z s z f z s P z t dz P s f x t dx

(b) For the expected survival time of the component:

( , )x z x ,

1 1 1 1

0 0

1 1 1

0

1 1 1 1

0

( , , ) { ( , , )} ( | ) ( | )

{ ( , , )} ( | ) ( | )

( | ) ( | ) ( | ) ( | ).

z

X Z

Z X

Z X Z X

e t s x e z t x s z f x t f z s dxdz

z e z t z s z f z s P z t dz


Thus,

1 2

1 1

[ ] [ ( , , ) ( ( , , ))] m m

T M j D j

S I j n j n

j m

E C nc c M t s c T e t s

. (5.33)

Equation 5.33 should be calculated for different values of n to find the optimal inspection

interval, which gives the minimum expected cost (optimal) over the cycle T .

Numerical example

Consider a repairable system with eight components. Five components have soft failure,

and three components have hard failure. The parameters of the power law processes, and

minimal repair and downtime costs for components with soft and hard failures are given in Table

5.5. The system is currently inspected periodically every 4 months. We are interested in finding


the optimal inspection interval, which minimizes the total cost incurred from soft and hard

failures over a 12-month cycle length. For simplicity we assume what the equipment is as good

as new at the beginning of the cycle, so 0t and 0s are used in the calculation. The

scheduled inspection cost $70IC is considered.

Component j j (months) Minimal repair

cost M

jc

Downtime penalty

cost/month D

jc

So

ft F

ail

ure

1 1.3 3.5 $70 $150

2 1.1 4.6 $45 $250

3 2.1 6 $100 $300

4 1.8 10 $75 $100

5 1.7 3.6 $150 $150

Hard

Fail

ure

1

2

3

1.5

1.2

1.7

11

7.2

2.8

---

---

---

---

---

---

Table 5.5. Parameters of the power law intensity functions, and costs for

components with soft and hard failure

We first developed a MATLAB procedure to calculate ( , , )j

kM t s and ( , , )j

ke t s for

1,...,k n , recursively using equations (5.31) and (5.32). Direct calculation of ( , , )j

kM t s and

( , , )j

ke t s would be tedious, except for small k . Therefore, Composite Simpson’s rule was

applied. To obtain an acceptable level of accuracy a very small step length h , such as 510h or

less should be considered. This increases intensively the calculation time. So, instead of using

the recursive procedure, we developed a simulation model to calculate ( , , )j

kM t s and

( , , )j

ke t s for inspection intervals 1,2,...,12 months. The expected values and their standard


errors were calculated for every value of using 1,000,000 simulated histories. The values of

0.0014, and 0.0011 were obtained as the average standard errors for the calculated expected

values ( , , )j

kM t s and ( , , )j

ke t s , respectively, for components with soft failure. The results of

simulation are given in Table M.1 in Appendix M.

The total expected costs for different inspection intervals over 12T months were

obtained using equation (5.33) and are given in Table 5.6. Relatively similar estimated expected

costs obtained for 2,...,12 are due to the low minimal repair and downtime costs considered

in this example. The result is shown in Figure 5.14.

Expected Cost

1 3681.05

2 3522.08

3 3509.65

4 3519.46

5 3526.75

6 3521.49

7 3521.60

8 3530.19

9 3526.87

10 3534.43

11 3543.85

12 3523.29

Table 5.6. Expected cost calculated for 1,2,...,12 in Figure 5.14


Figure 5.14. Expected costs for different inspection intervals ( ),

assuming soft and hard failures are minimally repaired

As shown, 3 is the optimal inspection interval with the expected cost of $3509.65,

without including the cost related to the hard failures ( [ ]T

HE C ). This means that the system

should be inspected four times in its life cycle of 12T months, or that inspections should be

performed every 3 months. Increase of the cost curve at some points such as at 5 or 8

can be explained as follows: the system under the inspection policy of 5 is inspected

periodically 3 times during its life cycle, first time after 5 and the second time after 10 months,

and the last inspection after 2 months, i.e. at the end of the life cycle of 12 months. So we have

two intervals of length 5, and one of length 2 months. With 4 , we inspect 3 times, that is

every 4 months. In the longer inspection intervals of length 5 months compared to intervals of 4

months, it is more likely to have soft failures, and consequently penalty cost arising from their

downtimes. However, in the case of 5 , in the last inspection interval of 2 months, it is less

likely to have a failed soft component, but still the cost of an additional periodic inspection is

incurred. Thus, the total expected cost of 5 is greater than of 4 . A similar justification


can be given for the increase in the cost curve at 8 compared to 7 . A sharp drop in the

cost curve at 12 is due to the inspection cost, which is incurred only once, compared to

11 when it is incurred twice, but still with almost the same expected downtime and repair

costs.

5.3.2. Minimal Repair and Replacement of Soft Failures

We again consider a system with the assumptions stated in Section 4.3 and 5.3. In

addition, we assume that the components with hard failure is minimally repaired, and the

components with soft failures are either minimally repaired or replaced with new ones with some

age dependent probabilities. We assume that component j is minimally repaired at age x with

probability ( )jr x , or is replaced with probability ( ) 1 ( )j jr x r x . Thus, ( )jr x is a function of the

component’s age.



Component with hard failures are minimally repaired.

Component with soft failures are either minimally repaired or replaced with some age

dependent probabilities.

Additional Notation

Cost of the component if found failed with minimal repair ( M

jc )

Cost of the component if found failed with replacement ( R

jc )


Penalty cost for the elapsed time of soft failure for component j ( D

jc )

Let jI denotes the status of component j at the current inspection (either periodic or

opportunistic):

0, minimal repair at current inspection

1, replacement at current inspection

2, no failure in the current inspection interval

jI

We will now derive recursive equations to find the conditional expectation of a

placeholder function ( , , )X Z I given the initial ages of the component with soft failure and the

“hard” subsystem are 0X t and 0Z s , respectively. The placeholder ( , , )X Z I here is a

function of three variables. For simplicity, we suppress index j . Let, as before,

1 0 0( , , ) [ | , ]G t s E V X t Z s .

From the assumptions of NHPP and minimal repair or replacement, we get the recursive

equations as follow (Appendix N):

1

1 1 1 1

0 0

1 1 1

0

1 1

0

( , , )

{[ ( , ,0) ( , , )] ( ) [ ( , ,1) ( ,0, )] ( )} ( | ) ( | )

{ ( , , 2) ( , , )} ( | ) ( | )

( | ) [ ( , ,0) ( ) ( , ,1) ( )] ( | )

z

X Z

X Z

Z X

G t s

x z G z t x s z r t x x z G z s z r t x f x t f z s dxdz

z z G z t z s z P z t f z s dz

P s x r t x x r t x f x t

1 1( | ) ( | ) ( , , 2),Z Xdx P s P t

(5.34)

and

( , , )kG t s


1 1

0 0

{[ ( , ,0) ( , , )] ( ) [ ( , ,1) ( ,0, )] ( )} ( | ) ( | )

z

X Z

k kx z G z t x s z r t x x z G z s z r t x f x t f z s dxdz

1 1

0

{ ( , ,2) ( , , )} ( | ) ( | )Z X


1 1 1 1

0

( | ) {[ ( , ,0) ( , , )] ( ) [ ( , ,1) ( ,0, )] ( )} ( | )Z X

k kP s x G t x s r t x x G s r t x f x t dx

1 1 1( | ) ( | ){ ( , , 2) ( , , )}Z X

kP s P t G t s

(5.35)

Then equations (5.34) and (5.35) can be applied to the special cases as follows:


1, and 0( , , )

0, or 0

x z ix z i

x z i

,

1 1 1 1

0 0

1 1 1 1 1

0 0

( , , ) {1 ( , , )} ( ) ( | ) ( | )

( , , ) ( | ) ( | ) ( | ) ( ) ( | )

z

X Z

Z X Z X

M t s M z t x s z r t x f x t f z s dxdz

M z t z s z f z s P z t dz P s r t x f x t dx

(b) For the expected number of replacements of the component:

1, and 1( , , )

0, or 1

x z ix z i

x z i

,

1 1 1 1

0 0

1 1 1 1 1

0 0

( , , ) {1 ( ,0, )} ( ) ( | ) ( | )

( , , ) ( | ) ( | ) ( | ) ( ) ( | )

z

X Z

Z X Z X

R t s R z s z r t x f x t f z s dxdz

R z t z s z f z s P z t dz P s r t x f x t dx


(c) For the expected survival time of the component:

( , , )x z i x ,

1 1 1 1 1

0 0

1 1 1

0

1 1 1 1

0

( , , ) { ( ) ( , , ) ( ) ( ,0, )} ( | ) ( | )

{ ( , , )} ( | ) ( | )

( | ) ( | ) ( | ) ( | ).

z

X Z

Z X

Z X Z X

e t s x r t x e z t x s z r t x e z s z f x t f z s dxdz

z e z t z s z f z s P z t dz


The total expected cost incurred over the cycle of length T with n scheduled inspections is

1 2

1 1

[ ] [ ( , , ) ( , , ) ( ( , , ))] m m

T M j R j D j

S I j n j n j n

j m

E C nc c M t s c R t s c T e t s

(5.36)

The optimal inspection interval is the value of which minimizes [ ]T

SE C .

Numerical example

Assume that

( ) , 0, 0, 0jb x

j j j jr x a e a b x

; using the same parameters presented in

the numerical example of Section 5.3.1, with the replacement

costs

given in the second column

of Table 5.7, and the parameters ja and jb of ( )jr x , the expected cost curve is calculated using

simulation and is given in Figure 5.15 for 1,2,...,12 . The estimated expected costs are

presented in Table 5.8. The expected number of minimal repairs and replacements, and the

expected uptimes and their standard errors for the components with soft failure for 1,2,...,12

are given in Table M.2 of Appendix M. The values of 0.0010, 0.0007, and 0.0011 were obtained

as the average standard errors for the calculated expected values ( , , )j

kM t s , ( , , )j

kR t s and

( , , )j

ke t s , respectively, for the components with soft failure.


Component

Replacement cost R

jc

used in Fig. 5.15

Replacement cost R

jc

used in Fig. 5.16

ja jb S

oft

Fail

ure

1 $700 $175 0.9 0.2317

2 $450 $112.5 0.9 0.1763

3 $1000 $250 0.9 0.1352

4 $750 $187.5 0.9 0.0811

5 $1500 $375 0.9 0.2253

Table 5.7. Costs and parameters of the probability function for minimal repair of components

with soft failure

Expected costs

calculated in Fig. 5.15

Expected costs

calculated in Fig. 5.16

1 8413.40 3814.39

2 7936.29 3642.98

3 7768.25 3629.11

4 7682.95 3633.87

5 7670.33 3653.15

6 7605.29 3642.51

7 7599.54 3649.80

8 7596.29 3658.20

9 7594.22 3661.44

10 7591.47 3669.37

11 7591.37 3678.89

12 7532.93 3649.39

Table 5.8. Expected costs calculated for 1,2,...,12 in Figures 5.15 and 5.16



assuming soft failures are either minimally repaired or replaced

As shown in Figure 5.15, 12 is the optimal inspection interval with the expected cost

of $7532.93. This means that the system should be inspected only once at the end of the life

cycle of length 12T months. The optimal inspection interval varies by changing the

replacement costs. For example, by considering the replacement costs given in the third column

of Table 5.7, optimal 3 months is obtained with the expected cost of $3629.105 (Figure

5.16). The estimated expected costs are presented in the last column of Table 5.8.


Figure 5.16. Expected costs for different inspection intervals ( ), assuming soft

failures are either minimally repaired or replaced with different replacement costs

5.3.3. Minimal Repair and Replacement of Soft and Hard Failures

We consider a system with the assumptions stated in Section 4.3 and 5.3, additional

assumptions are assumed as follows:



Component with soft and hard failures are either minimally repaired or replaced with some

age dependent probabilities.

If we assume that a component with hard failure, depending on its age is either replaced

or minimally repaired at failure, then each hard failure component should be considered

individually in the model. When there is a hard failure in an interval and the component is


replaced, only the initial age of the failed component becomes zero and the age of the other

component remains unchanged.

Let ( )j z is the intensity function of the thj component with hard failure, and j is its

initial age, 11,2,...,j m .

( | ) ( )j j j jz z .

Let J is the random variable which denotes the failed component with hard failure in

z Z z z , and let ( )jq z is the conditional probability that the failed component in

z Z z z is component j given initial ages of components with hard failure are 11,..., m .

So,

0( ) lim ( | , )j

zq z P J j z Z z z

,

where 11( ,..., )m .

Then

0 0

( | )( , | )( ) lim ( | , ) lim ,

( | ) ( | )

j jj

z zH

zP J j z Z z zq z P J j z Z z z

P z Z z z z

and

0

0

( | )

1

( | )

( | )( ) ( | ) ( | )

( | )

( | ) .

z

H

z

H

x dxj jj Z

H

H

x dx

j j

zq z dF z z e dz

z

z e dz

.

Let 1 min{ , }Z Z , so 1Z denotes either the time to the first hard failure, if there is any

in [0, ] , or the inspection interval of length , if all components with hard failure survive. Let

XI and ZI be the indicators for survival or failure with minimal repair or replacement of the


component with soft and hard failures (component j ), respectively and ( )Xr x and ( )Z

jr z be their

age dependent probability functions.

Let 1

1

1 2

0 1 0 2 0, ,...,m

mZ Z Z , and 1 11 1 0 1( , , ,..., ) [ | , ( ,..., ), ]m mG t E V X t U .

It can be shown (see Appendix O) that

1 11 1 0 1( , , ,..., ) [ | , ( ,..., ), ]m mG t E V X t U

1

11 1 2

10 0

[ ( ,0, , ,0) ( , , , ,..., ,..., )] ( )

z mX

j m

j

x z j G z t x z z z z r t x

11 1 2[ ( ,1, , ,0) ( ,0, , ,..., ,..., )] ( ) ( )X Z

j m j jx z j G z z z z z r t x r z

11 1 2[ ( ,0, , ,1) ( , , , ,...,0,..., )] ( )X

mx z j G z t x z z z r t x

11 1 2[ ( ,1, ,1) ( ,0, , ,...,0,..., )] ( ) ( )X Z

m j jx j G z z z z r t x r z

11 1 1 2( ) ( | , ) ( | ( , ,..., ))j X Z

mq z f x t z f z dxdz

1

11 1 2

10

{[ ( ,2, , ,0) ( , , , ,..., ,..., )] ( )m

Z

j m j j

j

z z j G z t z z z z z r z

11 1 2[ ( ,2, , ,1) ( , , , ,...,0,..., )] ( )}Z

m j jz z j G z t z z z z r z

11 1 1 2( ) ( | ) ( | ( , ,..., ))j X Z

mq z P z t f z dz

11 1 1 2

0

{ ( ,0, ,0,2) ( ) ( ,1, ,0,2) ( )} ( | , ) ( | ( , ,..., ))X X X Z

mx r t x x r t x f x t dxP

11 1 1 2( ,2, ,0,2) ( | ) ( | ( , ,..., )X Z

mP t P .

(5.37)

and for the general case, when the interval length is [0, ( 1) ]k , ( 1,2,..., )k n :

1 11 2 0 1 2( , , , ,..., ) [ | , ( , ,..., ), ( 1) ]k m mG t E V X t U k

11 0 1 2{ [ | ] | , ( , ,..., ), }mE E V Z X t U


1

11 2

10 0

[ ( ,0, , ,0) ( , , , ,..., ,..., )] ( )

z mX

k j m

j


1

1

1

1 2

1 2

1 2

[ ( ,1, , ,0) ( ,0, , ,..., ,..., )] ( ) ( )

[ ( ,0, , ,1) ( , , , ,...,0,..., )] ( )

[ ( ,1, ,1) ( ,0, , ,...,0,..., )] ( ) ( )

(

X Z

k j m j j

X

k m

X Z

k m j j

j

x z j G z z z z z r t x r z

x z j G z t x z z z r t x

x j G z z z z r t x r z

q z

11 1 1 2) ( | , ) ( | ( , ,..., ))X Z

mf x t z f z dxdz

1

11 2

10

{[ ( ,2, , ,0) ( , , , ,..., ,..., )] ( )m

Z

k j m j j

j


11 2[ ( ,2, , ,1) ( , , , ,...,0,..., )] ( )}Z

k m j jz z j G z t z z z z r z

11 1 1 2( ) ( | ) ( | ( , ,..., ))j X Z

mq z P z t f z dz

1

1 1

1 1

0

1 1 1 1 1 2

{[ ( ,0, ,0, 2) ( , , ,..., )] ( )

[ ( ,1, ,0, 2) ( ,0, ,..., )] ( )} ( | , ) ( | ( , ,..., ))

X

k m

X X Z

k m m

x G t x r t x

x G r t x f x t dxP

1 11 1 1 1 1 2[ ( ,2, ,0,2) ( , , ,..., )] ( | ) ( | ( , ,..., ))X Z

k m mG t P t P

(5.38)

The argument 0 in the function ( , , ,0, )X Zx i z i implies that 0j or none of the

components with hard failure has failed. Equations (5.37) and (5.38) can be applied to the special

cases as follows:


1, 0( , , , , )

0, 0

X

X Z

X

ix i z j i

i

,

11 1 2( , , , ,..., )mM t

1

11 1 2

10 0

[1 ( , , , ,..., ,..., )] ( )

z mX

j m

j

M z t x z z z z r t x

11 1 2( ,0, , ,..., ,..., ) ( ) ( )X Z

j m j jM z z z z z r t x r z


11 1 2[1 ( , , , ,...,0,..., )] ( )X

mM z t x z z z r t x

11 1 2( ,0, , ,...,0,..., ) ( ) ( )X Z

m j jM z z z z r t x r z

11 1 1 2( ) ( | , ) ( | ( , ,..., ))j X Z


1

11 1 2

10

{ ( , , , ,..., ,..., ) ( )m

Z

j m j j

j

M z t z z z z z r z

11 1 2( , , , ,...,0,..., ) ( )}Z

m j jM z t z z z z r z

11 1 1 2( ) ( | ) ( | ( , ,..., ))j X Z

mq z P z t f z dz

11 1 1 2

0

( ) ( | , ) ( | ( , ,..., ))X X Z

mr t x f x t dxP

(b) For the expected number of replacements of the component:

1, 1( , , , , )

0, 1

X

X Z

X

ix i z j i

i

,

11 1 2( , , , ,..., )mR t

1

11 1 2

10 0

( , , , ,..., ,..., ) ( )

z mX

j m

j

R z t x z z z z r t x

11 1 2[1 ( ,0, , ,..., ,..., )] ( ) ( )X Z

j m j jR z z z z z r t x r z

11 1 2( , , , ,...,0,..., ) ( )X

mR z t x z z z r t x

11 1 2[1 ( ,0, , ,...,0,..., )] ( ) ( )X Z

m j jR z z z z r t x r z

11 1 1 2( ) ( | , ) ( | ( , ,..., ))j X Z


1

11 1 2

10

{ ( , , , ,..., ,..., ) ( )m

Z

j m j j

j

R z t z z z z z r z

11 1 2( , , , ,...,0,..., ) ( )}Z

m j jR z t z z z z r z

11 1 1 2( ) ( | ) ( | ( , ,..., ))j X Z

mq z P z t f z dz


11 1 1 2

0

( ) ( | , ) ( | ( , ,..., ))X X Z

mr t x f x t dxP

(c) For the expected survival time of the component:

( , , , , )X Zx i z j i x

11 1 2( , , , ,..., )mE t

1

11 1 2

10 0

[ ( , , , ,..., ,..., )] ( )

z mX

j m

j

x e z t x z z z z r t x

11 1 2[ ( ,0, , ,..., ,..., )] ( ) ( )X Z

j m j jx e z z z z z r t x r z

11 1 2[ ( , , , ,...,0,..., )] ( )X

mx e z t x z z z r t x

11 1 2[ ( ,0, , ,...,0,..., )] ( ) ( )X Z

m j jx e z z z z r t x r z

11 1 1 2( ) ( | , ) ( | ( , ,..., ))j X Z


1

11 1 2

10

{[ ( , , , ,..., ,..., )] ( )m

Z

j m j j

j

z e z t z z z z z r z

11 1 2[ ( , , , ,...,0,..., )] ( )}Z

m j jz e z t z z z z r z

11 1 1 2( ) ( | ) ( | ( , ,..., ))j X Z

mq z P z t f z dz

11 1 1 2

0

( | , ) ( | ( , ,..., ))X Z

mxf x t dxP

11 1 1 2( | ) ( | ( , ,..., ))X Z

mP t P .

The optimal inspection interval can be obtained by minimizing the following:

1 2

1 1 1

1

1 2 1 2 1 2

1

[ ] [ ( , , , ,..., ) ( , , , ,..., ) ( ( , , , ,..., ))] m m

T M j R j D j

S I j n m j n m j n m

j m

E C nc c M t c R t c T e t

(5.39)


Numerical example

Using the parameters given in Tables 5.9 and 5.10, and the inspection cost $70IC , the

expected cost curve is calculated from equation (5.39) and is given in Figure 5.17 for

1,2,...,12 .

Component j j (months) Minimal repair

cost M

jc

Replacement

cost R

jc

Downtime penalty

cost/month D

jc

So

ft

Fail

ure

1 1.3 3.5 $70 $700 $150

2 1.1 4.6 $45 $450 $250

3 2.1 6 $100 $1000 $300

4 1.8 10 $75 $750 $100

5 1.7 3.6 $150 $1500 $150

Hard

Fail

ure

1 1.5 11 --- ---

2 1.2 7.2 --- ---

3 1.7 2.8 --- ---

Table 5.9. Parameters of the power law intensity functions, and costs for

components with soft and hard failure

Component ja jb

Soft Failure

1 0.9 0.2317

2 0.9 0.1763

3 0.9 0.1352

4 0.9 0.0811

5 0.9 0.2253

Hard Failure

1 0.8 0.06

2 0.5 0.02

3 0.9 0.28

Table 5.10. Parameters ja and jb for different components



assuming soft failures are either minimally repaired or replaced

As it is shown on Figure 5.17, the optimal value 12 is obtained with the total cost of

$7570.638. If replacement costs given in Table 5.11 are considered, the optimal value 2

months with the total cost of $4317.506 will be obtained (Figure 5.18).

Component Replacement cost

R

jc

Soft Failure

1 233.33

2 150.00

3 333.33

4 250.00

5 500.00

Table 5.11. Replacement costs considered in Figure 5.18


Figure 5.18. Expected costs for inspection intervals ( ), assuming soft and

hard failures are either minimally repaired or replaced with the replacement

costs given in Table 5.11

5.3.4. Preventive Replacement of Components with Hard Failures at Periodic Inspections

We consider again a system with the assumptions stated in Section 5.3.3, i.e. both

components with soft and hard failures are either minimally repaired or replaced with some age

dependent probabilities. In addition, we assume that the components with hard failures can be

preventively replaced at periodic inspections, so the most advanced case is considered (see the

structure shown in Figure 5.1). The preventive replacement of a component with hard failure still

depends on its age. In other words, the component may be replaced at a scheduled inspection or

no preventive action is performed.




Component with soft and hard failures are either minimally repaired or replaced with some

age dependent probabilities.

Components with hard failure are preventively replaced at periodic inspections depending on

some age dependent probabilities

Additional Notation

Cost of the preventive replacement of component j with hard failure ( PR

jc )

Probability that the component with soft failure is minimally repaired at age x ( ( )Xr x )

Probability that component j with hard failure is minimally repaired at age z ( ( )Z

jr z )

Components with hard failures are preventively replaced at scheduled inspections with

some age dependent probabilities with a lower cost, or they are repaired or replaced at failures

(non-scheduled) with a higher cost. Therefore, we need to calculate the expected number of

minimal repairs and replacements of hard components at both scheduled and non-scheduled

inspections.

The optimal inspection interval can be obtained by minimizing the following:

1 2

1 1 1

1

1

1

1 2 1 2 1 2

1

1

1

[ ] [ ( , , , ,..., ) ( , , , ,..., ) ( ( , , , ,..., ))]

[ ( , ) ( , )]

+ ( , ),

m mT M j R j D j

I j n m j n m j n m

j m

mM Hj R Hj

j n j j n j

j

mPR j

j n j

j

E C nc c M t c R t c T e t

c M c R

c PR

(5.40)


where 1

1

[ ( , ) ( , )] m

M Hj R Hj

j n j j n j

j

c M c R

is the total costs of minimal repairs and replacements

of the components with hard failures inside inspection intervals (non-scheduled inspections), and

1

1

( , )m

PR j

j n j

j

c PR

is the replacement costs of hard failures at scheduled inspections.

To calculate the required variables used in [ ]TE C , i.e. j

nM ,j

nR ,j

ne , Hj

nM , Hj

nR , and j

nPR ,

similarly to the methods described in the previous sections, we derive general recursive

equations to find the conditional expectation of a placeholder function ( , , , , )X Zx i z j i . In this

case, function 11 1 2( , , , ,..., )mG t will be identical to equation (5.37). For the general case,

when the interval length is [0, ( 1) ]k , ( 1,2,..., )k n a more complicated equation is

derived as follows:

11 2( , , , ,..., )k mG t

1

11 2

10 0

[ ( ,0, , ,0) ( , , , ,..., ,..., )] ( )

z mX

k j m

j


11 2[ ( ,1, , ,0) ( ,0, , ,..., ,..., )] ( ) ( )X Z

k j m j jx z j G z z z z z r t x r z

11 2[ ( ,0, , ,1) ( , , , ,...,0,..., )] ( )X

k mx z j G z t x z z z r t x

11 2[ ( ,1, ,1) ( ,0, , ,...,0,..., )] ( ) ( )X Z

k m j jx j G z z z z r t x r z

11 1 1 2( ) ( | , ) ( | ( , ,..., ))j X Z


1

11 2

10

{[ ( ,2, , ,0) ( , , , ,..., ,..., )] ( )m

Z

k j m j j

j


11 2[ ( ,2, , ,1) ( , , , ,...,0,..., )] ( )}Z

k m j jz z j G z t z z z z r z

11 1 1 2( ) ( | ) ( | ( , ,..., ))j X Z

mq z P z t f z dz


1 1 11

11

1 1 1 1 1

,...,0

( ,0, ,0,2) ( , , ( ),..., ( )) ( ,..., ) ( )m

X

k m m m mx G t x P V V r t x

1 1 11

11

1 1 1 1 1

,...,

( ,1, ,0,2) ( ,0, ( ),..., ( )) ( ,..., ) ( )m

X

k m m m mx G P V V r t x

11 1 1 2( | , ) ( | ( , ,..., ))X Z

mf x t z dxP

1 1 11

11

1 1 1 1 1

,...,

( ,2, ,0,2) ( , , ( ),..., ( )) ( ,..., )m

k m m m mG t x P V V

11 1 1 2( | ) ( | ( , ,..., ))X Z

mP t P ,

(5.41)

where

1

1, minimal repair of component , 1,...,

0, replacement of component i

ii m

i

and

( ), minimal repair of component ( )

( ), replacement of component

Z

i i

i i Z

i i

r z iP V

r z i

,

or

1 11

1( ) ( ) ( ) , 1,...,Z Z

i i i iP V r r i m

.

Also,

1 1 1 11 1 1 1( ,..., ) ( )... ( )m m m mP V V P V P V

1 1 1 1

1 1

11

1 1( ) ( ) ...( ) ( )m m

m m

Z Z Z Zr r r r

1G and kG can be simplified as 1

HG and H

kG for calculating the expected values inside

inspection intervals, and at the inspections for component j with hard failure. Let Z Z

jr r and

j , then


1

1 1 1

0

1

( , )

{[ ( ,0; ) ( , )] ( ) [ ( ,1; ) ( ,0)] ( )} ( | )

( | )[ ( ,0; ) ( ) ( ,1; ) ( )],

H

H Z H Z Z

Z Z Z

G

z G z z r z z G z r z f z dz

P r r

(5.42)

and

( , )H

kG

1

0

{[ ( ,0; ) ( , )] ( ) [ ( ,1; ) ( ,0)] ( )} ( | )H Z H Z Z

k kz G z z r z z G z r z f z dz

1 1 1{ ( ,0; ) ( , ) ( ) ( ,1; ) ( ,0) ( )} ( | )H Z H Z X

k kG r G r P

(5.43)

Variables 1M , 1R , and 1e in equation (5.40) can be calculated from the equations given in

Section 5.3.3, the other required variables 1

HM , 1

HR , and 1PR are the special cases of equation

(5.42):

(a) For the expected number of minimal repairs of the component in [0, ] :

1, , 0( , ; )

0, otherwise

z iz i

,

1 1 1 1

0

( , ) {[1 ( , )] ( ) ( ,0) ( )} ( | )H H Z H Z ZM s M z s z r s z M z r s z f z s dz

(b) For the expected number of replacements of the component in [0, ] :

1, , 1( , ; )

0, otherwise

z iz i

,

1 1 1 1

0

( , ) { ( , ) ( ) [1 ( ,0)] ( )} ( | )H H Z H Z ZR s R z s z r s z R z r s z f z s dz


(c) For the expected number of replacements at periodic inspections:

1, , 1( , ; )

0, otherwise

z iz i

,

1

1 1 1

0

1

( , )

{ ( , ) ( ) ( ,0) ( )} ( | )

( | ) ( )

Z Z Z

Z Z

PR s

PR z s z r s z PR z r s z f z s dz

P s r s

Numerical example

We use the power law parameters, the costs of components with soft failure, and

parameters ja and jb given in Tables 5.9 and Table 5.10. The costs of components with hard

failure are given in Table 5.12. Inspection cost $70IC is considered. We developed a

simulation model to estimate the required expected values for inspection intervals 1,2,...,12

months. The expected cost curve is calculated using equation (5.40) and is shown in Figure 5.19

for different values of . The optimal 12 is obtained with cost= S18012.5.

Component Minimal repair

cost M

jc

Replacement cost R

jc used in

Figure 5.19

Preventive replacement PR

jc

Replacement cost R

jc

used in

Figure 5.20

Ha

rd F

ail

ure

1 $100 $800 $500 $2400

2 $200 $1500 $1000 $4500

3 $150 $2000 $1500 $6000

Table 5.12. The costs for different components with hard failure



assuming preventive replacement of component with hard failure,

with the costs given in the third column of Table 5.12

With different replacement costs R

jc , given in the last column of Table 5.12 for the

components with hard failure, the optimal 3 is obtained with cost= $33959.92.

Figure 5.20. Expected costs for different inspection intervals ( ), assuming

preventive replacement of component with hard failure, with the costs given in the

last column of Table 5.12



A complex repairable system consists of various components with different types of

failure, hard and soft. So-called hard failures are revealed, i.e., either they are self-announcing or

the system stops when they take place. However, the system still can continue functioning when

soft failures occur. Both hard and soft failures should be incorporated in the general formulation

of an inspection optimization model for a repairable system. Soft and hard failures are important

in terms of the consequences which may result. Power plants, hospitals or any other organization

for which safety of instruments is a vital issue have to consider carefully all outcomes of their

systems’ failures. Unrevealed or soft failures may even result in a catastrophe if unattended.

This chapter proposes several inspection optimization models for a periodically inspected

repairable system under different assumptions. Two main cases of non-opportunistic and

opportunistic maintenance are considered. In all models, it is assumed that hard failures are

instantly rectified, and the failed component is inspected (non-scheduled) and fixed.

In the models proposed for the case of non-opportunistic maintenance, soft failures are

only rectified at periodic inspections. Therefore, occurrence of hard failures does not affect the

optimal periodic inspection interval, i.e. hard failures do not incorporate in the optimization

models. The optimal inspection interval minimizes the expected cost incurred only from the soft

failures.

We first assume that soft failures are minimally repaired, and develop two variants of the

model under this assumption. The first variant takes into account the elapsed times from soft

failures by considering downtime penalty costs for each component. The second variant sets up a

threshold for the total number of soft failures and considers penalty costs for exceeding the

threshold and for the number of failures, which exceed the threshold. A combined model is also


proposed to incorporate both the threshold on number of soft failures and the elapsed times of

failures. Due to the assumption of minimal repair, a non-homogeneous Poisson process is used as

the model for failure processes. Since the times of soft failures are not known (they are interval

censored by inspections), it is necessary to develop a recursive procedure to calculate

probabilities of failures and expected downtimes, which makes calculation of the optimum

inspection frequency computationally intensive. The procedure allows for considering initial age

of each component not necessarily equal to zero at the beginning of the cycle, and therefore,

possibility to perform incomplete repair at the beginning of the next cycle. In other words, some

components can be renewed, while the ages of others can be reduced. Then, the optimal

inspection frequency can be calculated for the next cycle. The recursive procedure that deals

with downtimes and minimal repairs in a NHPP is the main contribution of the proposed models.

Secondly, we extend the repair assumption of the soft failures to include both possibility

of minimal repair and replacement of a component, depending on some age dependent

probabilities. Under this assumption, two models are proposed for finding the optimal inspection

interval, one over finite and the other over infinite time horizon. The model over a finite time

horizon assumes that at an inspection a failed component may be replaced or minimally repaired.

The probability of minimal repair is a function of the component’s age and is assumed to be

decreasing by age. A penalty cost is incurred for the elapsed time from a failure to its detection

at the next inspection. In this model, we find the optimal inspection interval with minimum

expected cost. The second model uses the renewal-reward theorem (Tijms, 2003) and formulates

the objective function as the expected cost per unit time for the system, which is equal to the sum

of the expected costs per unit time of all components. A recursive procedure is developed to

calculate probabilities of failures, expected number of minimal repairs, and expected downtimes.


Then, the case of opportunistic maintenance is considered. It is assumed that hard failures

are detected and fixed instantaneously; however, soft failures are rectified at periodic or

opportunistic inspections. An opportunistic inspection is the moment of a hard failure, when all

components with soft failure are also inspected.

In the opportunistic case, it is first assumed that hard failures are only minimally repaired.

Two models are proposed; the first model assumes the minimal repair of the components with

soft failures. The second model is an extension of the first model and considers the possibility of

minimal repair and replacement of components with soft failures, depending on some age

dependent probabilities. In the proposed models, the repair cost of a hard failure and the

opportunistic inspection cost do not affect the optimal periodic inspection interval. However, the

occurrences of hard failures influence the detection of soft failures and then their expected

number of minimal repairs, replacements, and the expected downtime.

Moreover, the proposed models are extended to the case when components with hard

failures are also either minimally repaired or replaced. In addition, preventive replacement of

components with hard failures at periodic inspections is also included in the model. In the latter,

both repair/replacement cost of a hard failure and opportunistic inspection cost influence the

optimal periodic inspection interval.

The calculation of the excepted cost with delayed replacement or minimal repair of a

component at periodic or opportunistic inspections is technically solved in all proposed models.

Due to involved numerical calculation, simulation is used instead to calculate the cost function

and to find the optimal inspection interval.

178

6. CONCLUSION AND FUTURE RESEARCH

6.1. Conclusion

Hospitals’ equipment management program can be significantly improved by proper

employment of optimization techniques. Management of a large number of medical devices, if

fairly planned and executed, can be cost-effective, efficient and can assure reliability of

equipment, and safety of patients and operators.

In spite of the fact that reliability engineering techniques and maintenance optimization

models are well-established and used in other industries, their application to the medical field is

fairly new. Most research mainly proposes methods to assess or improve the reliability of

devices in their design or manufacturing phases. However, improving the management of

devices in their operating context has not been much considered. In the research reported in this

dissertation, we propose methods to improve current maintenance strategies in the healthcare

industry. We looked into the challenges that clinical engineers are currently facing, in particular,

for maintenance management of medical equipment. We attempted to propose some models and

methods that can be practically used. We constructed the models after taking into

6.1. Conclusion 179

account the actual procedures used for maintaining medical devices in hospitals, and considering

the evidences in maintenance data.

(a) Prioritization of medical devices- We presented a multi-criteria decision-making

model for prioritization of medical devices according to their criticality. The model can be

integrated into the CMMS of a hospital to prioritize medical devices. Devices with lower

criticality scores are excluded from the hospital’s MEMP. Devices with higher scores should be

investigated in detail to find the main causes of having such high criticality scores. Appropriate

actions, such as “preventive maintenance”, “user training”, “redesigning the device”, etc. should

be taken to reduce the risk.

Technically, Analytical Hierarchy Process was used to identify and decide on inclusion

of critical devices in the hospital’s equipment management program. The proposed model takes

six criteria, “Function”, “Mission criticality”, “Age”, “Risk”, “Recalls and hazard alerts”, and

“Maintenance requirements”, into account to evaluate the criticality of devices. The model

realistically estimates the total risk of a device by considering its individual failure modes. A risk

score is assigned to each failure mode after evaluating its frequency, detectability and

consequences. Pairwise comparison of the criteria was performed, and the consistency ratio was

calculated to produce more precise and consistent criteria’s weights. The relative measurement

method was applied to find more consistent values for the intensities of the criteria’s grades.

Moreover, some general guidelines were presented to select an appropriate maintenance

strategy for a device according to its individual score with respect to a criterion or a combination

of several criteria.

The proposed prioritization model is comprehensive and incorporates all important

criteria to assess the criticality of a device. Moreover, it is more consistent in terms of criteria’

6.1. Conclusion 180

weights and the intensities of the criteria’s grades. Acceptable and consistent results were

obtained from applying the model to 26 different medical devices.

On the other hand, the model requires involvement of experts in the process of applying

the model to medical devices, which might be expert intensive. In addition, the prioritization

results may not always be acceptable by clinical engineers and requires reassigning the criteria’s

weights and/or grades’ intensities.

(b) Field maintenance data- Then, we investigated the maintenance data from a hospital.

We categorized the failure types into hard and soft failures, and proposed policies to deal with

the censored and missing data at different levels of a device. Hard failures are self-announcing or

they make the system stop functioning. They are rectified and fixed immediately. However, the

system can still work in presence of soft failures, even though its performance may be reduced.

Soft failures are only rectified at inspections, so their failure times are not known. We applied the

proposed policies to general infusion pumps, and conducted reliability and trend analysis of the

data. For the trend analysis, midpoints of the censoring intervals for failures were calculated and

the Laplace trend analysis was applied. The results revealed that it is not uncommon to have

decreasing or increasing trend in the failure process of devices, mainly due to aging of the

mechanical parts of the device. This conclusion opposes the common belief of healthcare

societies that medical devices fail independent of age, following exponential distribution.

Obviously, more precise and reliable results can be obtained when more precise failure data is

available. Medical devices tend to be highly reliable; therefore, scarce failure data is always a

problem in statistical analysis.

Our approach in using the midpoints of censoring intervals may not always provide

precise results, particularly, when censoring intervals are long. We opted to use it because no

6.1. Conclusion 181

available trend test analysis in the literature incorporates interval censored data if repairs are not

instantaneous.

(c) Trend test- We then proposed a method for a system whose failure process follows a

non-homogeneous Poisson process (NHPP) with a power law intensity function. The assumption

of NHPP is common for the failure process of complex systems such as medical equipment. We

used the likelihood ratio test to check for trends in the failure data with censoring. The EM

algorithm and one of its modifications were used to find the parameters, which maximize the

data likelihood in the case of no trend and trend hypotheses. We developed a recursive procedure

to calculate the expected values in the E-step requiring numerical integration. The proposed

method was applied for several components of general infusion pump. We compared the results

of the EM algorithm, its modification, and the ad-hoc method of replacing unknown failure times

with the mid points of censoring intervals. We concluded that the parameters estimates obtained

from the complete EM and its modified version are very close. Moreover, these parameters

might be close to what is obtained from the mid-points method, if the censoring intervals are

short, but not always. At least, the parameters obtained from the mid-points method can be used

as initial values in the EM algorithm.

(d) Inspection interval- Moreover, we developed several optimization models to find the

optimal periodic inspection interval for a system with hard and soft failures. We assumed that the

system undergoes periodic inspections due to its simple implementation, especially for a large

number of devices in a hospital. We first assumed that the soft failures can only be rectified at

periodic (scheduled) inspections. Several models were constructed under this assumption. Then,

we considered also the possibility of opportunistic inspection of components with soft failure, at

non-scheduled inspections arising from the occurrence of hard failures. In addition, we extended

6.1. Conclusion 182

the models to the case when the components with hard failure are preventively replaced at

periodic inspections according to some age dependent probabilities.

In all models, the optimal inspection interval is found by minimizing the total expected

cost. We solved the main technical problem of calculating the excepted cost with delayed

replacement or minimal repair of a component at periodic or opportunistic inspections. For some

models, we recursively calculated the contributed variables in the cost model, such as expected

downtime, expected number of minimal repairs, etc. However, for more complicated models,

such as opportunistic maintenance models, simulation was used due to involved numerical

calculation. We obtained reliable results from both recursive calculation and simulation,

although the former might be more time consuming when optimization is performed for a system

with a large number of components, or for a longer life cycle with more periodic inspections.

The optimal inspection interval depends significantly on the input cost parameters, such

as inspection cost, repair cost, downtime penalty cost, etc. Changing one or some of the cost

parameters can result in a completely different optimal inspection interval.

(e) Remarks- As the final remarks in this section, we emphasize that the optimizing

maintenance strategies for medical equipment has not been much discussed in the literature.

Hospitals must assure that they are in compliance with the risk and maintenance standards for

medical equipment. Therefore, some hospitals include all medical devices in the MEMP, and just

follow manufacturers’ maintenance recommendations to avoid liability. In other words, there has

not been much tendency to explore new methods to improve the maintenance procedures in

hospitals, other than applying manufacturers’ guidelines. However, optimization techniques if

applied properly, not only can assure the safety of devices, but also can offer more efficient and

cost-effective maintenance procedures. The research in the area of maintenance of medical

6.2. Future Research 183

(f) devices, especially in their operating context is fairly new, which requires much

more work to be done in the future. In the next section, we discuss some new directions, which

can be considered to extend our current work.

6.2. Future Research

The models proposed in this research can be enriched by incorporating assumptions that

are more realistic or taking into account new considerations. Moreover, other techniques and

methods can be investigated and employed to improve the precision and calculation time of the

required variables in the current models.

(a) Prioritization of medical devices- In Section 3.2.7 of Chapter 3, we briefly discussed

that the score value assigned to a device with respect to a criterion, or a combination of several

criteria may be used to decide on appropriate maintenance strategy for the device. We presented

some initial heuristic guidelines in Table 3.7. However, more research is required to improve this

proposed multi-criteria methodology.

In addition, using Saaty’s crisp 1-9 scales, we constructed the pairwise comparison

matrices for the criteria and their intensities. Alternatively, fuzzy analytic hierarchy process

(Kwong and Bai, 2002) can be used, in which crisp values are substituted with fuzzy numbers for

scales. Thus, expert judgment is performed more confidently in comparing the criteria, especially

when there is an uncertainty in assigning the scale values.

(b) Trend test- Furthermore, in the proposed model for trend analysis of the failure data

discussed in Section 4.4 of Chapter 4, other iterative methods such as Simplex (Lagarias et al.,

1998) can be employed instead of the EM algorithm. Simplex is a direct search method, which

does not require numerical or analytic gradients. In addition, Newton-Raphson (McLachlan and

6.2. Future Research 184

Krishnan, 1996) can be used in both E-step and M-step of the EM algorithm. In this case, the

convergence of the parameters is achieved with less number of iterations; on the other hand,

calculating more expected values is required.

Other methods, such as Monte-Carlo simulation can be also used to estimate the

expected values required in the E-step of the EM (Levine and Casella, 2001). In this case, no

recursive procedure is required. Research can be conducted to compare all these numerical

algorithms, in terms of precision, number of iterations, and convergence time. In addition, failure

histories can be simulated to evaluate the performance of these algorithms for a larger number of

histories per unit.

(c) Inspection interval- Finally, the inspection optimization models proposed in Chapter 5

can be extended to consider the possibility of non-periodic inspections, and condition-based

inspections. Furthermore, the current models do not optimize any maintenance decision at

inspection for a component or the system. Maintenance decision can be also optimized in the

models to decide which components should be replaced at inspections. Another extension of the

current models is to assume that a soft failure or a combination of soft failures may eventually

convert to a hard failure, if left unattended. Moreover, different types of dependency can be also

considered for the system’s components which makes the optimization models much more

complicated, but more realistic. In this case, both maintained data and experts’ knowledge should

be used to incorporate the components’ structure. The CMMS of most hospitals keeps track of

the failures and maintenance actions at the component level, so this data can be used for

optimization models incorporating the components dependency.

185

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203

8. APPENDICES

APPENDIX A 204

APPENDIX A

Qualitative grades and intensities for criteria/sub-criteria

We explain how the intensities for grades of criterion “Function” can be obtained; the

intensities of other criteria’s grades are obtained using the same method.

Step 1. Pairwise comparison matrix of the grades is constructed using expert opinion ( ija for

1,...,5i , 1,...,5j ).

Life support Therapeutic Patient

Diagnostic

Analytical Miscellaneous

Life support 1.00 5.00 6.00 8.00 9.00

Therapeutic 0.20 1.00 1.60 1.40 1.80

Patient

Diagnostic 0.17 0.63 1.00 1.25 1.50

Analytical 0.13 0.71 0.80 1.00 1.29

Miscellaneous 0.11 0.56 0.67 0.78 1.00

Table A.1. Pairwise comparison matrix for the grades of criterion “Function”

Step 2. The weight of each grade can be obtained as follows

155

1

15 55

11

( ) , 1,...,5, 1,...,5.

( )

ijj

i

ijj

i

av i j

a

Step 3. The intensity of each grade can be obtained as follows

Intensity , 1,...,5.max( )

i

i

vi

v

APPENDIX A 205

5

1ij

ja

15

5

1( )ij

ja

15

5

1

155

5

11

( )

( )

ijj

i

ijj

i

a

v

a

Intensity

max( )

i

i

v

v

Life support 2160.00 4.64 0.62 1

Therapeutic 0.81 0.96 0.13 0.21

Patient

Diagnostic 0.20 0.72 0.1 0.16

Analytical 0.09 0.62 0.08 0.13

Miscellaneous 0.03 0.50 0.07 0.11

5

1i

= 7.45

Table A.2. Calculating intensities for the grades of criterion “Function”

Therefore,

Function

Grade Intensity

Life support 1.00

Therapeutic 0.21

Patient Diagnostic 0.16

Analytical 0.13

Miscellaneous 0.11

Table A.3. Function grades and intensities

The same method is employed for calculating the intensities of other criteria.

APPENDIX A 206

Utilization

Grade Description Intensity

High Usage hours per week 24 1.00

Medium 12 Usage hours per week < 24 0.34

Low 0 Usage hours per week < 12 0.15

Table A.4. Utilization grades and intensities

Availability of alternative devices


Low No of available alternatives 1 1.00

Medium 1 No of available alternatives 4 0.34

High No of available alternatives > 4 0.20

Table A.5. Availability of alternatives grades and intensities

Age


Old Actual life/Life span > 1 1.00

Almost old 0.75 Actual life/Life span 1 0.67

Average 0.5 Actual life/Life span 0.75 0.43

Almost new 0.25 Actual life/Life span 0.5 0.17

New 0 Actual life/Life span 0.25 0.12

Table A.6. Age grades and intensities

Failure frequency


Frequent Likely to occur (several occurrences in 1 year) 1.00

Occasional Probably will occur (several occurrences in 1-2 years) 0.33

Uncommon Possible to occur (one occurrence in 2-5 years) 0.20

Remote Unlikely to occur (one occurrence in 5-30 year) 0.15

Table A.7. Failure frequency grades and intensities

APPENDIX A 207

Failure detectability


Very low Not detected by regular inspection 1.00

Low Detected by inspection 0.33

Moderate Visible by naked eye 0.20

High Self-announcing 0.13

Table A.8. Failure detectability grades and intensities

Table A.9. Downtime grades and intensities

Cost of repair


High Total repair cost 2500 $ 1.00

Medium 500 $ Total repair cost < 2500 $ 0.22

Low 0 $ Total repair cost < 500 $ 0.17

Table A.10. Cost of repair grades and intensities

Safety and environment

Grade Intensity

Death 1.00

Injury 0.34

Inappropriate therapy or misdiagnosis 0.21

Delayed Treatment 0.14

No consequence 0.09

Table A.11. Safety and environment grades and intensities

Downtime


High Total waiting time 72 hrs per day 1.00

Medium 24 Total waiting time < 72 hrs per day 0.25

Low Total waiting time < 24 hrs per day 0.14

APPENDIX A 208

Recalls and hazard alerts


High Total number of class I or II recalls (in a year) >= 1 or Total number of Hazard

Alerts (in a year) >= 4 1.00

Medium 2 <= Total number of Hazard Alerts (in a year) < 4 0.21

Low Total number of Hazard Alerts (in a year) < 2 0.12

Null Total number of Hazard Alerts (in a year) = 0 0.00

Table A.12. Recalls and hazards grades and intensities

Maintenance requirements


High Equipment that is predominantly mechanical, pneumatic or fluidics

in nature often requires the most extensive maintenance. 1.00

Medium A device is considered to have average maintenance requirements

if it needs only performance verification and safety testing. 0.50

Low

Equipment that receives only visual inspection, a basic

performance check, and safety testing is classified as having

minimal maintenance requirements

0.17

Table A.13. Maintenance requirements grades and intensities

APPENDIX B 209

APPENDIX B

Calculating the lower bound for the total score value

Table B.1 shows how a score value of 0.1050 is obtained when a device gets the lowest

intensity value with respect to all assessment criteria.

Main Criteria (Weight)

Assigned/calculated score

Sub-Criteria

(Weight)

Assigned/calculated

score

Sub-Criteria

(Weight)

Assigned/calculated

score

Sub-Criteria (Weight)

Assigned/calculated

score

c1-Function

(0.45)

0.11

c2-Mission criticality

(0.10)

0.165

= 0.7*0.15

+0.3*0.2

c21-Utilization

(0.70)

0.15

c22 -Availability of

alternatives devices

(0.30)

0.2

c3-Age

(0.06)

0.12

c4-Total risk

(0.16)

0.124224

=0.30*0.15

+0.24*0.13

+0.46*0.1044

c41-Failure frequency

(0.30)

0.15

c42-Detectability

(0.24)

0.13

c43-Failure c431-Operational c4311- Downtime (1.00)

APPENDIX B 210

consequence (0.46)

0.1044

=(0.16*0.14)

+(0.08*0.17)

+(0.76*0.09)

(0.16)

0.14=1*0.14

0.14

c432-Non-operational

(0.08)

0.17=1*0.17

c4321-Cost of repair (1.00)

0.17

c433-Safety and

environment

(0.76)

0.09

c5-Recalls and hazard alerts

(0.16)

0

c6-Maintenance requirement

(0.07)

0.17

Table B.1. Calculating the minimum total score value

Thus, the minimum total score value is

(0.45*0.11)+(0.1*0.165)+(0.06*0.12)+(0.16*0.124224)+(0.16*0)+(0.07*0.17)=0.1050

APPENDIX C 211

APPENDIX C

Soft and hard failures


Component P F N/A Total P

%

F

%

Accessories 48 0 1154 1202 100 0

Cables 51 0 1151 1202 100 0

Casters/Brakes 49 0 1153 1202 100 0

Flow-Stop Mechanism 1202 0 0 1202 100 0

Lockout Interval (PCAs Only) 48 0 1154 1202 100 0


P F N/A Total P

%

F

%

35 0 1629 1664 100 0

38 0 1626 1664 100 0

33 0 1631 1664 100 0

1519 0 145 1664 100 0

34 0 1630 1664 100 0

Table C.1. Components/features that can be ignored since they have no failure reported



%

F

%

AC

Plug/Receptacles 1189 13 0 1202 98.92 1.08

Alarms 881 321 0 1202 73.29 26.71

Circuit

Breaker/Fuse 48 13 1141 1202 78.69 21.31

Controls/Switches 1097 105 0 1202 91.26 8.74

Indicators/Displays 1094 108 0 1202 91.01 8.99

Mount 1009 191 2 1202 84.08 15.92

Occlusion Alarm 1191 9 2 1202 99.25 0.75


P F N/A Total P

%

F

%

*Ratio

1510 9 145 1664 99.41 0.59 1.83

1268 251 145 1664 83.48 16.52 1.62

33 3 1628 1664 91.67 8.33 2.56

1433 86 145 1664 94.34 5.66 1.54

1383 136 145 1664 91.05 8.95 1

1319 200 145 1664 86.83 13.17 1.21

1507 12 145 1664 99.21 0.79 0.95

Table C.2. Components/features that have more failure at non-scheduled work orders

(hard failures)

APPENDIX C 212



%

F

%

Audible Signals 1105 97 0 1202 91.93 8.07

Battery/Charger 1177 25 0 1202 97.92 2.08

Chassis/Housing 700 502 0 1202 58.24 41.76

Fittings/Connectors 962 240 0 1202 80.03 19.97

Labeling 1090 112 0 1202 90.68 9.32

Strain Reliefs 1182 18 2 1202 98.5 1.5

Line Cord 1067 135 0 1202 88.77 11.23

Grounding

Resistance 1199 2 1 1202 99.83 0.17


P F N/A Total P

%

F

%

*Ratio

1250 269 145 1664 82.29 17.71 0.46

1321 198 145 1664 86.97 13.03 0.16

620 899 145 1664 40.82 59.18 0.71

1073 446 145 1664 70.64 29.36 0.68

1063 455 146 1664 70.03 29.97 0.31

1473 41 150 1664 97.29 2.71 0.55

1117 402 145 1664 73.54 26.46 0.42

1517 2 145 1664 99.87 0.13 1.31

Table C.3. Components/features that have more failures at scheduled work orders (soft failures)

*Ratio = No of failures identified at NS/ No of failures identified at PM

We included “Occlusion Alarm” in the hard group because of its similarity to the

“Alarms” component in the qualitative test. “Grounding Resistance” has only a few failures in

both scheduled and non-scheduled work orders, so we included it in the soft group.

APPENDIX D 213

APPENDIX D

Rules for analyzing failure data at the system level

We consider all non-scheduled work order starting times as system exact failure times

(complete data) since we assume that a non-scheduled work order is requested upon a hard

failure.

The time interval between two consecutive failures ending in a SPI with at least one failure

(scheduled work order) is treated as left censored data since we consider this detected failure

as a soft failure.

The time interval between an inspection time with “no problem found” and a SPI with at

least one failure is treated as interval censored data.

If the last available history ends in a failure and the time interval between this failure and

the end of the test is short (<=0.5*mean time between failures), we consider the time

interval as right censored data.

If the last available history is a failure and the time interval between this failure and the end

of the test is long (>0.5*mean time between failures), we consider 0.5*mean time between

failures as right censored data.

If the last available history is an inspection with “no failure found” and the time interval

between this inspection and the end of the test is short (<= 0.5*mean time between failures),

we consider the time interval between the last failure event and the end of the test as right

censored data.

APPENDIX D 214

If the last available history is an inspection with “no failure found” and the time interval

between this inspection and the end of the test is long (> 0.5*mean time between failures),

we consider the time interval between the last failure event + 0.5*mean time between

failures as right censored data.

APPENDIX E 215

APPENDIX E

Confidence limits estimation for (the Delta method)

The “Delta method” (Meeker and Escobar, 1998) can be applied to estimate ))((Var

g ,

if )(

g is a smooth nonlinear function of i

and can be approximated by a linear function of the

i

values in the region with no negligible likelihood. The Delta method is applied based on a

first order Taylor series expansion of )(

g about )](),...,([γ 1 rEE

.

If we apply the Delta method to the as a function of and , and assuming low

correlation between and , we obtain:

22

2 2 21 1ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ[ ( , )] ( ) ( ) ( ) ( ) (1 1/ ) ( ) (1 ) (1 ) /ˆ ˆ

Var Var Var Var

,

whered(log( ( )) d( ( ))/d

( ) digamma( )d ( )

x x xx x

x x

.

The )ˆ(Var and )ˆ(Var can be obtained from standard error estimates of SAS output.

APPENDIX F 216

APPENDIX F

Analysis of failure data

F.1. Hard failures

The results of the Weibull distribution fitted to the times between n-1th and nth event (failure/censoring)

n No of

observations

No of

non-

censored

No of

right

censors

No of

left

censors

No of

interval

censors

1ˆˆ ( 1)

ˆ

1 674 502 172 0 0 1.2982 1215.687 1123.1

2 502 317 185 0 0 1.1256 1077.554 1032.1

3 317 185 132 0 0 1.1007 924.7741 892.1

4 185 91 94 0 0 0.9864 768.3577 772.9

5 91 46 45 0 0 0.8626 701.4417 756.3

6 121 76 45 0 0 0.7785 310.3867 358.8

Table F.1. Estimated parameters of Weibull distribution for times between events – hard failures

LA=6.12 (systems are degrading)

days

S

a

m

p

l

e

M

C

F

0 500 1000 1500 2000 2500 3000

- 1

0

1

2

3

No. of Uni t s 674

No. of Event s 1217

Conf . Coef f . 95%

Figure F.1. The MCF plot of hard failures

APPENDIX F 217

F.2. Soft failures


n No of

observations

No of

non-

censored

No of

right

censors

No of

left

censors

No of

interval

censors

1ˆˆ ( 1)

ˆ

1 674 0 30 463 181 1.1162 350.7222 336.8

2 644 0 122 418 104 1.2525 254.8389 237.2

3 523 0 116 335 72 1.0424 247.6217 243.5

4 404 0 158 217 29 1.0465 303.4680 298

5 247 0 124 108 15 0.8027 409.1623 462.5

6 123 0 76 40 7 0.5665 701.3626 1142.3

7 82 0 46 26 10 0.7959 351.3244 399.5

Table F.2. Estimated parameters of Weibull distribution for times between events – soft

failures LA=11.08 (systems are degrading)

days

S

a

m

p

l

e

M

C

F

0 500 1000 1500 2000 2500

- 2

0

2

4

6

8

No. of Uni t s 674

No. of Event s 2027

Conf . Coef f . 95%

Figure F.2. The MCF plot of soft failures

APPENDIX F 218

F.3. Alarm failures


n No of

observations

No of

non-

censored

No of

right

censors

No of

left

censors

No of

interval

censors

1ˆˆ ( 1)

ˆ

1 674 306 368 0 0 1.1876 2564.701 2419.1

2 306 140 166 0 0 0.9368 2118.686 2183.1

3 140 64 76 0 0 0.8782 1353.484 1443.9

4 64 24 40 0 0 0.6894 1113.709 1430.8

5 61 38 23 0 0 0.6548 255.7360 346.6

Table F.3. Estimated parameters of Weibull distribution for times between events – Alarms

failures (hard) LA=4.13 (systems are degrading)

F.4. Chassis/Housing failures


n No of

observations

No of

non-

censored

No of

right

censors

No of

left

censors

No of

interval

censors

1ˆˆ ( 1)

ˆ

1 674 0 101 317 256 1.2662 645.6414 599.6

2 573 0 146 251 176 1.1804 497.9293 470.4

3 428 0 173 178 77 1.2267 477.8781 447.1

4 252 0 146 79 27 1.5471 473.8461 426.3

5 107 0 75 24 8 1.3434 597.5264 548.4

6 40 0 32 7 1 0.7905 1395.492 1594.9

Table F.4. Estimated parameters of Weibull distribution for times between events –

Chassis/Housing failures (soft) LA=11.40 (systems are degrading)

APPENDIX G 219

APPENDIX G

Proof of the recursive equation (4.11)

For the general case and the special case of ( ) 1n nC y 0 0y is considered. We first

want to calculate

11

1 1 1 1 1...

( , ,..., ) ... ( | )... ( ) ...n

n

n n Y n n Y nA A

P A A A f y y f y dy dy .

Let

1 1 1 1( ) ( ) ( | ) ( | )n n

n n

n n n n Y n n n Y n n n

A A

C y C y f y y dy f y y dy

Therefore, 1 1 1( | ) ( )n n n nP A y C y , and

1

1

1 2 1 1 2 1( , | ) ( | ) ( | )n n

n n

n n n Y n n Y n n n n

A A

P A A y f y y f y y dy dy

1

1

1 2 1 1( | )[ ( | ) ]n n

n n

Y n n Y n n n n

A A

f y y f y y dy dy

1

1

1 2 1 1 1 2 2( | ) ( ) ( )n

n

Y n n n n n n n

A

f y y C y dy C y

,

or

1 1 1 1 1( ) ( ) ( | ) ( ) ( | )i

i i

i i

u

i i i i Y i i i i i X i

A l

C y C y f y y dy C y x f x y dx ,

, 1,...,1i n n . Then 1 1 0( , ,..., ) (0)n nP A A A C .

In general, for a given function ( )g x over a set A ,

APPENDIX G 220

( ) ( )( ) ( ) ( )[ ( ) ( )]

[ ( ) | ]( ) ( ) ( )

A

A A

g x f x dxg x I x A f x dxE g X I A

E g X X AP X A f x dx f x dx

.

Therefore,

( )

( )

1 1

1 1

1

[ ( ) ( , ,..., )][ ( ) | , ,..., ]

( ,..., )

t

t

n n n

n n n

n

E g Y I A A AE g Y A A A

P A A

11

11

1 1 1...

1 1 1...

... ( ) ( | )... ( ) ...

... ( | )... ( ) ...

nn

nn

n Y n n Y nA A

Y n n Y nA A

g y f y y f y dy dy

f y y f y dy dy

.

Let in general ( ) ( )n n nB y g y . Similar to the previous proof, we have

( ) 1 1

1 1 1

[ ( ) ( ) | ] ( ) ( | )

( ) ( | ) ( ) ,

tn

n

n

n

n n n n Y n n n

A

n n Y n n n n n

A

E g Y I A y g y f y y dy

B y f y y dy B y

( )

1

1

1

1

1

1

1 2

1 1 2 1

1 2 1 1

1 2 1 1 1 2 2

[ ( ) ( , ) | ]

( ) ( | ) ( | )

( | )[ ( ) ( | ) ]

( | ) ( ) ( )

t

n n

n n

n n

n n

n

n

n n n n

n Y n n Y n n n n

A A

Y n n n Y n n n n

A A

Y n n n n n n n

A

E g Y I A A y

g y f y y f y y dy dy

f y y g y f y y dy dy

f y y B y dy B y

Thus, in general,

1 1 1 1( ) ( ) ( | )i

i

i

u

i i i i X i

l

B y B y x f x y dx

Therefore,

( ) 1 1 0 0[ ( ) | , ,..., ] (0) / (0)t n n nE g Y A A A B C .

APPENDIX H 221

APPENDIX H

Proof of the recursive equation (4.12)

( )1

1

1 1 1 1 1...1 1

[( ln ) ( , ,..., )] ... ( ln ) ( | )... ( ) ... .tn

n

n n

i n n i Y n n Y nA Ai i

E Y I A A A y f y y f y dy dy

Let ( ) 0n nD y , ( ) 1n nC y , and 1 1 1( ) ln ( | )n

n

n n n Y n n n

A

D y y f y y dy . Then

1

1

2 2 1 1 1 2 1( ) [ln ln ] ( | ) ( | )n n

n n

n n n n Y n n Y n n n n

A A

D y y y f y y f y y dy dy

1

1

1 1 1 1 2 1[ln ( | ) ln ( | ) ] ( | )n n n

n n n

n Y n n n n Y n n n Y n n n

A A A

y f y y dy y f y y dy f y y dy

1

1

1 1 1 1 1 1 2 1[ln ( ) ( )] ( | )n

n

n n n n n Y n n n

A

y C y D y f y y dy

Thus, in general,

1 1 1 1 1 1( ) [ln( ) ( ) ( )] ( | )i

i

i

u

i i i i i i i X i

l

D y y x C y x D y x f x y dx .

Therefore,

( ) 1 0 0 0

1

[ ln | , ,..., ] (0) / (0)t

n

i n n

i

E Y A A A D C

.

APPENDIX I 222

APPENDIX I

Proof of recursive equations (5.17) and (5.18)

From the assumptions of NHPP and minimal repair or replacement, we have

1 1 1 0 1 1 0

0

1 1 0 1 1

1 1

0

( ) [ ( , ) | ] [ ( ,0) Pr{ 0 | , }

( ,1) Pr{ 1| , }] ( | ) ( , 2) ( )

[ ( ,0) ( ) ( ,1) ( )] ( | ) ( , 2) ( ) .

t E X I X t x I X x X t

x I X x X t f x t dx P t

x r t x x r t x f x t dx P t

Also, for 1,2k ,...,n ,

0 1 0

1 0 1

0

1 0 1 1 0 1 0

0

1 1 0 1 1 1

0

( ) [ ( , ) | ] { [ ( , ) | ] | }

[ ( , ) | , ] ( | )

[ ( , ) | , ] ( | ) [ ( , ) | , ]Pr{ | }

{ [ ( , ) | ] | , } ( | ) ( ) (

k k k k k

k k

k k k k

k k k

t E X I X t E E X I X X t

E X I X x X t dF x t

E X I X x X t f x t dx E X I X X t X X t

E E X I I X x X t f x t dx t P

1 1 0 1 1 0

0

1 1 0 1 1 0 1 1 1

1 1 1 1 1

0

1 1

)

{ [ ( , ) | 0, , ]Pr{ 0 | , }

[ ( , ) | 1, , ]Pr{ 1| , }} ( | ) ( ) ( )

[ ( ) ( ) (0) ( )] ( | ) ( ) ( )

( ) ( ) (

k k

k k k

k k k

k

t

E X I I X x X t I X x X t

E X I I X x X t I X x X t f x t dx t P t

t x r t x r t x f x t dx t P t

t x r t x f x

1 1 1 1

0

| ) (0) ( ) ( ) ( ). k kt dx g t t P t

APPENDIX J 223

APPENDIX J


Following the same argument as in Appendix I, except that

1 1 1 1 0[ ( , ) ( ) | 1, , ] ( ,1)E X I T I X x X t x ,

we find that

1 1 1 0 1 1

0

( ) [ ( , ) | ] [ ( ,0) ( ) ( ,1) ( )] ( | ) ( ,2) ( )t E X I X t x r t x x r t x f x t dx P t

.

Also, for 1,2k ,...,n ,

0 0 1

0 1 1

0

0 1 1

0

0 1 1 0

( ) [ ( , ) ( )| ] { [ ( , ) ( )| , ]}

[ ( , ) ( )| , ] ( | )

[ ( , ) ( )| , ] ( | )

[ ( , ) ( )| , ]Pr{ | }

{ [ ( , )

k k k k k

k k

k k

k k

k k

t E X I I T k X t E E X I I T k X t X

E X I I T k X t X x dF x t

E X I I T k X t X x f x t dx

E X I I T k X t X X X t

E E X I

1 0 1 1 1 1

0

1 1 0 1 1 0

0

1 1 0 1 1 0 1 1 1

1

( ) | ]| , } ( | ) ( ) ( )

{ [ ( , ) ( ) | 0, , ]Pr{ 0 | , }

[ ( , ) ( ) | 1, , ]Pr{ 1| , }} ( | ) ( ) ( )

( ) (

k

k k

k k k

k

I T k I X t X x f x t dx t P t

E X I I T k I X x X t I X x X t

E X I I T k I X x X t I X x X t f x t dx t P t

t x r t x

1 1 1

0

) ( | ) ( ) ( ), kf x t dx t P t

because 1 1 0[ ( , ) ( ) | 1, , ] 0k kE X I I T k I X x X t .

APPENDIX K 224

APPENDIX K

Upper bound for 1

( ) ( )k

k

V t V t

Lemma 1: Let ( ) Pr{ | }kV t T k t and 2 20

( ) sup ( )x

V t V t x

. If 2 ( ) 1V t , 0t then also

( ) 1kV t , 3,4,...k , and 2 2

1 1( )

1 ( ) 1 (0)V t

V t V

.

Proof:

We will show that 1

2 1 2( ) ( ) ( ) [ ( )]k

k kV t V t V t V t

for 2k . We have 1( ) Pr{ | } 1V t T t

and obviously for 1,2k we have 1

2( ) [ ( )]k

kV t V t and for 2k , 2 2 1 2( ) ( ) ( ) ( )V t V t V t V t .

Let iX be the survival time on (( 1) , ], 1,2,...i i i , and 1

, ( | ) Pr{ | }k

k i k k

i

S X F x t S x t

.

Obviously kS k . Then for 3k

( 2)

2 2 2

0

( 2)

2 2 2

0

2 2

1

2 1 2

( ) [Pr{ ( 2) | , } ( ) ( | )

( ) Pr{ ( 2) | , } ( | )

( ) Pr{ ( 2) | } ( ) Pr{ ( 1) | }

( ) ( ) [ ( )] ,

k

k k k

k

k k

k

k

V t T k S x t V t x dF x t

V t T k S x t dF x t

V t T k t V t T k t

V t V t V t

and the result follows.

Lemma 2: If ( ) 1r x for 0x and nonincreasing for 0x , and ( ) 0x for 0x and

nondecreasing for 0x , then 2 ( ) 1V t for 0t .

Proof:

APPENDIX K 225

We will prove that for 2 2 1

0

( ) 1 ( ) ( ) ( | )V t V t r t x f x t dx

, 2 2 2

0( ) inf ( ) 1 ( ) 0

xV t V t x V t

.

From our assumptions, ( ) 1 ( )r u r u and 1( ) 1 exp[ ( ) ] 0

u

u

P u x dx

are nondecreasing

and positive functions for 0u , so that for 0t

2 1 1 1 1

0 0

( ) ( ) ( | ) ( ) ( | ) ( ) ( ) ( ) ( ) 0V t x r t x y f y t x dy r t f y t x dy r t P t x r t P t

.

Then 2 2 1

0( ) inf ( ) ( ) ( ) 0

xV t V t x r t P t

for 0t , but it is not clear whether 2 (0) 0V because

(0)r may not be positive. On the other hand, for any fixed 0 0x and 00 x x , and 0t ,

0

0

( )( )

2 0

0 0

( ) ( ) ( )e ( ) ( )e ( , ) 0

t x yt x y

t xt x

s dss ds

V t x r t x y t x y dy r t y t y dy g t x

.

Then

0 0

2 2 2 2 0 0 1 00 0

( ) inf ( ) min{ inf ( ), inf ( )} min{ ( , ), ( ) ( )} 0x x x x x

V t V t x V t x V t x g t x r t x P t x

for 0t .

Also, by using inequality 2 1 2( ) ( ) ( ) ( )[ ( )]k

n k n k nV t V t V t V t V t

, we have

22

1 1 1 1 1 2

( )( ) ( ) ( ) ( ) [ ( )] ( ) ( ) .

1 ( )

n n nk n

k k k n k n

k k k k n k

V tV t V t V t V t V t V t V t

V t

So, 12 2 22

1 1 2 2 2

( ) ( ) [ ( )]0 ( ) ( ) ( ) [ ( )]

1 ( ) 1 ( ) 1 ( )

nnn

k k n

k k

V t V t V tV t V t V t V t

V t V t V t

.

APPENDIX K 226

Therefore, criterion 2

2

( )( )

1 ( )n

V tV t

V t

can be used to decide about terminating calculation of

1

( )k

k

V t

.

Note: If ( )x , then ( )

1 1( ) exp{- lim ( )} 0,

t

t

s ds

xP t e P x t

Note also that 2 2 1

0

( ) ( ; ) ( ) ( | )V t V t r t x f x t dx

is a nondecreasing function in , as a function

of the upper limit of an integral of a nonnegative function. Then

2 1

0

lim ( ; ) ( ) ( | )V t r t x f x t dx

,

20

lim ( ; ) 0V t

.

Lemma 3: Let the placeholder function ( , )k kX I be bounded on finite intervals, that is,

| ( , ) |x i A for 0 x and all i . Under the same conditions as in Lemma 2,

1

( ) ( )k

k

G t t

converges.

Proof:

Obviously, from 0

1

( ) [ ( , ) ( ) | ]k k

k

G t E X I T k X t

,

0 0

1 1

| ( ) | {[| ( , ) | ( )] | } [ ( ) | ] ( )k k

k k

G t E X I T k X t A E T k X t A V t

.

For calculation of ( )G t we may use that

0| ( ) | {[| ( , ) | ( )]| } ( )k k k kt E X I T k X t A V t ,

so that

APPENDIX K 227

2 2

1 1 1 2 2

( ) [ ( )]| ( ) ( ) | ( ) ( )

1 ( ) 1 ( )

nn

k k k n

k k k n

V t V tt t A V t AV t A

V t V t

.

A somewhat better approximation can be obtained following the proof of Lemma 2. Let

1 10

( ) sup | ( ) |x

t t x A

. Then

0 1 0

1 0 1 0

1 0 1 1

( ) [ ( , ) ( )| ] [ ( , ) ( ) ( ( 1) )| , ]

[Pr{ ( 1) | , } [ ( , ) ( ) | ( 1) , , ]]

[Pr{ ( 1) | , } ( )] .

k k k k k k

k k k k

k k

t E X I T k X t E X I T k T k S X t

E T k S X t E X I T k T k S X t

E T k S X t t S

Then

1 0 1 1 1 1 0

1 0 2

| ( ) | [Pr{ ( 1) | , } | ( ) |] ( ) [Pr{ ( 1) | , }]

( )Pr{ | } ( ) ( ),

k k k k

k

t E T k S X t t S t E T k S X t

t T k X t t V t

and

2 21 1 1

1 1 1 2 2

( ) [ ( )]| ( ) ( ) | ( ) ( ) ( ) ( ) ( )

1 ( ) 1 ( )

nn

k k k n

k k k n

V t V tt t t V t t V t t

V t V t

.

Alternatively, if 2 20

( ) sup | ( ) |x

t t x

, we also can prove that 2 1| ( ) | ( ) ( ),k kt t V t

and then

1

22 1 2 2

1 1 1 2 2

1 [ ( )]| ( ) ( ) | ( ) ( ) ( ) ( ) ( )

1 ( ) 1 ( )

nn

k k k n

k k k n

V tt t t V t t V t t

V t V t

,

which gives a better error bound if 2 1 2( ) ( ) ( )t t V t .

APPENDIX L 228

APPENDIX L


1 0 0 1 0 0( , , ) [ | , , ] { [ | ] | , , }G t s E V X t Z s U E E V Z X t Z s U

1 0 0 1

0

[ | , , , ] ( | )ZE V Z z X t Z s U dF z s

1 0 0 1

0

[ | , , , ] ( | )ZE V Z z X t Z s U f z s dz

1 0 0 1 0[ | , , , ] ( | )E V Z X t Z s U P Z Z s

1 1 0 0 1 1

0 0

[ | , , , , ] ( | , ) ( | )

z

X ZE V X x Z z X t Z s U dF x s z f z s dz

1 1 0 0 1 1

0

[ | , , , , ] ( | , ) ( | )X ZE V X x Z X t Z s U dF x s z P s

0 0 1 1

0 0

{ ( , ) | , , } ( | ) ( | )

z

X Zx z E V X t x Z s z U z f x t f z s dxdz

0 0 1 1

0

{ ( , ) [ | , , ]} ( | ) ( | )X Zz z E V X t z Z s z U z P z t f z s dz

1 1 1 1 1

0

( | ) ( , ) ( | ) ( | ) ( | ) ( , )Z X Z XP s x f x t dx P s P t

1 1 1

0 0

{ ( , ) ( , , )} ( | ) ( | )

z

X Zx z G z t x s z f x t f z s dxdz

1 1 1

0

{ ( , ) ( , , )} ( | ) ( | )Z Xz z G z t z s z f z s P z t dz

1 1 1 1

0

( | ) ( , ) ( | ) ( | ) ( | ) ( , )Z X Z XP s x f x t dx P s P t

.

So,

APPENDIX L 229

1 1 1 1

0 0

1 1 1

0

1 1 1 1

0

( , , ) { ( , ) ( , , )} ( | ) ( | )

{ ( , ) ( , , )} ( | ) ( | )

( | ) ( , ) ( | ) ( | ) ( | ) ( , )

z

X Z

Z X

Z X Z X

G t s x z G z t x s z f x t f z s dxdz

z z G z t z s z f z s P z t dz


.

Also, for 1,2k ,...,n ,

0 0 1 0 0( , , ) [ | , , ( 1) ] { [ | ] | , , ( 1) }kG t s E V X t Z s U k E E V Z X t Z s U k

1 0 0 1

0

[ | , , , ( 1) ] ( | )ZE V Z z X t Z s U k dF z s

1 0 0 1

0

[ | , , , ( 1) ] ( | )ZE V Z z X t Z s U k f z s dz

1 0 0 1 0[ | , , , ( 1) ] ( | )E V Z X t Z s U k P Z Z s

1 1 0 0 1 1

0 0

[ | , , , , ( 1) ] ( | , ) ( | )

z

X ZE V X x Z z X t Z s U k dF x s z f z s dz

1 1 0 0 1 1

0

[ | , , , , ( 1) ] ( | , ) ( | )X ZE V X x Z X t Z s U k dF x s z P s

0 0 1 1

0 0

{ ( , ) | , , ( 1) } ( | ) ( | )

z

X Zx z E V X t x Z s z U z k f x t f z s dxdz

0 0 1 1

0

{ ( , ) [ | , , ( 1) ]} ( | ) ( | )X Zz z E V X t z Z s z U z k P z t f z s dz

1 1 0 0 1

0

( | ) { ( , ) [ | , , ( 1) ]} ( | )Z XP s x E V X t x Z s U k f x t dx

1 1 0 0( | ) ( | ){ ( , ) [ | , , ( 1) ]}Z XP s P t E V X t Z s U k

1 1

0 0

{ ( , ) ( , , )} ( | ) ( | )

z

X Z

kx z G z t x s z f x t f z s dxdz

APPENDIX L 230

1 1

0

{ ( , ) ( , , )} ( | ) ( | )Z X


1 1 1

0

( | ) { ( , ) ( , , )} ( | )Z X

kP s x G t x s f x t dx

1 1 1( | ) ( | ){ ( , ) ( , , )}Z X

kP s P t G t s .

APPENDIX M 231

APPENDIX M

Simulation results for the case of opportunistic maintenance

Com

pon

ent

j

nM STDE j

ne STDE j

nM STDE j

ne STDE

1

1 4.20469 0.00175 10.55941 0.00077

5

3.71371 0.00152 9.57365 0.00136

2 2.63350 0.00149 11.08058 0.00066 2.43269 0.00134 10.30819 0.00131

3 3.49796 0.00155 10.87594 0.00064 3.14052 0.00138 10.28621 0.00105

4 1.30242 0.00107 11.58187 0.00044 1.25090 0.00101 11.29642 0.00082

5 5.76685 0.00185 10.06897 0.00079 4.87907 0.00161 9.08153 0.00125

2

1 3.93490 0.00161 10.02193 0.001074

6

3.65831 0.00150 9.46029 0.00142

2 2.52556 0.00141 10.67942 0.000978 2.41023 0.00133 10.22146 0.00137

3 3.27938 0.00144 10.52293 0.000867 3.08069 0.00136 10.18970 0.00110

4 1.27156 0.00104 11.42082 0.000637 1.24007 0.00100 11.24983 0.00087

5 5.23924 0.00168 9.499829 0.001036 4.76565 0.00158 8.95299 0.00130

3

1 3.80515 0.00155 9.76110 0.00123

7

3.65322 0.00150 9.44591 0.00143

2 2.47272 0.00137 10.46855 0.00116 2.40548 0.00133 10.20979 0.00139

3 3.18800 0.00140 10.36780 0.00097 3.08267 0.00136 10.19024 0.00111

4 1.25867 0.00102 11.34221 0.00074 1.23986 0.00100 11.24699 0.00088

5 5.01651 0.00163 9.24678 0.00115 4.76138 0.00158 8.94826 0.00130

4

1 3.73093 0.00153 9.61045 0.00132

8

3.64357 0.00150 9.43404 0.00144

2 2.44157 0.00135 10.34499 0.00126 2.39936 0.00132 10.19978 0.00140

3 3.13618 0.00138 10.28436 0.00103 3.08106 0.00136 10.18518 0.00112

4 1.25003 0.00101 11.29777 0.00080 1.24138 0.00100 11.24205 0.00089

5 4.89567 0.00160 9.10409 0.00122 4.74971 0.00158 8.93921 0.00131

APPENDIX M 232

9

1 3.64466 0.00150 9.42816 0.00145

11

3.63324 0.00149 9.40450 0.00146

2 2.40384 0.00133 10.19222 0.00141 2.39655 0.00132 10.17171 0.00142

3 3.08397 0.00136 10.18958 0.00113 3.07487 0.00136 10.17609 0.00114

4 1.23875 0.00100 11.24675 0.00090 1.23854 0.00100 11.23419 0.00091

5 4.74847 0.00159 8.93599 0.00132 4.73604 0.00159 8.91416 0.00133

10

1 3.63849 0.00150 9.41760 0.00145

12

3.59383 0.00147 9.32175 0.00149

2 2.39854 0.00132 10.18375 0.00141 2.38107 0.00131 10.11942 0.00145

3 3.07808 0.00136 10.18301 0.00113 3.02159 0.00133 10.08842 0.00117

4 1.23963 0.00100 11.24053 0.00090 1.23047 0.00099 11.19550 0.00094

5 4.74084 0.00158 8.92746 0.00132 4.63851 0.00156 8.80968 0.00135

Table M.1. Estimated expected number of minimal repairs and uptime and their standard

errors calculated for the components with soft failure, for 1,2,...,12 (used in Figure 5.14)

APPENDIX M 233

Com

pon

ent

j

nM STDE j

nR STDE j

ne STDE

1

1 1.73970 0.00126 1.72640 0.00081 10.80080 0.00068

2 1.33541 0.00113 1.16907 0.00079 11.12388 0.00064

3 1.05839 0.00101 1.16444 0.00057 11.26892 0.00050

4 0.62560 0.00078 0.50968 0.00056 11.63172 0.00039

5 1.76728 0.00126 1.90050 0.00074 10.75595 0.00064

2

1 1.72182 0.00122 1.58775 0.00077 10.29554 0.00101

2 1.32769 0.00110 1.09298 0.00075 10.72552 0.00096

3 1.05631 0.00099 1.11518 0.00055 10.97570 0.00074

4 0.62764 0.00077 0.49383 0.00055 11.48155 0.00059

5 1.75537 0.00122 1.75706 0.00070 10.25216 0.00094

3

1 1.71051 0.00119 1.52252 0.00075 10.03774 0.00118

2 1.32093 0.00108 1.04993 0.00074 10.51729 0.00114

3 1.06034 0.00099 1.08942 0.00055 10.83429 0.00086

4 0.62970 0.00077 0.48489 0.00055 11.40584 0.00069

5 1.75234 0.00120 1.68503 0.00069 10.00030 0.00110

4

1 1.70358 0.00117 1.48372 0.00075 9.88248 0.00129

2 1.31828 0.00107 1.02662 0.00073 10.39371 0.00125

3 1.06286 0.00098 1.07222 0.00055 10.75228 0.00094

4 0.63053 0.00077 0.47983 0.00055 11.36136 0.00075

5 1.74773 0.00118 1.64407 0.00069 9.85005 0.00120

APPENDIX M 234

5

1 1.70212 0.00117 1.47007 0.00075 9.83526 0.00133

2 1.31653 0.00107 1.01741 0.00073 10.35484 0.00130

3 1.06790 0.00098 1.06725 0.00055 10.73303 0.00097

4 0.63061 0.00077 0.47735 0.00055 11.35863 0.00078

5 1.74628 0.00118 1.62940 0.00069 9.80369 0.00124

6

1 1.69221 0.00116 1.44687 0.00074 9.73069 0.00139

2 1.31015 0.00106 1.00474 0.00073 10.27396 0.00136

3 1.06666 0.00098 1.05579 0.00055 10.65917 0.00103

4 0.63205 0.00077 0.47448 0.00054 11.31237 0.00083

5 1.73825 0.00117 1.60474 0.00068 9.69034 0.00130

7

1 1.68884 0.00116 1.44271 0.00074 9.71284 0.00141

2 1.30930 0.00106 1.00197 0.00073 10.25961 0.00137

3 1.06703 0.00098 1.05194 0.00054 10.65030 0.00104

4 0.63039 0.00077 0.47359 0.00054 11.30917 0.00084

5 1.73382 0.00117 1.59894 0.00069 9.66754 0.00131

8

1 1.68722 0.00116 1.43884 0.00074 9.69748 0.00142

2 1.30919 0.00106 0.99941 0.00073 10.24423 0.00139

3 1.06628 0.00097 1.04898 0.00054 10.63604 0.00105

4 0.62967 0.00077 0.47249 0.00054 11.30636 0.00085

5 1.72890 0.00117 1.59368 0.00068 9.65047 0.00132

9

1 1.68580 0.00116 1.43667 0.00074 9.69196 0.00142

2 1.31067 0.00106 0.99883 0.00073 10.24017 0.00140

3 1.06628 0.00098 1.04800 0.00054 10.63040 0.00106

4 0.63047 0.00076 0.47237 0.00054 11.30785 0.00086

5 1.73050 0.00117 1.59086 0.00069 9.64148 0.00133

APPENDIX M 235

10

1 1.68338 0.00115 1.43143 0.00074 9.67960 0.00143

2 1.30974 0.00105 0.99585 0.00073 10.22906 0.00140

3 1.06649 0.00098 1.04616 0.00054 10.61472 0.00107

4 0.63013 0.00076 0.47151 0.00054 11.30310 0.00086

5 1.73014 0.00116 1.58636 0.00069 9.62845 0.00133

11

1 1.68305 0.00115 1.42867 0.00074 9.66239 0.00144

2 1.30948 0.00105 0.99264 0.00072 10.21715 0.00141

3 1.06232 0.00097 1.04441 0.00054 10.60004 0.00108

4 0.63118 0.00076 0.46892 0.00054 11.29200 0.00088

5 1.72427 0.00116 1.58253 0.00068 9.60910 0.00134

12

1 1.67696 0.00115 1.41586 0.00073 9.59723 0.00146

2 1.30717 0.00105 0.98783 0.00072 10.16794 0.00143

3 1.06357 0.00097 1.03820 0.00054 10.54750 0.00111

4 0.62976 0.00076 0.46851 0.00054 11.25938 0.00090

5 1.72137 0.00116 1.56856 0.00068 9.54242 0.00137

Table M.2. Expected number of minimal repairs and replacements, and the uptimes and

their standard errors calculated for the components with soft failure, for 1,2,...,12

(used in Figures 5.15 and 5.16)

APPENDIX N 236

APPENDIX N


Following the same argument as in Appendix L:

1 1 0 0 1 0 0( , , ) ( , , ) [ | , , ] { [ | ] | , , }G t s G t s E V X t Z s U E E V Z X t Z s U

0 0 1 1 0

0 0

0 0 1 1 0 1 1

( , ,0) | , , ( 1) Pr{ 0 | , }

( , ,1) | 0, 0, ( 1) Pr{ 1| , } ( | ) ( | )

z

X Z

x z E V X t x Z s z U z k I X x X t

x z E V X Z U z k I X x X t f x t f z s dxdz

0 0 1 1

0

{ ( , ,2) [ | , , ( 1) ]} ( | ) ( | )X Zz z E V X t z Z s z U z k P z t f z s dz

1 0 0 1 1 0

0

0 0 1 1 0 1

( | ) ( , ,0) [ | , , ( 1) ] Pr{ 0 | , }

( , ,1) [ | 0, , ( 1) ] Pr{ 1| , } ( | )

Z

X

P s x E V X t x Z s U k I X x X t

x E V X Z s U k I X x X t f x t dx

1 1 0 0( | ) ( | ) ( , ,2) [ | , , ( 1) ]Z XP s P t E V X t Z s U k

1 1 1 1

0 0

1 1 1

0

1 1 1

0

{[ ( , ,0) ( , , )] ( ) [ ( , ,1) ( ,0, )] ( )} ( | ) ( | )

{ ( , , 2) ( , , )} ( | ) ( | )

( | ) [ ( , ,0) ( ) ( , ,1) ( )] ( | ) ( |

z

X Z

X Z

Z X Z

x z G z t x s z r t x x z G z s z r t x f x t f z s dxdz

z z G z t z s z P z t f z s dz

P s x r t x x r t x f x t dx P

1) ( | ) ( , , 2).Xs P t

Also, for 1,2k ,...,n ,

0 0 1 0 0( , , ) [ | , , ( 1) ] { [ | ] | , , ( 1) }kG t s E V X t Z s U k E E V Z X t Z s U k

0 0 1 1 0

0 0

0 0 1 1 0 1 1

( , ,0) | , , ( 1) Pr{ 0 | , }

( , ,1) | 0, , ( 1) Pr{ 1| , } ( | ) ( | )

z

X Z

x z E V X t x Z s z U z k I X x X t

x z E V X Z s z U z k I X x X t f x t f z s dxdz

0 0 1 1

0

{ ( , ) [ | , , ( 1) ]} ( | ) ( | )X Zz z E V X t z Z s z U z k P z t f z s dz

APPENDIX N 237

1 0 0 1 1 0

0

0 0 1 1 0 1

( | ) ( , ,0) [ | , , ( 1) ] Pr{ 0 | , }

( , ,1) [ | 0, , ( 1) ] Pr{ 1| , } ( | )

Z

X

P s x E V X t x Z s U k I X x X t

x E V X Z s U k I X x X t f x t dx

1 1 0 0( | ) ( | ){ ( , ) [ | , , ( 1) ]}Z XP s P t E V X t Z s U k

1 1

0 0

{[ ( , ,0) ( , , )] ( ) [ ( , ,1) ( ,0, )] ( )} ( | ) ( | )

z

X Z

k kx z G z t x s z r t x x z G z s z r t x f x t f z s dxdz

1 1

0

{ ( , ,2) ( , , )} ( | ) ( | )Z X


1 1 1 1

0

( | ) {[ ( , ,0) ( , , )] ( ) [ ( , ,1) ( ,0, )] ( )} ( | )Z X

k kP s x G t x s r t x x G s r t x f x t dx

1 1 1( | ) ( | ){ ( , , 2) ( , , )}Z X

kP s P t G t s .

APPENDIX O 238

APPENDIX O

Proof of recursive equation (5.37)

1 11 1 2 0 1 2( , , , ,..., ) [ | , ( , ,..., ), ]m mG t E V X t U

11 0 1 2{ [ | ] | , ( , ,..., ), }mE E V Z X t U

1 11 0 1 2 1 1 2

0

[ | , , ( , ,..., ), ] ( | ( , ,..., ))Z

m mE V Z z X t U dF z

1 11 0 1 2 1 1 2

0

[ [ | ] | , , ( , ,..., ), ] ( | ( , ,..., ))Z

m mE E V J Z z X t U dF z

1

1 11 0 1 2 1 1 2

10

[ | , , , ( , ,..., ), ] ( ) ( | ( , ,..., ))m

j Z

m m

j

E V Z z J j X t U q z dF z

1

1 11 0 1 2 1 1 2

10

[ | , , , ( , ,..., ), ] ( ) ( | ( , ,..., ))m

j Z

m m

j

E V Z z J j X t U q z f z dz

1 11 0 1 2 1 1 2[ | , , ( , ,..., ), ] ( | ( , ,..., ))m mE V Z X t U P Z

1

1 11 0 1 2 1 1 2

10

[ [ | ] | , , , ( , ,..., ), ] ( ) ( | ( , ,..., ))m

Z j Z

m m

j

E E V I Z z J j X t U q z f z dz

1 11 0 1 2 1 1 2[ | , , ( , ,..., ), ] ( | ( , ,..., ))m mE V Z X t U P Z

1

1

1 1

1 0 1 2

10

1 0 1 2 1 1 2

{ [ | , , , 0, ( , ,..., ), ] ( )

[ | , , , 1, ( , ,..., ), ] ( )} ( ) ( | ( , ,..., ))

mZ Z

j m j j

j

Z Z j Z

j m j j m

E V Z z J j X t I U r z

E V Z z J j X t I U r s z q z f z dz

1 11 0 1 2 1 1 2[ | , ,( , ,..., ), ] ( | ( , ,..., ))Z

m mE V Z X t U P

1

1

1

1

1 1 0 1 2

10 0

1 1 0 1 2

1 1 1 2

{ [ | , , , , 0, ( , ,..., ), ] ( )

[ | , , , , 1, ( , ,..., ), ] ( )}

( ) ( | , ) ( | ( , ,..., ))

z mZ Z

j m j j

j

Z Z

j m j j

j X Z

m

E V Z z J j X x X t I U r z


q z dF x t z f z dz

1 11 1 0 1 2 1 1 1 2

0

[ | , , , ( , ,..., ), ] ( | , ) ( | ( , ,..., ))X Z

m mE V Z X x X t U dF x t P

APPENDIX O 239

1

1

1

1

1 1 0 1 2

10 0

1 1 0 1 2

1 1 1 2

{ [ | , , , , 0, ( , ,..., ), ] ( )

[ | , , , , 1, ( , ,..., ), ] ( )}

( ) ( | , ) ( | ( , ,..., ))

z mZ Z

j m j j

j

Z Z

j m j j

j X Z

m



q z f x t z f z dxdz

1

1

1

1

1 1 0 1 2

10

1 1 0 1 2

1 1 1 2

{ [ | , , , , 0, ( , ,..., ), ] ( )

[ | , , , , 1, ( , ,..., ), ] ( )}

( ) ( | ) ( | , ,..., )

mZ Z

j m j j

j

Z Z

j m j j

j X Z

m

E V Z z J j X z X t I U r z


q z P z t f z dz

1 11 1 0 1 2 1 1 1 2

0

[ | , , , ( , ,..., ), ] ( | , ) ( | ( , ,..., )X Z

m mE V Z X x X t U f x t dxP

1 11 1 0 1 2 1 1 1 2[ | , , ,( , ,..., ), ] ( | ) ( | ( , ,..., ))X Z

m mE V Z X X t U P t P

1

1

1

1

1 1 0 1 2

10 0

1 1 0 1 2

1 1 0 1 2

[ | , , , , 0, 0, ( , ,..., ), ] ( )

[ | , , , , 0, 1, ( , ,..., ), ] ( ) ( )

[ | , , , , 1, 0, ( , ,..., ), ]

z mZ X X

j m

j

Z X X Z

j m j j

Z X

j m

E V Z z J j X x X t I I U r t x

E V Z z J j X x X t I I U r t x r z

E V Z z J j X x X t I I U

1

1

1 1 0 1 2

1 1 1 2

( )

[ | , , , , 1, 1, ( , ,..., ), ] ( ) ( )

( ) ( | , ) ( | ( , ,..., ))

X

Z X X Z

j m j j

j X Z

m

r t x

E V Z z J j X x X t I I U r t x r z


1

1

1

1

1 1 0 1 2

10

1 1 0 1 2

1 1 1 2

{ [ | , , , , 0, ( , ,..., ), ] ( )

[ | , , , , 1, ( , ,..., ), ] ( )}

( ) ( | ) ( | ( , ,..., ))

mZ Z

j m j j

j

Z Z

j m j j

j X Z

m



q z P z t f z dz

1

1 1

1 1 1 2

0

1 1 0 1 2 1 1 1 2

{ [ | , , 0, ( , ,..., ), ] ( )

[ | , , , 1, ( , ,..., ), ] ( )} ( | , ) ( | ( , ,..., ))

X X

m

X X X Z

m m

E V Z X x I U r t x

E V Z X x X t I U r t x f x t dxP

1 11 1 0 1 2 1 1 1 2[ | , , ,( , ,..., ), ] ( | ) ( | ( , ,..., ))X Z

m mE V Z X X t U P t P

APPENDIX O 240

1

1

1

1

1 1 2

10 0

1 1 2

1 1 2

[ ( ,0, , ,0) ( , , , ,..., ,..., )] ( )

[ ( ,1, , ,0) ( ,0, , ,..., ,..., )] ( ) ( )

[ ( ,0, , ,1) ( , , , ,...,0,..., )]

z mX

j m

j

X Z

j m j j

X

m


x z j G z z z z z r t x r z

x z j G z t x z z z r

1

1

1 1 2

1 1 1 2

( )

[ ( ,1, ,1) ( ,0, , ,...,0,..., )] ( ) ( )

( ) ( | , ) ( | ( , ,..., ))

X Z

m j j

j X Z

m

t x

x j G z z z z r t x r z


1

1

1

1

1 1 2

10

1 1 2

1 1 1 2

{[ ( , 2, , ,0) ( , , , ,..., ,..., )] ( )

[ ( , 2, , ,1) ( , , , ,...,0,..., )] ( )}

( ) ( | ) ( | ( , ,..., ))

mZ

j m j j

j

Z

m j j

j X Z

m


z z j G z t z z z z r z

q z P z t f z dz

11 1 1 2

0

{ ( ,0, ,0,2) ( ) ( ,1, ,0,2) ( )} ( | , ) ( | ( , ,..., ))X X X Z

mx r t x x r t x f x t z dxP

11 1 1 2( ,2, ,0,2) ( | ) ( | ( , ,..., ))X Z

mP t P .

APPENDIX P 241

APPENDIX P

Audible Signal and Housing/Chassis Datasets

Tables P.1 and P.2 present the data history for audible signal and housing/chassis units,

respectively. The data include unit no, and the lower and upper bounds for the censoring

intervals. The number in the brackets indicates the history number of the unit. The origin for

measuring , 1j il and , 1j iu is the previous failure detection time ( ,j iu ). jiu indicates the last

(right) censoring.

Unit

No.

j

jil

(years)

jiu

(years)

Unit

No.

j

jil

(years)

jiu

(years)

Unit

No.

j

jil

(years)

jiu

(years)

Unit

No.

j

jil

(years)

jiu

(years)

1 1.289 24(1) 1.211 2.389 49(1) 0.000 1.522 68(1) 1.439 2.583

2 1.289 (2) 3.917 (2) 4.783 (2) 3.722

3 1.289 25(1) 1.661 3.586 50(1) 1.786 3.844 69 1.289

4 1.289 (2) 2.667 (2) 2.461 70(1) 0.000 1.814

5 1.289 26 1.289 51 1.289 (2) 4.492

6 1.289 27(1) 1.419 2.200 52 1.289 71 1.289

7 1.289 (2) 3.892 53 1.289 72(1) 0.000 0.997

8 1.289 28(1) 0.000 1.500 54 1.289 (2) 1.875 2.392

9 (1) 2.331 2.369 (2) 4.806 55(1) 0.000 1.433 (3) 2.917

(2) 5.397 29 1.289 (2) 1.586 2.081 73(1) 0.000 1.542

10 1.289 30 1.289 (3) 2.792 (2) 4.764

11 1.289 31 1.289 56 1.289 74(1) 1.439 1.906

12(1) 1.736 2.608 32 1.289 57(1) 0.000 1.503 (2) 4.400

(2) 3.697 33 1.289 (2) 0.000 0.308 75 1.289

APPENDIX P 242

13 1.289 34 1.289 (3) 4.494 76(1) 0.000 1.500

14(1) 0.000 1.300 35 1.289 58(1) 0.519 1.983 (2) 0.000 1.689

(2) 2.886 3.597 36 1.289 (2) 4.322 (3) 3.117

(3) 1.408 37 1.289 59(1) 0.000 1.478 77(1) 1.564 3.553

15 1.289 38 1.289 (2) 4.828 (2) 2.753

16(1) 1.831 3.853 39 1.289 60(1) 1.842 3.814 78(1) 1.733 3.908

(2) 2.156 40 1.289 (2) 2.492 (2) 2.397

17(1) 1.436 1.961 41(1) 0.000 1.942 61(1) 1.531 1.828 79(1) 0.000 1.947

(2) 4.344 (2) 4.364 (2) 4.478 (2) 4.358

18 1.289 42 1.289 62(1) 1.711 2.158 80(1) 0.000 1.717

19(1) 1.367 2.275 43 1.289 (2) 4.147 (2) 4.589

(2) 3.664 44(1) 1.906 3.419 63(1) 1.744 3.517

20 (1) 3.106 4.397 (2) 2.886 (2) 2.692

(2) 1.289 45(1) 3.669 5.697 64 1.289

21(1) 3.292 4.744 (2) 0.608 65(1) 2.628 3.417

(2) 1.561 46 1.289 (2) 2.889

22 1.289 47 1.289 66(1) 1.742 3.658

23(1) 1.394 2.350 48(1) 0.000 1.558 (2) 2.647

(2) 3.956 (2) 0.000 2.311 67 1.289

(3) 2.436

Table P.1. Audible signal dataset for 80 units’ failure and censoring events

APPENDIX P 243

Unit

No.

j

jil

(years)

jiu

(years)

Unit

No.

j

jil

(years)

jiu

(years)

Unit

No.

j

jil

(years)

jiu

(years)

Unit

No.

j

jil

(years)

jiu

(years)

1(1) 2.183 3.739 12(1) 1.206 2.128 21(1) 0.001 2.258 29(1) 0.001 1.089

(2) 0.833 1.925 (2) 1.031 1.978 (2) 0.000 1.308 (2) 0.000 0.328

(3) 0.000 0.697 (3) 0.000 0.542 (3) 0.439 2.594 (3) 0.506 1.967

(4) 1.406 (4) 0.000 1.461 (4) 0.000 1.078 (4) 0.000 0.833

2(1) 0.544 1.092 (5) 0.956 (5) 0.528 (5) 0.086 0.317

(2) 0.000 1.092 13(1) 0.001 1.853 22(1) 0.001 2.294 (6) 1.772

(3) 3.436 3.867 (2) 0.283 1.400 (2) 0.000 1.364 30(1) 0.001 2.047

(4) 0.956 (3) 0.000 0.794 (3) 1.261 1.717 (2) 2.622 4.125

3(1) 0.000 2.214 (4) 0.000 2.278 (4) 0.000 1.100 (3) 0.956

(2) 1.431 1.931 (5) 0.000 0.542 (5) 0.000 0.953 31(1) 0.000 2.028

(3) 0.000 1.244 (6) 0.900 (6) 0.339 (2) 0.000 1.803

(4) 0.000 1.008 14(1) 0.000 0.544 23(1) 0.000 2.136 (3) 0.117 1.378

(5) 0.000 1.017 (2) 0.000 1.075 (2) 0.000 0.194 (4) 0.000 1.992

(6) 0.353 (3) 0.000 0.867 (3) 0.000 0.039 (5) 0.567

4(1) 0.000 0.544 (4) 1.256 1.933 (4) 2.322 3.017 32(1) 0.000 1.736

(2) 0.756 1.497 (5) 0.953 1.336 (5) 0.000 1.319 (2) 0.000 0.872

(3) 2.031 4.072 (6) 1.167 1.711 (6) 0.000 1.033 (3) 0.000 1.050

(4) 0.956 (7) 0.000 0.036 (7) 0.028 (4) 0.000 2.408

5(1) 0.000 5.239 (8) 0.264 24(1) 0.000 2.178 (5) 0.239

(2) 0.000 1.156 15(1) 0.000 2.294 (2) 1.894 2.725 33(1) 0.000 1.942

(3) 1.372 (2) 0.000 1.519 (3) 0.000 1.228 (2) 1.961 2.981

APPENDIX P 244

6(1) 0.000 2.119 (3) 0.211 2.347 (4) 0.000 0.761 (3) 1.383

(2) 0.000 1.286 (4) 0.000 0.742 (5) 0.172 0.731 34(1) 0.000 1.478

(3) 0.636 2.778 (5) 0.864 (6) 0.144 (2) 0.083 1.783

(4) 0.956 16(1) 0.000 2.203 25(1) 0.000 2.214 (3) 0.181 2.186

7(1) 2.058 3.819 (2) 0.000 1.617 (2) 0.000 1.431 (4) 0.858

(2) 0.292 1.381 (3) 1.644 3.247 (3) 0.369 2.508 35(1) 0.000 1.386

(3) 0.000 1.997 (4) 0.700 (4) 0.956 (2) 0.425 2.469

(4) 0.569 17(1) 0.000 2.047 26(1) 0.000 2.042 (3) 0.000 2.150

8(1) 1.206 2.136 (2) 3.600 5.108 (2) 0.000 1.786 (4) 0.300

(2) 3.139 4.578 (3) 0.611 (3) 1.819 3.044 36(1) 0.000 1.300

(3) 1.053 18(1) 0.000 1.444 (4) 0.894 (2) 0.211 0.975

9(1) 2.008 3.197 (2) 2.014 3.317 27(1) 0.000 1.203 (3) 1.911 2.622

(2) 3.658 (3) 1.544 (2) 0.233 0.758 (4) 0.000 1.264

10(1) 0.000 1.572 19(1) 1.825 3.408 (3) 0.000 1.311 (5) 0.144

(2) 0.000 2.003 (2) 0.000 1.106 (4) 0.117 0.656 37(1) 3.669 5.697

(3) 0.000 1.961 (3) 0.000 0.992 (5) 0.000 1.497 (2) 0.000 0.542

(4) 0.769 (4) 0.800 (6) 0.881 (3) 0.067

11(1) 3.419 4.567 20(1) 1.031 1.831 28(1) 0.000 1.386 38(1) 3.869 5.831

(2) 0.000 0.219 (2) 2.022 2.889 (2) 0.000 1.100 (2) 0.475

(3) 1.519 (3) 0.956 (3) 0.911 3.111

(4) 0.708

Table P.2. Housing/Chassis dataset for 38 units’ failure and censoring events

APPENDIX Q 245

APPENDIX Q

Source Codes

All codes are developed in MATLAB version 7.7.0.471 (R2008b).

Q.1. Non-Opportunistic Maintenance, Recursive Procedures

Q.1.1. Finite Time Horizon

In the optimization model over a finite time horizon we need to calculate the following

variables. The name and arguments of the MATLAB function which calculates the required

variable is also given:

The probability that a component survives in the ith

inspection interval, where i=1,2,…,k.

function intout =Simp_int_integral_GeneralP(k,tow,sigma,beta,eta,h,a,b)

The probability that a component fails in the ith

inspection interval, and is minimally repaired,

where i=1,2,…,k.

function intout=Simp_int_integral_GeneralM(k,tow,sigma,beta,eta,h,a,b)

The probability that a component fails in the ith

inspection interval, and is replaced, where

i=1,2,…,k.

function intout=Simp_int_integral_GeneralR(k,tow,sigma,beta,eta,h,a,b)

APPENDIX Q 246

The expected downtime of a component in the ith

inspection interval, where i=1,2,…,k.

function intout=Simp_int_integral_GeneralR(k,tow,sigma,beta,eta,h,a,b)

In addition, other functions are developed to call the required functions for calculating the

probabilities and expected values for all components. For example function

generalprobs(n,tow,sigma) calls function Simp_int_integral_GeneralP for all components, for a

given n, tow and sigma.

Q.1.2. Infinite Time Horizon

In the optimization model over infinite time horizon we need to calculate the following

variables. The name and arguments of the MATLAB function, which calculates the required

variable, is also given:

The expected number of inspection intervals in a cycle for a component, i.e., number of

inspection intervals until the component eventually is replaced.

function intout =Simp_int_integral_GeneralInfinite_V(tow,beta,eta,h,a,b,epsilon)

The expected number of minimal repairs for a component in the cycle.

function intout =Simp_int_integral_GeneralInfinite_n(tow,beta,eta,h,a,b,epsilon)

The expected downtime for a component in the cycle.

function intout =Simp_int_integral_GeneralInfinite_e(tow,beta,eta,h,a,b,epsilon)

APPENDIX Q 247

In addition, other functions are developed to call the required functions for calculating the

probabilities and expected values for all components. For example function

GN = generalVs(tow) calls function Simp_int_integral_GeneralInfinite_V for all components, for a

given tow.

Q.2. Opportunistic Maintenance

Four cases are considered as the opportunistic maintenance, as follows:

1. Minimal repair of components with soft and hard failures.

2. Minimal repair of hard failures, and minimal repair or replacement of components with soft

failure.

3. Minimal repair or replacement of both components with soft and hard failure.

4. In addition to 3, preventive replacement of components with hard failure at periodic

inspections.

Q.2.1. Recursive Procedures-Opportunistic Maintenance

We developed recursive procedures in MATLAB to calculate the required variables for

each case. However, calculating accurately the multi-dimensional integrals requires to consider a

very small value for step h in applying Simpson’s rule. This increases intensively the

APPENDIX Q 248

calculatation time of the procedure. We even improved the performance of the code, but still the

running time is too long. Therefore, eventually we developed a simulation model for each

opportunistic case to calculate the required variables of the model.

In the first case, we need to calculate the expected number of minimal repairs and the

expected downtime of each component over its life cycle, in presence of opportunistic

inspections. We present the function developed for recursively calculating the expected number

of minimal repairs for the first opportunistic case.

function GCalculator_Minimal_Hard_improve3(totaltime,tau,t,beta1,eta1,s1,

beta2,eta2,s2,beta3,eta3,h)

Q.2.2. Simulation Models-Opportunistic Maintenance

For each opportunistic case, a simulation model was developed in MATLAB to obtain

the required variables for the optimization model.

Case 1. The expected number of minimal repairs and the expected downtime of components with

soft failure should be calculated.

function SO = SimulationOpportunisticFile(TotalTime,Tau)

Case 2. The expected number of minimal repairs and replacements, and the expected downtime

of components with soft failure should be calculated.

APPENDIX Q 249

function SO = SimulationOpportunisticMRSoftFile(TotalTime,Tau)


of components with soft failure should be calculated. In the simulation model developed for this

case, the expected number of minimal repairs and replacements of components with hard failure

are also calculated, but they do not contribute in the cost model.

function SO = SimulationOpportunisticMRSoftHardFile(TotalTime,Tau)


of components with soft failure should be calculated. In addition, the expected number of

minimal repairs and replacements of components with hard failures within inspection intervals,

and their expected number of replacements at periodic inspections should be calculated.

function SO = SimulationOpportunisticMRSoftHardPMFile(TotalTime,Tau)

Q.3. Maximum Likelihood Estimation

Q.3.1. Estimating Parameters of a Power Law Process Using a Modified Version of EM

To estimate the parameters using the modified version of EM we need to calculate

functions C , 1B , 2B , 3B , and 4B (shown below) at each iteration. The iterations continue until

the convergence of the parameters.

1 1

1 1

1,...,

... ( ,..., ) ...i i i i i

Y n ny l y y ui n

C f y y dy dy

1 1

1 1 1

1,...,

... ( ,..., ) ...i i i i i

n Y n ny l y y ui n

B y f y y dy dy

APPENDIX Q 250

1 1

2 1 1

1,...,

... log ( ,..., ) ...i i i i i

n n Y n ny l y y ui n

B y y f y y dy dy

1 1

2

3 1 1

1,...,

... (log ) ( ,..., ) ...i i i i i

n n Y n ny l y y ui n

B y y f y y dy dy

1 1

4 1 1

11,...,

... ( ,..., ) ...i i i i i

n

i Y n ny l y y u

ii n

B y f y y dy dy

A function is required to estimate the parameters using the modified version of EM.

function GN = generalLike()

A function is needed to calculate the required C , 1B , 2B , 3B , and 4B at each iteration of the

algorithm.

function [C,B1,B2,B3,B4] =Simp_int_integral_GeneralLikelihoodX(k,beta0,alpha0,h,L,U)

Q.3.2. Estimating Parameters of a Power Law Process Using Complete EM

To estimate the parameters using complete EM we need to calculate functions

C ,

1B , 2B , 3B , and 4B at

each iteration. The iterations continue until the convergence of the

parameters.

A function is required to estimate the parameters using complete EM.

function GN = generalLikeEM()

A function is needed to calculate the required functions C , 1B , 2B , 3B , and 4B at each

iteration of the EM algorithm.

function [C,B1,B2,B3,B4]=

APPENDIX Q 251

Simp_int_integral_GeneralLikelihoodEMX(k,beta0,alpha0,beta1,alpha1,h,L,U)

Q.3.3. Calculating Information Matrix and Standard Errors

To calculate the standard errors of the parameters, which are calculated using any

methods (modified EM and complete EM), we developed a function to calculate the Hessian

matrix, or the second order derivatives of the likelihood, and the standard errors.

function GN = InformationMatrix()

Q.3.4. Calculating the Parameters Using Mid-Points

In order to compare the parameters estimated from different methods (modified EM and

complete EM) with the case that the midpoint of each failure interval is considered as an exact

failure time, we developed a function to estimate the parameters under the assumption of having

complete data.

function GN = generalLike_Mid()

Q.3.5. Estimating Parameter When 1

To perform the likelihood ratio test we need to find parameter iteratively assuming that

1 .

GN = generalLike_Beta1_2()

To find parameter iteratively, calculating function C , 1B , and logC is required.

APPENDIX Q 252

function [C,B1,logC] =Simp_int_integral_GeneralLikelihoodBeta1_exact(alpha0,L,U)

Q.1. Non-Opportunistic Maintenance, Recursive Procedures

Q.1.1. Finite Time Horizon

% function generalprobs(n,tow,sigma) calls function % Simp_int_integral_GeneralP for all components, for a given n, tow and sigma % for example, when T=12, tow=5, then sigma=2 and n=3

function GN = generalprobs(n,tow,sigma)

beta=[1.3,1.1,2.1,1.8,1.7]; eta=[3.5,4.6,6,10,3.6]; % prob of minimal repair at etha=0.6 a=[0.9,0.9,0.9,0.9,0.9]; b=[0.2317,0.1763,0.1352,0.0811,0.2253];

for i=1:numel(beta); N(i,1)=i; intout

=Simp_int_integral_GeneralP(n,tow,sigma,beta(i),eta(i),0.0001,a(i),b(i)) N(i,2:numel(intout)+1)=intout(1:n); end; GN=N; xlswrite('C:\Project\paper 4\excel\probs\Generalprobs5.xls',N); end

% Simp_int_integral_GeneralP calculates the probability of survival of % a given component in the i-th inspection interval where i=1,2,...,k.

function intout =Simp_int_integral_GeneralP(k,tow,sigma,beta,eta,h,a,b) if k==1 intout(k)=FirstFunction(0,tow,beta,eta); else for j=1:k-1; i=0; stopcon=(k-j)*tow; for x=0:h:stopcon; i=i+1; if j==1 IndArray(i)=x; PArray(i)= FirstFunction(x,tow,beta,eta); if sigma ~= 0 PSigmaArray(i)= FirstFunction(x,sigma,beta,eta); end; if x==0 intout(j)=PArray(i);

APPENDIX Q 253

end; else TmpPArray=PArray(fix(x/h+1):fix((x+tow)/h+1)); if sigma ~= 0 TmpPSigmaArray= PSigmaArray(fix(x/h+1):fix((x+tow)/h+1)); end; TmpIndArray=IndArray(fix(x/h+1):fix((x+tow)/h+1)); if x==0 PArrayZero=PArray(1); if sigma ~= 0 PSigmaArrayZero=PSigmaArray(1); end; end; PArray(i)=TmpPArray(numel(TmpPArray))*exp(-

1*(((x+tow)/eta)^beta-(x/eta)^beta));

if sigma ~= 0

PSigmaArray(i)=TmpPSigmaArray(numel(TmpPSigmaArray))*exp(-

1*(((x+tow)/eta)^beta-(x/eta)^beta)); end;

for r=1:numel(TmpIndArray); rf=a*exp(-b*TmpIndArray(r)); TmpPArray(r)=(TmpPArray(r)*rf+PArrayZero*(1-

rf))*(beta/eta)*((TmpIndArray(r)/eta)^(beta-1))*exp(-

1*((TmpIndArray(r)/eta)^beta-(x/eta)^beta)); if sigma ~= 0

TmpPSigmaArray(r)=(TmpPSigmaArray(r)*rf+PSigmaArrayZero*(1-


1*((TmpIndArray(r)/eta)^beta-(x/eta)^beta)); end; end;

PArray(i)=PArray(i)+h*(TmpPArray(1)+4*sum(TmpPArray(2:2:(numel(TmpPArray)-

1)))+2*sum(TmpPArray(3:2:(numel(TmpPArray)-

2)))+TmpPArray(numel(TmpPArray)))/3; if sigma ~= 0

PSigmaArray(i)=PSigmaArray(i)+h*(TmpPSigmaArray(1)+4*sum(TmpPSigmaArray(2:2:(

numel(TmpPSigmaArray)-1)))+2*sum(TmpPSigmaArray(3:2:(numel(TmpPSigmaArray)-

2)))+TmpPSigmaArray(numel(TmpPSigmaArray)))/3; end; IndArray(i)=x;

if x==0 intout(j)=PArray(i); end;

APPENDIX Q 254

end; end; end;

if sigma == 0 TmpPArray=PArray(1:fix(tow/h+1)); TmpIndArray=IndArray(1:fix(tow/h+1)); intout(k)=TmpPArray(numel(TmpPArray))*exp(-1*((tow/eta)^beta)); PArrayZero=PArray(1); for r=1:numel(TmpIndArray); rf=a*exp(-b*TmpIndArray(r)); TmpPArray(r)=(TmpPArray(r)*rf+PArrayZero*(1-


1*(TmpIndArray(r)/eta)^beta); end

intout(k)=intout(k)+h*(TmpPArray(1)+4*sum(TmpPArray(2:2:(numel(TmpPArray)-


2)))+TmpPArray(numel(TmpPArray)))/3; else TmpPSigmaArray=PSigmaArray(1:fix(tow/h+1)); TmpIndArray=IndArray(1:fix(tow/h+1)); intout(k)=TmpPSigmaArray(numel(TmpPSigmaArray))*exp(-

1*((tow/eta)^beta)); PSigmaArrayZero=PSigmaArray(1); for r=1:numel(TmpIndArray); rf=a*exp(-b*TmpIndArray(r)); TmpPSigmaArray(r)=(TmpPSigmaArray(r)*rf+PSigmaArrayZero*(1-


1*(TmpIndArray(r)/eta)^beta); end

intout(k)=intout(k)+h*(TmpPSigmaArray(1)+4*sum(TmpPSigmaArray(2:2:(numel(TmpP

SigmaArray)-1)))+2*sum(TmpPSigmaArray(3:2:(numel(TmpPSigmaArray)-

2)))+TmpPSigmaArray(numel(TmpPSigmaArray)))/3; end; end end

%for P1(0)

function first_func=FirstFunction(x,tow,beta,eta) if x==0 first_func=exp(-1*(tow/eta)^beta); else first_func=exp(-1*(((x+tow)/eta)^beta-(x/eta)^beta)); end; end

% Simp_int_integral_GeneralM calculates the probability that a component

APPENDIX Q 255

% fails in the i-th inspection interval where i=1,2,...,k, and is minimally % repaired

% This function is similar to Simp_int_integral_GeneralP, the difference is

% their FirstFunction

function intout =Simp_int_integral_GeneralM(k,tow,sigma,beta,eta,h,a,b)

…

…

…

end

function first_func=FirstFunction(x,tow,beta,eta,a,b) if x==0 F=@(y) (a*(beta/eta).*(y/eta).^(beta-1)).*exp((-b.*y)-(y/eta).^beta); else F=@(y) (a*(beta/eta).*((x+y)/eta).^(beta-1)).*exp((-b.*(x+y))-

((x+y)/eta).^beta+(x/eta).^beta); end; first_func=quad(F,0,tow); end

% Simp_int_integral_GeneralR calculates the probability that a component % fails in the i-th inspection interval where i=1,2,...,k, and is replaced % This function is similar to Simp_int_integral_GeneralP, the difference is % their FirstFunction

function intout =Simp_int_integral_GeneralR(k,tow,sigma,beta,eta,h,a,b) …

…

…

end

% for R1(0)

function first_func=FirstFunction(x,tow,beta,eta,a,b) if x==0 F=@(y) (1-a*exp(-1*b*y))*(beta/eta).*(y/eta).^(beta-1).*exp(-

(y/eta).^beta); else F=@(y) ((1-a*exp(-1*b*(x+y)))*(beta/eta).*((x+y)/eta).^(beta-1)).*exp(-

((x+y)/eta).^beta+(x/eta).^beta); end; first_func=quad(F,0,tow);

APPENDIX Q 256

end

% Simp_int_integral_GeneralE calculates the expected survival time of a % component in the i-th inspection interval where i=1,2,...,k % This function is similar to Simp_int_integral_GeneralP, the difference is % their FirstFunction

function intout =Simp_int_integral_GeneralE(k,tow,sigma,beta,eta,h,a,b) …

…

…

end

% for e1(0) function first_func=FirstFunction(x,tow,beta,eta) if x==0 F=@(y) exp(-1.*(y/eta).^beta); else F=@(y) exp(-1.*(((y+x)/eta).^beta-(x/eta).^beta)); end; first_func=quad(F,0,tow); end

Q.1.2. Infinite Time Horizon

% function generalVs(tow) calls function % Simp_int_integral_GeneralInfinite_V for all components for a given tow

function GN = generalVs(tow)

beta=[1.3,1.1,2.1,1.8,1.7]; eta=[3.5,4.6,6,10,3.6]; % prob of minimal repair at etha=0.6 a=[0.9,0.9,0.9,0.9,0.9]; b=[0.2317,0.1763,0.1352,0.0811,0.2253];

for i=1:numel(beta); intout

=Simp_int_integral_GeneralInfinite_V(tow,beta(i),eta(i),0.001,a(i),b(i),0.000

1) N(i)=intout; end; GN=N; xlswrite('C:\Project\paper 4\excel\Vs\Generalv12.xls',N); end

APPENDIX Q 257

% function Simp_int_integral_GeneralInfinite_V calculates stopping value k % for a given component (number of inspection intervals till eventually the % component is replaced) for given tow. Also the function calculates % Sum(vi) for i=1,...,k which is the expected number of inspection intervals % in a cycle for the component. vi is the probability that the cycle length % of the component is at least i*tow

function intout

=Simp_int_integral_GeneralInfinite_V(tow,beta,eta,h,a,b,epsilon) tic; ContWhile=true; k=1; while ContWhile %ContWhile i=0; stopcon=tow; for x=0:h:stopcon; i=i+1; if k==1 cell{k}(i)=1; %FirstFunction(x,tow,beta,eta); else if x > 0 for l=1:1:k-1 y=(k-l)*tow+x; if l==1 cell{l}(round(y/h+1))=1; %FirstFunction(y,tow,beta,eta); else TmpPArray=cell{l-1}(round(y/h+1):round((y+tow)/h+1)); TmpIndArray=[y:h:y+tow]; cell{l}(round(y/h+1))=TmpPArray(end)*exp(-

1*(((y+tow)/eta)^beta-(y/eta)^beta)); for r=1:numel(TmpIndArray); rf=a*exp(-b*TmpIndArray(r));

TmpPArray(r)=(TmpPArray(r)*rf)*(beta/eta)*((TmpIndArray(r)/eta)^(beta-

1))*exp(-1*((TmpIndArray(r)/eta)^beta-(y/eta)^beta)); end;

cell{l}(round(y/h+1))=cell{l}(round(y/h+1))+h*(TmpPArray(1)+4*sum(TmpPArray(2

:2:(numel(TmpPArray)-1)))+2*sum(TmpPArray(3:2:(numel(TmpPArray)-

2)))+TmpPArray(numel(TmpPArray)))/3; end end % for l=1:1:k-1 end % if x > 0

TmpPArray=cell{k-1}(round(x/h+1):round((x+tow)/h+1)); TmpIndArray=[x:h:x+tow];

APPENDIX Q 258

cell{k}(i)=TmpPArray(end)*exp(-1*(((x+tow)/eta)^beta-

(x/eta)^beta));

for r=1:numel(TmpIndArray); rf=a*exp(-b*TmpIndArray(r)); % rf=1;


1))*exp(-1*((TmpIndArray(r)/eta)^beta-(x/eta)^beta)); end;

cell{k}(i)=cell{k}(i)+h*(TmpPArray(1)+4*sum(TmpPArray(2:2:(numel(TmpPArray)-


2)))+TmpPArray(numel(TmpPArray)))/3;

end; % if k==1 end; % for x=0:h:stopcon;

cell{k}(1)

if k > 2 checkterm= cell{k}(1)*cell{2}(1) /(1-cell{2}(1)); if checkterm < epsilon ContWhile=false; Sumterms=0; for cnt=1:1:k; Sumterms=Sumterms+cell{cnt}(1); end; k intout =Sumterms end end k=k+1; end % while

toc; end % function

% function Simp_int_integral_GeneralInfinite_n calculates stopping value k % for a given component and tow (k is the value where Sum(ni) for i=1,...,k % becomes smaller than a given epsilon. Also the function calculates % Sum(ni) for i=1,...,k which is the expected number of minimal repairs % in a cycle for the component.

function intout

=Simp_int_integral_GeneralInfinite_n(tow,beta,eta,h,a,b,epsilon,V2) tic; ContWhile=true;

APPENDIX Q 259

k=1; while ContWhile %ContWhile i=0; stopcon=tow; for x=0:h:stopcon; i=i+1; if k==1 cell{k}(i)=FirstFunction(x,tow,beta,eta,a,b); else if x > 0 for l=1:1:k-1 y=(k-l)*tow+x; if l==1 cell{l}(round(y/h+1))=FirstFunction(y,tow,beta,eta,a,b); else TmpPArray=cell{l-1}(round(y/h+1):round((y+tow)/h+1)); TmpIndArray=[y:h:y+tow]; cell{l}(round(y/h+1))=TmpPArray(end)*exp(-

1*(((y+tow)/eta)^beta-(y/eta)^beta)); for r=1:numel(TmpIndArray); rf=a*exp(-b*TmpIndArray(r));


1))*exp(-1*((TmpIndArray(r)/eta)^beta-(y/eta)^beta)); end;

cell{l}(round(y/h+1))=cell{l}(round(y/h+1))+h*(TmpPArray(1)+4*sum(TmpPArray(2

:2:(numel(TmpPArray)-1)))+2*sum(TmpPArray(3:2:(numel(TmpPArray)-

2)))+TmpPArray(numel(TmpPArray)))/3; end end % for l=1:1:k-1 end % if x > 0

TmpPArray=cell{k-1}(round(x/h+1):round((x+tow)/h+1)); TmpIndArray=[x:h:x+tow];

cell{k}(i)=TmpPArray(end)*exp(-1*(((x+tow)/eta)^beta-

(x/eta)^beta));

for r=1:numel(TmpIndArray); rf=a*exp(-b*TmpIndArray(r));


1))*exp(-1*((TmpIndArray(r)/eta)^beta-(x/eta)^beta)); end;

cell{k}(i)=cell{k}(i)+h*(TmpPArray(1)+4*sum(TmpPArray(2:2:(numel(TmpPArray)-


2)))+TmpPArray(numel(TmpPArray)))/3;

end; % if k==1

APPENDIX Q 260

end; % for x=0:h:stopcon;

cell{k}(1)

if k > 2 checkterm= cell{k}(1)*V2 /(1-V2); if checkterm < epsilon ContWhile=false; Sumterms=0; for cnt=1:1:k; Sumterms=Sumterms+cell{cnt}(1); end; k intout =Sumterms end end

k=k+1; end % while

toc; end % function

function first_func=FirstFunction(x,tow,beta,eta,a,b) if x==0 F=@(y) (a*(beta/eta).*(y/eta).^(beta-1)).*exp((-b.*y)-(y/eta).^beta); else F=@(y) (a*(beta/eta).*((x+y)/eta).^(beta-1)).*exp((-b.*(x+y))-

((x+y)/eta).^beta+(x/eta).^beta); end; first_func=quad(F,0,tow); end

% function Simp_int_integral_GeneralInfinite_e calculates stopping value k % for a given component and tow (k is the value where Sum(di) for i=1,...,k % becomes smaller than a given epsilon. Also the function calculates % Sum(di) for i=1,...,k which is the expected downtime for a given % component in a cycle

% This function is similar to Simp_int_integral_GeneralInfinite_n, the

difference is in their first function, which is as follows

function intout

=Simp_int_integral_GeneralInfinite_e(tow,beta,eta,h,a,b,epsilon,V2)

…

…

… end % function

% for e1(0) for downtime

APPENDIX Q 261

function first_func=FirstFunction(x,tow,beta,eta) if x==0 F=@(y) exp(-1.*(y/eta).^beta); else F=@(y) exp(-1.*(((y+x)/eta).^beta-(x/eta).^beta)); end; first_func=tow-quad(F,0,tow); end

Q.2. Opportunistic Maintenance

Q.2.1. Recursive Procedures-Opportunistic Maintenance

% Function GCalculator_Minimal_Hard_improve3(totaltime,tau,t,beta1,eta1,s1, % beta2,eta2,s2,beta3,eta3,h) is the function which its performance has % been improved to calculate faster the expected number of minimal repairs % of a component with soft failure in the case of opportunistic % inspections at hard failures. This function calculates recursively % the expected value of minimal repairs. Simpson's rule is used to % calculate the multi-dimenstional intergrals, however, to have a good % precision, smaller value of h is required which needs a long execution % time for this function.

function intout

=GCalculator_Minimal_Hard_improve3(totaltime,tau,t,beta1,eta1,s1,beta2,eta2,s

2,beta3,eta3,h)

function [out1,out2]=makeFxMatrix(l,Tauh)

expfX=zeros(Tauh,l+1); fX=zeros(Tauh,l+1);

j = 0:1:l;

for i=0:Tauh;

iarray=repmat(i,1,l+1); expfX(i+1,:)= exp((j*h/eta1).^beta1-((iarray+j)*h/eta1).^beta1); fX(i+1,:)=(beta1/eta1)*((iarray+j)*h/eta1).^(beta1-1).*expfX(i+1,:); end;

out1=fX; out2=expfX; end;

function [out1,out2]=makeFzMatrix(l,Tauh) expfZ=zeros(l,l); fZ=zeros(l,l);

APPENDIX Q 262

j = 0:1:l-1; for i=1:l;

iarray=repmat(i,1,l); expfZ(i,:)=exp(((j*h)/eta2).^beta2-

(((iarray+j)*h)/eta2).^beta2+((j*h)/eta3).^beta3-

(((iarray+j)*h)/eta3).^beta3); fZ(i,:)= ((beta2/eta2)*(((iarray+j)*h)/eta2).^(beta2-

1)+(beta3/eta3)*(((iarray+j)*h)/eta3).^(beta3-1)).*expfZ(i,:); end; out1=fZ; out2=expfZ; end;

function out=Gk(k,inspinterval,t,s1,s2,Gi,Gj,h)

function out=CaldblIntegral(t,s1,s2,i,Gi,j) Func_Z=zeros(1,i+1); Func_X=zeros(1,i+1); for i1=1:i; if (i1==i) for i2=0:i1; Func_X(i2+1)=fXmatrix(i2+1,j+1); end; else for i2=0:i1; Func_X(i2+1)=(1+M1(i-i1,i2+1))*fXmatrix(i2+1,j+1); end; end

switch i1+1 case 2 Func_Z(i1+1)=(h/2)*(Func_X(1)+Func_X(2)); case 3 Func_Z(i1+1)=(h/3)*(Func_X(1)+4*Func_X(2)+Func_X(3)); case 4

Func_Z(i1+1)=(3*h/8)*(Func_X(1)+3*Func_X(2)+3*Func_X(3)+Func_X(4)); otherwise if mod(i1+1,2)==0 %even

Func_Z(i1+1)=h/3*(Func_X(1)+4*sum(Func_X(2:2:(i1)))+2*sum(Func_X(3:2:(i1-

1)))+Func_X(i1+1)); else % odd Func_Z(i1+1)=h/3*(Func_X(1)+4*sum(Func_X(2:2:(i1-

3)))+2*sum(Func_X(3:2:(i1-4)))+Func_X(i1-2))+(3*h/8)*(Func_X(i1-

2)+3*Func_X(i1-1)+3*Func_X(i1)+Func_X(i1+1)); end; end

Func_Z(i1+1)=Func_Z(i1+1)*fZmatrix(i1,Gi); end;

APPENDIX Q 263

out=Simpson(Func_Z,h); end;

function out=CalSingleIntegralGk(t,s1,s2,l,Gi,j,h) Func_Z=zeros(1,l+1);

for i=1:l-1;

Func_Z(i+1)=M1(l-i,i)*fZmatrix(i,Gi)*expfXmatrix(i+1,j+1); end;


function out=CalNoIntegralGk(k,t,s1,s2,l,Gi,Gj,h) Func_X=zeros(1,l+1); for j=0:l; Func_X(j+1)=(1+G(k-1,j+1))*fXmatrix(j+1,Gj+1); end;

switch l+1 case 2 out1=(h/2)*(Func_X(1)+Func_X(2)); case 3 out1=(h/3)*(Func_X(1)+4*Func_X(2)+Func_X(3)); case 4 out1=(3*h/8)*(Func_X(1)+3*Func_X(2)+3*Func_X(3)+Func_X(4)); otherwise if mod(l+1,2)==0 %even

out1=h/3*(Func_X(1)+4*sum(Func_X(2:2:(l)))+2*sum(Func_X(3:2:(l-

1)))+Func_X(l+1)); else % odd out1=h/3*(Func_X(1)+4*sum(Func_X(2:2:(l-

3)))+2*sum(Func_X(3:2:(l-4)))+Func_X(l-2))+(3*h/8)*(Func_X(l-2)+3*Func_X(l-

1)+3*Func_X(l)+Func_X(l+1)); end; end

PZSigma_given_S=expfZmatrix(l,Gi);

out1=out1*PZSigma_given_S;

PXSigma_given_t=expfXmatrix(l+1,Gj+1);

out2=PZSigma_given_S*PXSigma_given_t*(G(k-1,l+1));

out=out1+out2;

APPENDIX Q 264

end;

if k==1 out=G1(inspinterval,t,s1,s2,Gi,Gj,h); else l=inspinterval/h;

M1=zeros(l,l);

for i=1:l; for j=0:l-i; M1(i,j+1)=CaldblIntegral(t+j*h,s1+(l-i)*h,s2+(l-i)*h,i,Gi+l-

i+1,Gj+j); end; end;

out=M1(l,1);

SingleIntegral=CalSingleIntegralGk(t,s1,s2,l,Gi+1,Gj,h); NoIntegral=CalNoIntegralGk(k,t,s1,s2,l,Gi+1,Gj,h);

out=out+SingleIntegral+NoIntegral;

Denominator=((beta2/eta2)*(s1/eta2)^(beta2-

1))+((beta3/eta3)*(s2/eta3)^(beta3-1));

switch l+1 case 2 out=out/(1-Denominator*h/2); case 3 out=out/(1-Denominator*h/3); case 4 out=out/(1-Denominator*h*3/8); otherwise out=out/(1-Denominator*h/3); end;

end;

end;

function out=Si(funccall,x,z) switch funccall case 1 if x < z

APPENDIX Q 265

out=1; else out=0; end; otherwise end; end;

function Simps= Simpson(InpArray,h) Dimension=numel(InpArray); switch Dimension case 2 Simps=(h/2)*(InpArray(1)+InpArray(2)); case 3 Simps=(h/3)*(InpArray(1)+4*InpArray(2)+InpArray(3)); case 4

Simps=(3*h/8)*(InpArray(1)+3*InpArray(2)+3*InpArray(3)+InpArray(4)); otherwise if mod(Dimension,2)==0 %even Simps=h/3*(InpArray(1)+4*sum(InpArray(2:2:(Dimension-

1)))+2*sum(InpArray(3:2:(Dimension-2)))+InpArray(Dimension)); else % odd Simps=h/3*(InpArray(1)+4*sum(InpArray(2:2:(Dimension-

4)))+2*sum(InpArray(3:2:(Dimension-5)))+InpArray(Dimension-

3))+(3*h/8)*(InpArray(Dimension-3)+3*InpArray(Dimension-

2)+3*InpArray(Dimension-1)+InpArray(Dimension)); end; end end;

function out=G1(sigma,t,s1,s2,Gi,Gj,h)

function out=CaldblIntegral(t,s1,s2,i,Gi,Gj,h) Func_Z=zeros(1,i+1); Func_X=zeros(1,i+1); for i1=1:i; if (i1==i) for i2=0:i1; Func_X(i2+1)=fXmatrix(i2+1,Gj+1); end; else for i2=0:i1; Func_X(i2+1)=(1+M1(i-i1,i2+1))*fXmatrix(i2+1,Gj+1); end; end

switch i1+1 case 2 Func_Z(i1+1)=(h/2)*(Func_X(1)+Func_X(2)); case 3

APPENDIX Q 266

Func_Z(i1+1)=(h/3)*(Func_X(1)+4*Func_X(2)+Func_X(3)); case 4

Func_Z(i1+1)=(3*h/8)*(Func_X(1)+3*Func_X(2)+3*Func_X(3)+Func_X(4)); otherwise if mod(i1+1,2)==0 %even

Func_Z(i1+1)=h/3*(Func_X(1)+4*sum(Func_X(2:2:(i1)))+2*sum(Func_X(3:2:(i1-

1)))+Func_X(i1+1)); else % odd Func_Z(i1+1)=h/3*(Func_X(1)+4*sum(Func_X(2:2:(i1-

3)))+2*sum(Func_X(3:2:(i1-4)))+Func_X(i1-2))+(3*h/8)*(Func_X(i1-

2)+3*Func_X(i1-1)+3*Func_X(i1)+Func_X(i1+1)); end; end

Func_Z(i1+1)=Func_Z(i1+1)*fZmatrix(i1,Gi); end; out=Simpson(Func_Z,h); end;

function out=CalSingleIntegral(t,s1,s2,l,Gi,j,h) Func_Z=zeros(1,l+1);

for i=1:l-1; Func_Z(i+1)=M1(l-i,i)*fZmatrix(i,Gi)*expfXmatrix(i+1,j+1); end;


function out=CalNoIntegral(sigma,t,s1,s2,Gi,h) F=@(x) (beta1/eta1).*((t+x)/eta1).^(beta1-1).*exp((t/eta1).^beta1-

((t+x)/eta1).^beta1); out=quad(F,0,sigma); out=out*expfZmatrix(sigma/h,Gi); end;

l=sigma/h;

M1=zeros(l,l);

for i=1:l; for j=0:l-i; M1(i,j+1)=CaldblIntegral(t+j*h,s1+(l-i)*h,s2+(l-i)*h,i,Gi+l-

i+1,Gj+j,h); end; end;

SingleIntegral=CalSingleIntegral(t,s1,s2,l,Gi+1,Gj,h); NoIntegral=CalNoIntegral(sigma,t,s1,s2,Gi+1,h);

APPENDIX Q 267

out=M1(l,1)+SingleIntegral+NoIntegral;

Denominator= ((beta2/eta2)*(s1/eta2)^(beta2-

1))+((beta3/eta3)*(s2/eta3)^(beta3-1));

switch l+1 case 2 out=out/(1-Denominator*h/2); case 3 out=out/(1-Denominator*h/3); case 4 out=out/(1-Denominator*h*3/8); otherwise out=out/(1-Denominator*h/3); end; end;

% for k

tStart=tic;

k=fix(totaltime/tau); sigma=rem(totaltime,tau); if sigma > 0 k=k+1; end

l3=tau/h;

[fXmatrix,expfXmatrix]=makeFxMatrix(totaltime/h,l3);

[fZmatrix,expfZmatrix]=makeFzMatrix(totaltime/h,l3);

G=zeros(k,(k-1)*l3);

for i3=1:k; if sigma > 0 && i3==1 inspinterval=sigma; else inspinterval=tau; end; for j3=0:(k-i3)*l3; G(i3,j3+1)= Gk(i3,inspinterval,t+j3*h,s1+(k-i3)*tau,s2+(k-i3)*tau,(k-

i3)*l3,j3,h); end; end;

APPENDIX Q 268

xlswrite('C:\opportunistic

maintenance\matlab\Opportunistic\bothMinimal\Excel\Out.xls',G); intout=G(k,1);

tElapsed=toc(tStart) end

Q.2.2. Simulation Models-Opportunistic Maintenance

% This function calls the simulation function % SimulationOpportunisticMRSoftFile(TotalTime,Tau) % for Tau=1,2,...,12, and T=12 months.

function out=RunAll() TotalTime=12; for Tau=TotalTime:-1:1; SO = SimulationOpportunisticMRSoftFile(TotalTime,Tau) end out=Tau; end

% Function SimulationOpportunisticMRSoftFile(TotalTime,Tau) runs the % simulation model OneSimulationRun(), which is the model for the % opportunistic case with the assumption of minimal and replacements of % soft failures, and minimal repair of hard failures 1,000,000 times. % The outputs which are the expected number of minimal repairs, % replacements, and uptime for all components with soft failures are % exported to the excel files

function SO = SimulationOpportunisticMRSoftFile(TotalTime,Tau)

function Out=OneSimulationRun()

NextFailuresSoft=zeros(1,SoftCompNo); CurrentAgesSoft=zeros(1,SoftCompNo); NextFailuresHard=zeros(1,HardCompNo); CurrentAgesHard=zeros(1,HardCompNo);

OutSoft=zeros(SoftCompNo,3); % OutSoft(i,1) # of minimal repairs ,

OutSoft(i,2) # of replacements , OutSoft(i,3) uptime

for i=1:SoftCompNo; z= random('uniform',0,1);

APPENDIX Q 269

NextFailuresSoft(i)=SoftParams(i,2)*power(power(CurrentAgesSoft(i)/SoftParams

(i,2),SoftParams(i,1))-log(z),1/SoftParams(i,1))-CurrentAgesSoft(i); i=i+1; end;

for i=1:HardCompNo; z= random('uniform',0,1);

NextFailuresHard(i)=HardParams(i,2)*power(power(CurrentAgesHard(i)/HardParams

(i,2),HardParams(i,1))-log(z),1/HardParams(i,1))-CurrentAgesHard(i); i=i+1; end;

CurrentTime=0; NextInspection=Tau;

while CurrentTime < TotalTime [MinHard,MinIdx]=min(NextFailuresHard);

while MinHard < NextInspection for i=1:SoftCompNo; if NextFailuresSoft(i) < MinHard

OutSoft(i,3)=OutSoft(i,3)+NextFailuresSoft(i);

rf=SoftParams(i,3)*exp(-SoftParams(i,4)*CurrentAgesSoft(i)); repairz= random('uniform',0,1);

if repairz <= rf % minimal repair OutSoft(i,1)=OutSoft(i,1)+1;

CurrentAgesSoft(i)=CurrentAgesSoft(i)+NextFailuresSoft(i); else % replacement OutSoft(i,2)=OutSoft(i,2)+1; CurrentAgesSoft(i)=0; end

NextFailuresSoft(i)=GenerateNextFailure(SoftParams(i,1),SoftParams(i,2),Curre

ntAgesSoft(i)); else OutSoft(i,3)=OutSoft(i,3)+MinHard; CurrentAgesSoft(i)=CurrentAgesSoft(i)+MinHard; NextFailuresSoft(i)=NextFailuresSoft(i)-MinHard; end end CurrentTime=CurrentTime+MinHard; CurrentAgesHard=CurrentAgesHard+MinHard;

APPENDIX Q 270

NextfailureofHardfailed=GenerateNextFailure(HardParams(MinIdx,1),HardParams(M

inIdx,2),CurrentAgesHard(MinIdx));

NextFailuresHard=NextFailuresHard-MinHard;

NextFailuresHard(MinIdx)=NextfailureofHardfailed;

NextInspection=NextInspection-MinHard; [MinHard,MinIdx]=min(NextFailuresHard); end

for i=1:SoftCompNo; if NextFailuresSoft(i) < NextInspection



if repairz <= rf % minimal repair OutSoft(i,1)=OutSoft(i,1)+1; CurrentAgesSoft(i)=CurrentAgesSoft(i)+NextFailuresSoft(i); else % replacement OutSoft(i,2)=OutSoft(i,2)+1; CurrentAgesSoft(i)=0; end


ntAgesSoft(i)); else OutSoft(i,3)=OutSoft(i,3)+NextInspection; CurrentAgesSoft(i)=CurrentAgesSoft(i)+NextInspection; NextFailuresSoft(i)=NextFailuresSoft(i)-NextInspection; end end CurrentAgesHard=CurrentAgesHard+NextInspection; CurrentTime=CurrentTime+NextInspection; NextFailuresHard=NextFailuresHard-NextInspection;

if CurrentTime - Tau*(CurrentTime/Tau)== 0 Sigma=TotalTime-CurrentTime; if Sigma < Tau NextInspection=Sigma; else NextInspection=Tau; end end end

Out=OutSoft;

APPENDIX Q 271

end

function nextfailure=GenerateNextFailure(beta,eta,CurrentAge) z= random('uniform',0,1); nextfailure=eta*power(power(CurrentAge/eta,beta)-log(z),1/beta)-

CurrentAge; end

Data = xlsread('C:\opportunistic

maintenance\matlab\Opportunistic\Simulation\Minimal_Replacement\Excel\SoftCom

ponents.xls');

RecordsNo=numel(Data(:,2));

rowindx=0;

while (rowindx < RecordsNo) rowindx=rowindx+1; SoftParams(rowindx,1)=Data(rowindx,1); SoftParams(rowindx,2)=Data(rowindx,2); SoftParams(rowindx,3)=Data(rowindx,3); SoftParams(rowindx,4)=Data(rowindx,4); end

SoftCompNo=rowindx;


maintenance\matlab\Opportunistic\Simulation\Minimal_Replacement\Excel\HardCom

ponents.xls');


rowindx=0;

while (rowindx < RecordsNo) rowindx=rowindx+1; HardParams(rowindx,1)=Data(rowindx,1); HardParams(rowindx,2)=Data(rowindx,2); end

HardCompNo=rowindx;

SimOut=zeros(SoftCompNo,3);

SimulationRuns=1000000;

APPENDIX Q 272

MinimalMatrix=zeros(SimulationRuns,SoftCompNo); ReplacementMatrix=zeros(SimulationRuns,SoftCompNo); UpMatrix=zeros(SimulationRuns,SoftCompNo);

for iteration=1:SimulationRuns; Out=OneSimulationRun(); MinimalMatrix(iteration,:)=Out(:,1)'; ReplacementMatrix(iteration,:)=Out(:,2)'; UpMatrix(iteration,:)=Out(:,3)'; SimOut=SimOut+Out; end;

for i=1:SoftCompNo; AvgSTDMinimals(i,1)=mean(MinimalMatrix(:,i)); AvgSTDMinimals(i,2)=std(MinimalMatrix(:,i)); AvgSTDMinimals(i,3)=std(MinimalMatrix(:,i))/sqrt(SimulationRuns);

AvgSTDReplacement(i,1)=mean(ReplacementMatrix(:,i)); AvgSTDReplacement(i,2)=std(ReplacementMatrix(:,i)); AvgSTDReplacement(i,3)=std(ReplacementMatrix(:,i))/sqrt(SimulationRuns);

AvgSTDUp(i,1)=mean(UpMatrix(:,i)); AvgSTDUp(i,2)=std(UpMatrix(:,i)); AvgSTDUp(i,3)=std(UpMatrix(:,i))/sqrt(SimulationRuns);

end;

SimOut=SimOut/SimulationRuns;

SO=SimOut;

f1=strcat('AvgSTDMinimals-tau-',num2str(Tau),'.xls'); f = fullfile('C:', 'opportunistic maintenance', 'matlab',

'Opportunistic','Simulation','Minimal_Replacement','Excel', f1); xlswrite(f,AvgSTDMinimals);

f1=strcat('AvgSTDReplacement-tau-',num2str(Tau),'.xls'); f = fullfile('C:', 'opportunistic maintenance', 'matlab',

'Opportunistic','Simulation','Minimal_Replacement','Excel', f1); xlswrite(f,AvgSTDReplacement);

f1=strcat('AvgSTDUp-tau-',num2str(Tau),'.xls'); f = fullfile('C:', 'opportunistic maintenance', 'matlab',

'Opportunistic','Simulation','Minimal_Replacement','Excel', f1); xlswrite(f,AvgSTDUp);

end

APPENDIX Q 273

% Function SimulationOpportunisticFile(TotalTime,Tau) % is similar to function SimulationOpportunisticMRSoftFile, except that % it runs the simulation model for the opportunistic case with the % assumption of minimal repair of the components with soft and hard % failures. % The outputs which are the expected number of minimal repairs and uptime % for all components with soft failures are exported to the excel files

% The main difference in the code of this function is in function

% OneSimulationRun()

function SO = SimulationOpportunisticFile(TotalTime,Tau)

function Out=OneSimulationRun()


OutSoft=zeros(SoftCompNo,2); % # of minimal repairs, total uptime









while MinHard < NextInspection for i=1:SoftCompNo; if NextFailuresSoft(i) < MinHard OutSoft(i,1)=OutSoft(i,1)+1; OutSoft(i,2)=OutSoft(i,2)+NextFailuresSoft(i);

APPENDIX Q 274

CurrentAgesSoft(i)=CurrentAgesSoft(i)+NextFailuresSoft(i);


ntAgesSoft(i)); else OutSoft(i,2)=OutSoft(i,2)+MinHard; CurrentAgesSoft(i)=CurrentAgesSoft(i)+MinHard; NextFailuresSoft(i)=NextFailuresSoft(i)-MinHard; end end CurrentTime=CurrentTime+MinHard; CurrentAgesHard=CurrentAgesHard+MinHard;






for i=1:SoftCompNo; if NextFailuresSoft(i) < NextInspection OutSoft(i,1)=OutSoft(i,1)+1; OutSoft(i,2)=OutSoft(i,2)+NextFailuresSoft(i); CurrentAgesSoft(i)=CurrentAgesSoft(i)+NextFailuresSoft(i);



if mod(CurrentTime,Tau)== 0 Sigma=TotalTime-CurrentTime; if Sigma < Tau NextInspection=Sigma; else NextInspection=Tau; end end end

Out=OutSoft;

APPENDIX Q 275

end

end

% Function SimulationOpportunisticMRSoftHardFile(TotalTime,Tau) runs the % simulation model OneSimulationRun(), which is the model for the % opportunistic case with the assumption of minimal and replacements of % soft and hard failures 1,000,000 times. % The outputs which are the expected number of minimal repairs, % replacements, and uptime for all components with soft failures, and the % expected numbers of minimal repairs and replacement for hard failures % are exported to the excel files

function SO = SimulationOpportunisticMRSoftHardFile(TotalTime,Tau)

function [Out1,Out2]=OneSimulationRun()



OutSoft(i,2) # of replacements , OutSoft(i,3) uptime OutHard=zeros(HardCompNo,2); % OutHard(i,1) # of minimal repairs ,

OutHard(i,2) # of replacements








APPENDIX Q 276






CurrentAgesSoft(i)=CurrentAgesSoft(i)+NextFailuresSoft(i); else % replacement OutSoft(i,2)=OutSoft(i,2)+1; CurrentAgesSoft(i)=0; end


ntAgesSoft(i)); else OutSoft(i,3)=OutSoft(i,3)+MinHard; CurrentAgesSoft(i)=CurrentAgesSoft(i)+MinHard; NextFailuresSoft(i)=NextFailuresSoft(i)-MinHard; end end CurrentTime=CurrentTime+MinHard;

CurrentAgesHard=CurrentAgesHard+MinHard;

rf=HardParams(MinIdx,3)*exp(-

HardParams(MinIdx,4)*CurrentAgesHard(MinIdx)); repairz= random('uniform',0,1);

if repairz <= rf % minimal repair OutHard(MinIdx,1)=OutHard(MinIdx,1)+1; CurrentAgesHard(MinIdx)=CurrentAgesHard(MinIdx)+MinHard; else % replacement OutHard(MinIdx,2)=OutHard(MinIdx,2)+1; CurrentAgesHard(MinIdx)=0; end

APPENDIX Q 277




NextFailuresHard(MinIdx)=NextfailureofHardfailed; NextInspection=NextInspection-MinHard; [MinHard,MinIdx]=min(NextFailuresHard); end




if repairz <= rf % minimal repair OutSoft(i,1)=OutSoft(i,1)+1; CurrentAgesSoft(i)=CurrentAgesSoft(i)+NextFailuresSoft(i); else % replacement OutSoft(i,2)=OutSoft(i,2)+1; CurrentAgesSoft(i)=0; end



if CurrentTime - Tau*(CurrentTime/Tau)== 0 Sigma=TotalTime-CurrentTime; if Sigma < Tau NextInspection=Sigma; else NextInspection=Tau; end end end

Out1=OutSoft; Out2=OutHard;

APPENDIX Q 278

end

function nextfailure=GenerateNextFailure(beta,eta,CurrentAge) z= random('uniform',0,1); nextfailure=eta*power(power(CurrentAge/eta,beta)-log(z),1/beta)-

CurrentAge; end


maintenance\matlab\Opportunistic\Simulation\Minimal_Replacement_Hard\Excel\So

ftComponents.xls');


rowindx=0;

while (rowindx < RecordsNo) rowindx=rowindx+1; SoftParams(rowindx,1)=Data(rowindx,1); SoftParams(rowindx,2)=Data(rowindx,2); SoftParams(rowindx,3)=Data(rowindx,3); SoftParams(rowindx,4)=Data(rowindx,4); end

SoftCompNo=rowindx;


maintenance\matlab\Opportunistic\Simulation\Minimal_Replacement_Hard\Excel\Ha

rdComponents.xls');


rowindx=0;

while (rowindx < RecordsNo) rowindx=rowindx+1; HardParams(rowindx,1)=Data(rowindx,1); HardParams(rowindx,2)=Data(rowindx,2); HardParams(rowindx,3)=Data(rowindx,3); HardParams(rowindx,4)=Data(rowindx,4); end

HardCompNo=rowindx;

SimOutSoft=zeros(SoftCompNo,3); SimOutHard=zeros(HardCompNo,2);

APPENDIX Q 279

SimulationRuns=1000000;

MinimalMatrix=zeros(SimulationRuns,SoftCompNo); ReplacementMatrix=zeros(SimulationRuns,SoftCompNo); UpMatrix=zeros(SimulationRuns,SoftCompNo);

MinimalMatrixHard=zeros(SimulationRuns,HardCompNo); ReplacementMatrixHard=zeros(SimulationRuns,HardCompNo);

for iteration=1:SimulationRuns; [out1,out2]=OneSimulationRun(); MinimalMatrix(iteration,:)=out1(:,1)'; ReplacementMatrix(iteration,:)=out1(:,2)'; UpMatrix(iteration,:)=out1(:,3)'; SimOutSoft=SimOutSoft+out1;

MinimalMatrixHard(iteration,:)=out2(:,1)'; ReplacementMatrixHard(iteration,:)=out2(:,2)'; SimOutHard=SimOutHard+out2; end;

for i=1:SoftCompNo; AvgSTDMinimals(i,1)=mean(MinimalMatrix(:,i)); AvgSTDMinimals(i,2)=std(MinimalMatrix(:,i)); AvgSTDMinimals(i,3)=std(MinimalMatrix(:,i))/sqrt(SimulationRuns);

AvgSTDReplacement(i,1)=mean(ReplacementMatrix(:,i)); AvgSTDReplacement(i,2)=std(ReplacementMatrix(:,i)); AvgSTDReplacement(i,3)=std(ReplacementMatrix(:,i))/sqrt(SimulationRuns);

AvgSTDUp(i,1)=mean(UpMatrix(:,i)); AvgSTDUp(i,2)=std(UpMatrix(:,i)); AvgSTDUp(i,3)=std(UpMatrix(:,i))/sqrt(SimulationRuns);

end;

for i=1:HardCompNo; AvgSTDMinimalsHard(i,1)=mean(MinimalMatrixHard(:,i)); AvgSTDMinimalsHard(i,2)=std(MinimalMatrixHard(:,i)); AvgSTDMinimalsHard(i,3)=std(MinimalMatrixHard(:,i))/sqrt(SimulationRuns);

AvgSTDReplacementHard(i,1)=mean(ReplacementMatrixHard(:,i)); AvgSTDReplacementHard(i,2)=std(ReplacementMatrixHard(:,i));

AvgSTDReplacementHard(i,3)=std(ReplacementMatrixHard(:,i))/sqrt(SimulationRun

s);

end;

APPENDIX Q 280

SimOutSoft=SimOutSoft/SimulationRuns;

SO=SimOutSoft;

f1=strcat('AvgSTDMinimals-tau-',num2str(Tau),'.xls'); f = fullfile('C:', 'opportunistic maintenance', 'matlab',

'Opportunistic','Simulation','Minimal_Replacement_Hard','Excel', f1); xlswrite(f,AvgSTDMinimals);

f1=strcat('AvgSTDReplacement-tau-',num2str(Tau),'.xls'); f = fullfile('C:', 'opportunistic maintenance', 'matlab',

'Opportunistic','Simulation','Minimal_Replacement_Hard','Excel', f1); xlswrite(f,AvgSTDReplacement);

f1=strcat('AvgSTDUp-tau-',num2str(Tau),'.xls'); f = fullfile('C:', 'opportunistic maintenance', 'matlab',

'Opportunistic','Simulation','Minimal_Replacement_Hard','Excel', f1); xlswrite(f,AvgSTDUp);

f1=strcat('AvgSTDMinimalsHard-tau-',num2str(Tau),'.xls'); f = fullfile('C:', 'opportunistic maintenance', 'matlab',

'Opportunistic','Simulation','Minimal_Replacement_Hard','Excel', f1); xlswrite(f,AvgSTDMinimalsHard);

f1=strcat('AvgSTDReplacementHard-tau-',num2str(Tau),'.xls'); f = fullfile('C:', 'opportunistic maintenance', 'matlab',

'Opportunistic','Simulation','Minimal_Replacement_Hard','Excel', f1); xlswrite(f,AvgSTDReplacementHard);

end

% Function SimulationOpportunisticMRSoftHardPMFile(TotalTime,Tau) % is similar to function SimulationOpportunisticMRSoftHardFile, except that % it runs the simulation model for the opportunistic case with the % additional assumption of preventive replacement of the components with % hard failures at periodic inspections. % The outputs which are the expected number of minimal repairs, % replacements, and uptime for all components with soft failures, and the % expected numbers of minimal repairs and replacements inside, and at the % inspection intervals for hard failures are exported to the excel files

% The main addition in the code of this function is in function

% OneSimulationRun()

function SO = SimulationOpportunisticMRSoftHardPMFile(TotalTime,Tau)

function [Out1,Out2]=OneSimulationRun()

APPENDIX Q 281



OutSoft(i,2) # of replacements , OutSoft(i,3) uptime OutHard=zeros(HardCompNo,3); % OutHard(i,1) # of minimal repairs ,

OutHard(i,2) # of replacements , OutHard(i,3) # of PM replacements













CurrentAgesSoft(i)=CurrentAgesSoft(i)+NextFailuresSoft(i); else % replacement OutSoft(i,2)=OutSoft(i,2)+1;

APPENDIX Q 282

CurrentAgesSoft(i)=0; end


ntAgesSoft(i)); else OutSoft(i,3)=OutSoft(i,3)+MinHard; CurrentAgesSoft(i)=CurrentAgesSoft(i)+MinHard; NextFailuresSoft(i)=NextFailuresSoft(i)-MinHard; end end CurrentTime=CurrentTime+MinHard;

CurrentAgesHard=CurrentAgesHard+MinHard;

rf=HardParams(MinIdx,3)*exp(-

HardParams(MinIdx,4)*CurrentAgesHard(MinIdx)); repairz= random('uniform',0,1);

if repairz <= rf % minimal repair OutHard(MinIdx,1)=OutHard(MinIdx,1)+1; CurrentAgesHard(MinIdx)=CurrentAgesHard(MinIdx)+MinHard; else % replacement OutHard(MinIdx,2)=OutHard(MinIdx,2)+1; CurrentAgesHard(MinIdx)=0; end









if repairz <= rf % minimal repair

APPENDIX Q 283

OutSoft(i,1)=OutSoft(i,1)+1; CurrentAgesSoft(i)=CurrentAgesSoft(i)+NextFailuresSoft(i); else % replacement OutSoft(i,2)=OutSoft(i,2)+1; CurrentAgesSoft(i)=0; end



if CurrentTime - Tau*(CurrentTime/Tau)== 0 % Applying PM for components with hard failure for i=1:HardCompNo;

rf=HardParams(i,3)*exp(-HardParams(i,4)*CurrentAgesSoft(i)); repairz= random('uniform',0,1);

if repairz > rf % replacement OutHard(i,3)=OutHard(i,3)+1; CurrentAgesHard(i)=0;

NextFailuresHard(i)=GenerateNextFailure(HardParams(i,1),HardParams(i,2),Curre

ntAgesHard(i)); end end

Sigma=TotalTime-CurrentTime; if Sigma < Tau NextInspection=Sigma; else NextInspection=Tau; end end end

Out1=OutSoft; Out2=OutHard; end

end

APPENDIX Q 284

Q.3. Maximum Likelihood Estimation

Q.3.1. Estimating Parameters of a Power Law Process Using a Modified Version of EM

% This function calculates the parameters of a Power Law process using a % modified version of EM, which has one iteration of Newton Raphson, % several functions including C, B1,B2,B3,B4, and B5 for each iteration % should be calculated, which is done by calling function % Simp_int_integral_GeneralLikelihoodX

function GN = generalLike()

First2=true; First3=true; First4=true;

tStart2=tic; tStart3=tic; tStart4=tic; tStart5=tic;

Params(1,1)=-0.7; Params(1,2)=1;

Data = xlsread('C:\Paper 5 new\excel\BatterySubset3.xls')


rowindx=1; UnitNo=1;

while (rowindx <= RecordsNo)

aHistoryL(1,1)=Data(rowindx,2); aHistoryU(1,1)=Data(rowindx,3); k=2; rowindx=rowindx+1; while (rowindx <= RecordsNo) & (Data(rowindx,1) ~= 1) aHistoryL(1,k)=Data(rowindx,2); aHistoryU(1,k)=Data(rowindx,3); k=k+1; rowindx=rowindx+1; end L{UnitNo}=aHistoryL(1:k-1); U{UnitNo}=aHistoryU(1:k-1);

UnitNo=UnitNo+1; end

APPENDIX Q 285

NoofSystems=UnitNo-1;%3;

l=1; AllParams(l,:)=Params(1,:); ContWhile=true; Oncemore=0;

while ContWhile %ContWhile

if Oncemore==1 ContWhile=false; end; %

% N(1)='Ci'; % N(2)='(yn)^beta'; % N(3)='((yn)^beta)*log(yn)'; % N(4)='((yn)^beta)*[log(yn)]^2';

Sumni=0;

for i=1:NoofSystems;

Sumni=Sumni+numel(L{i});

[C,B1,B2,B3,B4] =

Simp_int_integral_GeneralLikelihoodX(numel(L{i}),Params(1,2),Params(1,1),0.00

1,L{i},U{i});

N(1,i)=C; N(2,i)=B1; N(3,i)=B2; N(4,i)=B3; N(5,i)=B4;

end % Construct Q' array Q_Prime(1,1)=Sumni-exp(Params(1,1))*sum(N(2,:)); Q_Prime(2,1)=(1/Params(1,2))*Sumni+sum(N(5,:))-exp(Params(1,1))*sum(N(3,:));

% Construct Q'' matrix Q_Second(1,1)=-exp(Params(1,1))*sum(N(2,:)); Q_Second(1,2)=-exp(Params(1,1))*sum(N(3,:)); Q_Second(2,1)=Q_Second(1,2); Q_Second(2,2)=(-1/(Params(1,2)^2))*Sumni-exp(Params(1,1))*sum(N(4,:));

NewParams=Params'-inv(Q_Second)*Q_Prime

l=l+1;

APPENDIX Q 286

AllParams(l,1)=NewParams(1,1); AllParams(l,2)=NewParams(2,1); AllParams(l,3)=abs(NewParams(1,1)-Params(1,1)); AllParams(l,4)=abs(Params(1,2)-NewParams(2,1)); AllParams(l,5)=Q_Prime(1,1); AllParams(l,6)=Q_Prime(2,1); if NewParams(2,1) < 0 NewParams(2,1)=0.1; end

if First2 & (abs(Params(1,1)-NewParams(1,1)) < 10^-2) & (abs(Params(1,2)-

NewParams(2,1)) < 10^-2) AllParams(l,7)=toc(tStart2); First2=false; end





if (abs(Params(1,1)-NewParams(1,1)) < 10^-5) & (abs(Params(1,2)-

NewParams(2,1)) < 10^-5) Oncemore=1; AllParams(l,7)=toc(tStart5); end

Params(1,:)=NewParams(:,1)';

end; % while GN=N; xlswrite('C:\Paper 5 new\excel\NewSet\OneStep\Expected-

BatterySubset3.xls',N); xlswrite('C:\Paper 5 new\excel\NewSet\OneStep\AllParams-

BatterySubset3.xls',AllParams);

end

% This function calculates the recursive functions required to estimate the % parameters of a Power Law process using a modified version of EM, which % has one iteration of Newton Raphson, the function which are calculaed

APPENDIX Q 287

% here are C1, B1, B2, B3, and B4 with the following definitions % C1 1 % B1 (yn)^beta % B2 ((yn)^beta)*log(yn) % B3 ((yn)^beta)*[log(yn)]^2 % B4 Sum(yi) i=1,...,n

function [C,B1,B2,B3,B4]

=Simp_int_integral_GeneralLikelihoodX(k,beta0,alpha0,h,L,U) beta1=beta0; alpha1=alpha0;

%{ %

Ccell:

C1 1

Bcell:

{1} B1 (yn)^beta {2} B2 ((yn)^beta)*log(yn) {3} B3 ((yn)^beta)*[log(yn)]^2 {4} B4 Sum(yi) i=1,...,n

%}

if k==1 intout(1)=FirstFunctionC(0,L(k),U(k),beta0,alpha0); intout(2)=FirstFunctionB1(0,L(k),U(k),beta1,alpha1,beta0,alpha0); intout(3)=FirstFunctionB2(0,L(k),U(k),beta1,alpha1,beta0,alpha0); intout(4)=FirstFunctionB3(0,L(k),U(k),beta1,alpha1,beta0,alpha0); intout(5)=FirstFunctionB4(0,L(k),U(k),beta1,alpha1,beta0,alpha0);

else for j=k-1:-1:1; i=0; startcon=sum(L(1:j)); stopcon=sum(U(1:j)); for x=startcon:h:stopcon; i=i+1; if j==k-1 IndArray(i)=x; Ccell{1}(i)= FirstFunctionC(x,L(j+1),U(j+1),beta0,alpha0); Bcell{1}(i)=

FirstFunctionB1(x,L(j+1),U(j+1),beta1,alpha1,beta0,alpha0); Bcell{2}(i)=


FirstFunctionB3(x,L(j+1),U(j+1),beta1,alpha1,beta0,alpha0);

APPENDIX Q 288

Bcell{4}(i)=

FirstFunctionB4(x,L(j+1),U(j+1),beta1,alpha1,beta0,alpha0); else

INDICES=find(IndArray >= x+L(j+1) & IndArray <= x+U(j+1)); TmpIndArray=IndArray(INDICES);

TmpPArrayC=Ccell{1}(INDICES); TmpPArrayB1=Bcell{1}(INDICES); TmpPArrayB2=Bcell{2}(INDICES); TmpPArrayB3=Bcell{3}(INDICES); TmpPArrayB4=Bcell{4}(INDICES);

for r=1:numel(TmpIndArray);

TmpPArrayB4(r)=(log(TmpIndArray(r))*TmpPArrayC(r)+TmpPArrayB4(r))*(beta0*exp(

alpha0))*(TmpIndArray(r)^(beta0-1))*exp(exp(alpha0)*(x^beta0-

TmpIndArray(r)^beta0));

TmpPArrayC(r)=TmpPArrayC(r)*(beta0*exp(alpha0))*(TmpIndArray(r)^(beta0-

1))*exp(exp(alpha0)*(x^beta0-TmpIndArray(r)^beta0));

TmpPArrayB1(r)=TmpPArrayB1(r)*(beta0*exp(alpha0))*(TmpIndArray(r)^(beta0-






end;

Ccell{1}(i)=h*(TmpPArrayC(1)+4*sum(TmpPArrayC(2:2:(numel(TmpPArrayC)-

1)))+2*sum(TmpPArrayC(3:2:(numel(TmpPArrayC)-

2)))+TmpPArrayC(numel(TmpPArrayC)))/3;

Bcell{1}(i)=h*(TmpPArrayB1(1)+4*sum(TmpPArrayB1(2:2:(numel(TmpPArrayB1)-

1)))+2*sum(TmpPArrayB1(3:2:(numel(TmpPArrayB1)-

2)))+TmpPArrayB1(numel(TmpPArrayB1)))/3;








APPENDIX Q 289


2)))+TmpPArrayB4(numel(TmpPArrayB4)))/3; IndArray(i)=x;

end; % if j==k-1 end; % for x=startcon:h:stopcon;

end; %for j=k-1:-1:1;

INDICES=find(IndArray >= L(1) & IndArray <= U(1));


TmpIndArray=IndArray(INDICES);



alpha0))*(TmpIndArray(r)^(beta0-1))*exp(-1*exp(alpha0)*TmpIndArray(r)^beta0);


1))*exp(-1*exp(alpha0)*TmpIndArray(r)^beta0);







end intout(1)=h*(TmpPArrayC(1)+4*sum(TmpPArrayC(2:2:(numel(TmpPArrayC)-


2)))+TmpPArrayC(numel(TmpPArrayC)))/3; intout(2)=h*(TmpPArrayB1(1)+4*sum(TmpPArrayB1(2:2:(numel(TmpPArrayB1)-


2)))+TmpPArrayB1(numel(TmpPArrayB1)))/3; intout(3)=h*(TmpPArrayB2(1)+4*sum(TmpPArrayB2(2:2:(numel(TmpPArrayB2)-



APPENDIX Q 290

intout(4)=h*(TmpPArrayB3(1)+4*sum(TmpPArrayB3(2:2:(numel(TmpPArrayB3)-





end % if k==1 C=intout(1); B1=intout(2)/C; B2=intout(3)/C; B3=intout(4)/C; B4=intout(5)/C; end

function first_func=FirstFunctionC(prev_y,lower,upper,beta0,alpha0) if prev_y==0

F=@(x) (beta0.*exp(alpha0)).*(x.^(beta0-1)).*exp(-

1.*exp(alpha0).*x.^beta0);

else F=@(x) (beta0.*exp(alpha0)).*((prev_y+x).^(beta0-

1)).*exp(exp(alpha0).*(prev_y.^beta0-(prev_y+x).^beta0)); end; if upper ~= -1 first_func=quadgk(F,lower,upper); else first_func=quadgk(F,lower,inf); % Case of infinity upper limit end; end

% for B1(0)

function

first_func=FirstFunctionB1(prev_y,lower,upper,beta1,alpha1,beta0,alpha0) if prev_y==0

F=@(x) (x.^beta1).*(beta0.*exp(alpha0)).*(x.^(beta0-1)).*exp(-


else F=@(x) ((prev_y+x).^beta1).*(beta0.*exp(alpha0)).*((prev_y+x).^(beta0-

1)).*exp(exp(alpha0).*(prev_y.^beta0-(prev_y+x).^beta0));

end; if upper ~= -1 first_func=quadgk(F,lower,upper); else

APPENDIX Q 291

first_func=quadgk(F,lower,inf); % Case of infinity upper limit end; end

function


F=@(x) (x.^beta1).*log(x).*(beta0.*exp(alpha0)).*(x.^(beta0-1)).*exp(-


else F=@(x)

((prev_y+x).^beta1).*log(prev_y+x).*(beta0.*exp(alpha0)).*((prev_y+x).^(beta0

-1)).*exp(exp(alpha0).*(prev_y.^beta0-(prev_y+x).^beta0)); end; if upper ~= -1 first_func=quadgk(F,lower,upper); else first_func=quadgk(F,lower,inf); % Case of infinity upper limit end; end

function


F=@(x) (x.^beta1).*(log(x).^2).*(beta0.*exp(alpha0)).*(x.^(beta0-

1)).*exp(-1.*exp(alpha0).*x.^beta0);

else F=@(x)

((prev_y+x).^beta1).*(log(prev_y+x).^2).*(beta0.*exp(alpha0)).*((prev_y+x).^(

beta0-1)).*exp(exp(alpha0).*(prev_y.^beta0-(prev_y+x).^beta0)); end; if upper ~= -1 first_func=quadgk(F,lower,upper); else first_func=quadgk(F,lower,inf); % Case of infinity upper limit end; end

function


F=@(x) log(x).*(beta0.*exp(alpha0)).*(x.^(beta0-1)).*exp(-


else F=@(x) log(prev_y+x).*(beta0.*exp(alpha0)).*((prev_y+x).^(beta0-


APPENDIX Q 292

end; if upper ~= -1 first_func=quadgk(F,lower,upper); else first_func=quadgk(F,lower,inf); % Case of infinity upper limit end; end

Q.3.2. Estimating Parameters of a Power Law Process Using Complete EM

% This function calculates the parameters of a Power Law process using % complete EM. Several functions including C, B1,B2,B3, and B4 for each % iteration should be calculated, which is done by calling function % Simp_int_integral_GeneralLikelihoodEMX

function GN = generalLikeEM()

First2=true; First3=true; First4=true;

tStart2=tic; tStart3=tic; tStart4=tic; tStart5=tic;

Params(1,1)=-0.7; Params(1,2)=1;

Data = xlsread('C:\Paper 5 new\excel\ChassisSubset164.xls')





APPENDIX Q 293


NoofSystems=UnitNo-1;

l=1; AllParams(l,:)=Params(1,:); ContWhile=true;

beta0=Params(1,2); alpha0=Params(1,1);

beta1=beta0; alpha1=alpha0;

Oncemore=0; while ContWhile %ContWhile

if Oncemore==1 ContWhile=false; end;

Params(1,2)=beta1; Params(1,1)=alpha1; % N(1)='Ci'; % N(2)='(yn)^beta'; % N(3)='((yn)^beta)*log(yn)'; % N(4)='((yn)^beta)*[log(yn)]^2';

Sumni=0;



[C,B1,B2,B3,B4,B5,B6,B7,B8,B9,B10] =

Simp_int_integral_GeneralLikelihoodEMX(numel(L{i}),beta0,alpha0,beta1,alpha1,

0.001,L{i},U{i});

N(1,i)=C; N(2,i)=B1; N(3,i)=B2; N(4,i)=B3; N(5,i)=B4;

end % Construct Q' array

APPENDIX Q 294

Q_Prime(1,1)=Sumni-exp(alpha1)*sum(N(2,:)); Q_Prime(2,1)=(1/beta1)*Sumni+sum(N(5,:))-exp(alpha1)*sum(N(3,:));

% Construct Q'' matrix Q_Second(1,1)=-exp(alpha1)*sum(N(2,:)); Q_Second(1,2)=-exp(alpha1)*sum(N(3,:)); Q_Second(2,1)=Q_Second(1,2); Q_Second(2,2)=(-1/(beta1^2))*Sumni-exp(alpha1)*sum(N(4,:));


l=l+1;

if NewParams(2,1) < 0 NewParams(2,1)=0.1; end

AllParams(l,1)=NewParams(1,1); AllParams(l,2)=NewParams(2,1);

AllParams(l,7)=Q_Prime(1,1); AllParams(l,8)=Q_Prime(2,1);

if (abs(alpha1-NewParams(1,1)) < 10^-4) & (abs(beta1-NewParams(2,1)) < 10^-4)

% Interna' AllParams(l,3)=abs(NewParams(1,1)-alpha0); AllParams(l,4)=abs(beta0-NewParams(2,1)); AllParams(l,9)=toc(tStart5); if (abs(alpha0-NewParams(1,1)) < 10^-5) & (abs(beta0-NewParams(2,1)) <

10^-5) % External Oncemore=1;

else beta0=NewParams(2,1); alpha0=NewParams(1,1); beta1=beta0; alpha1=alpha0; end else AllParams(l,5)=abs(NewParams(1,1)-alpha1); AllParams(l,6)=abs(beta1-NewParams(2,1));

beta1=NewParams(2,1); alpha1=NewParams(1,1); end

end; % while GN=N; xlswrite('C:\Paper 5 new\excel\NewSet\EM\Expected-ChassisSubset164-5-

4.xls',N); % first external-then internal

APPENDIX Q 295

xlswrite('C:\Paper 5 new\excel\NewSet\EM\AllParams-ChassisSubset164-5-

4.xls',AllParams);

end

% This function calculates the recursive functions required to estimate the % parameters of a Power Law process using complete EM. The function which % are calculaed here C1, B1, B2, B3, and B4 with the following definitions % C1 1 % B1 (yn)^beta % B2 ((yn)^beta)*log(yn) % B3 ((yn)^beta)*[log(yn)]^2 % B4 Sum(yi) i=1,...,n

function [C,B1,B2,B3,B4]

=Simp_int_integral_GeneralLikelihoodEMX(k,beta0,alpha0,beta1,alpha1,h,L,U)

%{ %

Ccell:

C1 1 Bcell:

{1} B1 (yn)^beta {2} B2 ((yn)^beta)*log(yn) {3} B3 ((yn)^beta)*[log(yn)]^2 {4} B4 Sum(yi) i=1,...,n

%}

if k==1 intout(1)=FirstFunctionC(0,L(k),U(k),beta0,alpha0); intout(2)=FirstFunctionB1(0,L(k),U(k),beta1,alpha1,beta0,alpha0); intout(3)=FirstFunctionB2(0,L(k),U(k),beta1,alpha1,beta0,alpha0); intout(4)=FirstFunctionB3(0,L(k),U(k),beta1,alpha1,beta0,alpha0); intout(5)=FirstFunctionB4(0,L(k),U(k),beta1,alpha1,beta0,alpha0);

else for j=k-1:-1:1; i=0; startcon=sum(L(1:j)); stopcon=sum(U(1:j)); for x=startcon:h:stopcon; i=i+1; if j==k-1 IndArray(i)=x; Ccell{1}(i)= FirstFunctionC(x,L(j+1),U(j+1),beta0,alpha0);

APPENDIX Q 296

Bcell{1}(i)=




FirstFunctionB4(x,L(j+1),U(j+1),beta1,alpha1,beta0,alpha0); else

INDICES=find(IndArray >= x+L(j+1) & IndArray <= x+U(j+1)); TmpIndArray=IndArray(INDICES);

TmpPArrayC=Ccell{1}(INDICES); TmpPArrayB1=Bcell{1}(INDICES); TmpPArrayB2=Bcell{2}(INDICES); TmpPArrayB3=Bcell{3}(INDICES); TmpPArrayB4=Bcell{4}(INDICES); for r=1:numel(TmpIndArray);


alpha0))*(TmpIndArray(r)^(beta0-1))*exp(exp(alpha0)*(x^beta0-

TmpIndArray(r)^beta0));









end;

Ccell{1}(i)=h*(TmpPArrayC(1)+4*sum(TmpPArrayC(2:2:(numel(TmpPArrayC)-


2)))+TmpPArrayC(numel(TmpPArrayC)))/3;







APPENDIX Q 297







IndArray(i)=x;

end; % if j==k-1 end; % for x=startcon:h:stopcon;

end; %for j=k-1:-1:1;

INDICES=find(IndArray >= L(1) & IndArray <= U(1));


TmpIndArray=IndArray(INDICES);



alpha0))*(TmpIndArray(r)^(beta0-1))*exp(-1*exp(alpha0)*TmpIndArray(r)^beta0);









end intout(1)=h*(TmpPArrayC(1)+4*sum(TmpPArrayC(2:2:(numel(TmpPArrayC)-


2)))+TmpPArrayC(numel(TmpPArrayC)))/3; intout(2)=h*(TmpPArrayB1(1)+4*sum(TmpPArrayB1(2:2:(numel(TmpPArrayB1)-



APPENDIX Q 298

intout(3)=h*(TmpPArrayB2(1)+4*sum(TmpPArrayB2(2:2:(numel(TmpPArrayB2)-






2)))+TmpPArrayB4(numel(TmpPArrayB4)))/3; end % if k==1 C=intout(1); B1=intout(2)/C; B2=intout(3)/C; B3=intout(4)/C; B4=intout(5)/C; end

function first_func=FirstFunctionC(prev_y,lower,upper,beta0,alpha0) if prev_y==0

F=@(x) (beta0.*exp(alpha0)).*(x.^(beta0-1)).*exp(-


else F=@(x) (beta0.*exp(alpha0)).*((prev_y+x).^(beta0-


% for B1(0)

function


F=@(x) (x.^beta1).*(beta0.*exp(alpha0)).*(x.^(beta0-1)).*exp(-


else F=@(x) ((prev_y+x).^beta1).*(beta0.*exp(alpha0)).*((prev_y+x).^(beta0-


end; if upper ~= -1

APPENDIX Q 299

first_func=quadgk(F,lower,upper); else first_func=quadgk(F,lower,inf); % Case of infinity upper limit end; end

function

first_func=FirstFunctionB2(prev_y,lower,upper,beta1,alpha1,beta0,alpha0) if prev_y==0 F=@(x) (x.^beta1).*log(x).*(beta0.*exp(alpha0)).*(x.^(beta0-1)).*exp(-


else F=@(x)

((prev_y+x).^beta1).*log(prev_y+x).*(beta0.*exp(alpha0)).*((prev_y+x).^(beta0

-1)).*exp(exp(alpha0).*(prev_y.^beta0-(prev_y+x).^beta0)); end; if upper ~= -1 first_func=quadgk(F,lower,upper); else first_func=quadgk(F,lower,inf); % Case of infinity upper limit end; end

function


F=@(x) (x.^beta1).*(log(x).^2).*(beta0.*exp(alpha0)).*(x.^(beta0-

1)).*exp(-1.*exp(alpha0).*x.^beta0);

else F=@(x)

((prev_y+x).^beta1).*(log(prev_y+x).^2).*(beta0.*exp(alpha0)).*((prev_y+x).^(

beta0-1)).*exp(exp(alpha0).*(prev_y.^beta0-(prev_y+x).^beta0)); end; if upper ~= -1 first_func=quadgk(F,lower,upper); else first_func=quadgk(F,lower,inf); % Case of infinity upper limit end; end

function


F=@(x) log(x).*(beta0.*exp(alpha0)).*(x.^(beta0-1)).*exp(-


else

APPENDIX Q 300

F=@(x) log(prev_y+x).*(beta0.*exp(alpha0)).*((prev_y+x).^(beta0-


Q.3.3. Calculating Information Matrix and Standard Errors

% This function calculates the InformationMatrix and the standard erors of % the parameters which are estimated using any methods (modified EM, % complete EM, Newton Raphson, and Simplex)

function GN = InformationMatrix()

alpha=-0.140950133; beta=0.915588406;

Data = xlsread('C:\Paper 5 new\excel\ChassisSubset164.xls')







APPENDIX Q 301

Sumni=0;



[C,B1,B2,B3,B4,B5,B6,B7,B8,B9,B10] =

Simp_int_integral_GeneralLikelihood(numel(L{i}),beta,alpha,0.001,L{i},U{i}); % [C,B1,B2,B3,B4,B5,B6,B7,B8,B9,B10] =

Simp_int_integral_GeneralLikelihoodEM(numel(L{i}),beta0,alpha0,beta1,alpha1,0

.001,L{i},U{i});

N(1,i)=C; N(2,i)=B1; N(3,i)=B2; N(4,i)=B3; N(5,i)=B4; N(6,i)=B5; N(7,i)=B6; N(8,i)=B7; N(9,i)=B8; N(10,i)=B9; N(11,i)=B10; end

% Construct Q' array Q_Prime(1,1)=Sumni-exp(alpha)*sum(N(2,:)); Q_Prime(2,1)=(1/beta)*Sumni+sum(N(5,:))-exp(alpha)*sum(N(3,:));

% Construct Q'' matrix Q_Second(1,1)=-exp(alpha)*sum(N(2,:)); Q_Second(1,2)=-exp(alpha)*sum(N(3,:)); Q_Second(2,1)=Q_Second(1,2); Q_Second(2,2)=(-1/(beta^2))*Sumni-exp(alpha)*sum(N(4,:));

Variance(1,1)=exp(2*alpha)*sum(N(8,:))-exp(2*alpha)*sum(N(2,:).^2); Variance(1,2)=-

exp(alpha)*sum(N(11,:))+exp(alpha)*sum(N(2,:).*N(5,:))+exp(2*alpha)*sum(N(10,

:))-exp(2*alpha)*sum(N(2,:).*N(3,:)); Variance(2,1)=Variance(1,2); Variance(2,2)=sum(N(7,:))-2*exp(alpha)*sum(N(6,:))+exp(2*alpha)*sum(N(9,:))-

sum(N(5,:).^2)+2*exp(alpha)*sum(N(5,:).*N(3,:))-exp(2*alpha)*sum(N(3,:).^2);

Q_Second_Complete=Q_Second+Variance;

FisherReverse=inv(-1*Q_Second_Complete);

APPENDIX Q 302

Error(1,1)=sqrt(FisherReverse(1,1)); Error(2,1)=sqrt(FisherReverse(2,2)); GN=Error; xlswrite('C:\Paper 5 new\excel\NewSet\OneStep\InfoMatrix\Error-OneNR-

ChassisSubset164-3.xls',GN); xlswrite('C:\Paper 5 new\excel\NewSet\OneStep\InfoMatrix\QP-OneNR-

ChassisSubset164-3.xls',Q_Prime); end

Q.3.4. Calculating the Parameters Using Mid-Points

% This function estimates the parameters which maxime the likelihood with % the assumption that the midpoints of failure intervals are exact failure % times. The input file is an excel sheet where an exact failure time % is given as an interval with the same lower and upper bounds. The last % right censoring interval is specified as an interval with the lower bound % equal to the right censoring length, and the upper bound equal to -1.

function GN = generalLike_Mid()

Params(1,2)=0.5; Params(1,1)=-0.3;

Data = xlsread('C:\Paper 5 new\excel\ChassisSubset164-Mid.xls')






APPENDIX Q 303

NoofSystems=UnitNo-1;%3;

l=1; AllParams(l,:)=Params(1,:); ContWhile=true;


Sumni=0;


if U{numel(L{i})} ~= -1 Sumni=Sumni+numel(L{i}); else Sumni=Sumni+numel(L{i})-1; end;

[C,B1,B2,B3,B4,logC]

=Simp_int_integral_GeneralLikelihood_Midpoints(Params(1,2),Params(1,1),L{i},U

{i});

N(1,i)=C; N(2,i)=B1; N(3,i)=B2; N(4,i)=B3; N(5,i)=B4; N(6,i)=logC;

end

% Construct Q' array Q_Prime(1,1)=Sumni-exp(Params(1,1))*sum(N(2,:)); Q_Prime(2,1)=(1/Params(1,2))*Sumni+sum(N(5,:))-exp(Params(1,1))*sum(N(3,:));

% Construct Q'' matrix Q_Second(1,1)=-exp(Params(1,1))*sum(N(2,:)); Q_Second(1,2)=-exp(Params(1,1))*sum(N(3,:)); Q_Second(2,1)=Q_Second(1,2); Q_Second(2,2)=(-1/(Params(1,2)^2))*Sumni-exp(Params(1,1))*sum(N(4,:));


l=l+1; AllParams(l,:)=NewParams(:,1)';

if NewParams(2,1) < 0 NewParams(2,1)=0.1;

APPENDIX Q 304

end

if (abs(Params(1,1)-NewParams(1,1)) < 0.001) & (abs(Params(1,2)-

NewParams(2,1)) < 0.001) ContWhile=false; end


end; % while GN=N; xlswrite('C:\Paper 5 new\excel\ExpectedChassisSubset164-Mid-New.xls',N); xlswrite('C:\Paper 5 new\excel\AllParamsChassisSubset164-Mid-

New.xls',AllParams);

end

Q.3.5. Estimating Parameter When 1

% This function calculates iteratively the parameter alpha given that beta=1

function GN = generalLike_Beta1_2()

Params(1,2)=1; Params(1,1)=-0.3;

Data = xlsread('C:\Paper 5 new\excel\AudibleSubset125.xls')





APPENDIX Q 305



l=1;

AllParams(l,1)=Params(1,1);

ContWhile=true;


Sumni=0;



[C,B1,logC] =

Simp_int_integral_GeneralLikelihoodBeta1_exact(Params(1,1),L{i},U{i});

N(1,i)=C; N(2,i)=B1; N(3,i)=logC;

end

Q_Prime=Sumni-exp(Params(1,1))*sum(N(2,:));

NewParams(1,1)= log(Sumni/sum(N(2,:))); % alpha NewParams(2,1)=1; % beta

l=l+1;

AllParams(l,2)=NewParams(1,1); AllParams(l,3)=Q_Prime;

if (abs(Params(1,1)-NewParams(1,1)) > 0.3) NewParams(1,1)=Params(1,1)-0.3; end

if (abs(Params(1,1)-NewParams(1,1)) < 10^-7)

APPENDIX Q 306

ContWhile=false; end


end; % while GN=N; xlswrite('C:\Paper 5 new\excel\Expected-beta1exactAudibleSubset125-10-

7.xls',N); xlswrite('C:\Paper 5 new\excel\AllParams-beta1exactAudibleSubset125-10-

7.xls',AllParams); end

% This function calculates C, B1 and LogC which is required to find % parameter alpha in the case when beta=1

function [C,B1,logC]

=Simp_int_integral_GeneralLikelihoodBeta1_exact(alpha0,L,U)

lambda=exp(alpha0);

C=1; B1=0; logC=0; for i=1:numel(L); if U(i) ~= -1 Denom=exp(-1*exp(alpha0)*L(i))-exp(-1*exp(alpha0)*U(i)); else Denom=exp(-1*exp(alpha0)*L(i)); end;

C=C*Denom; logC=logC+log(Denom);

if U(i) ~= -1 B1=B1+(1/lambda)+(L(i)*exp(-lambda*L(i))-U(i)*exp(-

lambda*U(i)))/(exp(-lambda*L(i))-exp(-lambda*U(i))); else B1=B1+(1/lambda)+L(i); end;

end;

end

307

INDEX

A

absolute comparison, 28

age dependent probabilities, 5, 128, 154, 160, 168, 169,

176, 177, 182

AHP. See Analytical Hierarchy Process

Analytical Hierarchy Process, 27, 179

Availability, xvi, 32, 33, 34, 35, 38, 188, 206, 209

C

classification, iii, 4, 33, 34, 37, 39, 53, 54, 68, 98, 190

CMMS. See Computerized Maintenance Management

System

Computerized Maintenance Management System, 3, 188

corrective maintenance, 13, 14, 16, 21, 55, 67, 68, 104

Criteria, iii, xi, xii, xiv, xxi, 3, 13, 14, 20, 27, 28, 29, 30, 31,

32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 52, 53, 54,

55, 56, 57, 58, 109, 179, 180, 183, 185, 194, 204, 205,

209

Criticality score, 29, 30, 31, 42, 43, 56, 58

D

delayed replacement, 5, 177, 182

detectability, xvi, 25, 26, 35, 36, 52, 58, 179, 207

downtime, xv, xix, 5, 55, 105, 109, 117, 118, 121, 126, 127,

128, 138, 139, 140, 141, 144, 145, 146, 150, 152, 154,

175, 177, 182, 246, 248, 249, 260

F

failure consequence, 42, 57, 121, 122

Failure frequency, xvi, 35, 38, 206, 209

failure mode, 25, 35, 36, 40, 41, 142, 179, 189

Failure Mode and Effect Analysis, xxiv, 25

failure rate, 55, 60, 62, 102, 104, 106, 108

failures data, 73, 74

FDA. See Food and Drug Administration

Finite Time Horizon, xii, 128, 245, 252

FMEA. See Failure Mode and Effect Analysis

Food and Drug Administration, 37

G

grade, 30, 31, 39, 40, 41, 43, 204

INDEX 308

H

Hard failure, xiv, xv, xx, 5, 67, 68, 69, 77, 81, 122, 141, 144,

145, 146, 147, 150, 151, 154, 160, 161, 163, 166, 168,

169, 171, 173, 174, 177, 182, 184, 213, 247, 249, 283

hazard rate, 101

hidden failures, 4, 16, 57, 62, 90, 101, 103, 104

I

infinite time horizon, 104, 136, 176, 246

infusion pump, 4, 21, 22, 63, 78, 90, 92, 94, 98, 99, 181

inspection interval, iii, xix, 110, 111, 115, 130, 133, 134,

135, 136, 139, 144, 146, 147, 148, 149, 150, 151, 153,

157, 159, 161, 165, 169, 175, 176, 177, 182, 245, 246,

252, 255, 256

intensity, xv, 4, 30, 31, 39, 54, 61, 62, 81, 82, 98, 110, 115,

116, 134, 142, 145, 151, 166, 181, 204, 209

intensity function, 61, 145, 161

J

JACAHO. See Joint Commission on Accreditation of

Healthcare Organization

Joint Commission on Accreditation of Healthcare

Organizations, xxiii, 2, 26, 191

L

Laplace trend test, 72, 73, 75, 76, 77, 78, 90, 98

likelihood ratio, 4, 73, 81, 82, 83, 92, 93, 96, 98, 181, 251

M

MADM. See Multi-Attribute Decision Making

maintenance requirement, 25, 37

Maintenance requirements, xvi, 33, 37, 58, 179, 208

maximum likelihood, 61, 62, 82, 86, 89, 90

MCDM. See Multi-Criteria Decision Making

Medical Equipment Management Program, xxiii, 1, 9, 24

MEMP. See Medical Equipment Management Program,

See Medical Equipment Management Program

minimal repair, xv, 5, 78, 101, 102, 106, 108, 110, 111, 112,

115, 118, 129, 131, 132, 134, 143, 144, 150, 152, 154,

155, 158, 161, 176, 177, 182, 186, 187, 196, 222, 247,

252, 256, 268, 269, 270, 273, 276, 277, 281, 282

Mission criticality, xiv, 32, 33, 34, 38, 46, 58, 179, 209

Multi-Attribute Decision Making, xxiii, 27

Multi-Criteria Decision Making, xxiii, 27, 200

N

Newton-Raphson, 81, 87, 90, 96, 183

NHPP. See Non-homogenous Poisson Process

non-homogenous Poisson process, 4, 72, 98, 110, 142,

176, 181

non-scheduled inspections, xix, 5, 101, 141, 142, 144, 169,

170, 181

O

Opportunistic inspection, 104, 177, 181

opportunistic maintenance, xiii, xxi, 64, 104, 105, 141, 175,

177, 182, 202, 231, 247, 268, 271, 272, 278, 280

P

pairwise comparison, 28, 29, 30, 53, 58, 59, 183

penalty cost, xix, 110, 114, 117, 118, 119, 125, 126, 128,

129, 134, 140, 144, 151, 153, 166, 176, 182

periodic inspection, 4, 5, 23, 100, 104, 106, 107, 128, 133,

144, 153, 175, 177, 181

Periodic preventive maintenance, 100

placeholder function, 131, 138, 148, 155, 170

Poisson process, xxiii, 61, 83

power law, xv, 4, 61, 62, 81, 82, 98, 115, 116, 134, 150,

151, 166, 173, 181

INDEX 309

predictive maintenance, 14, 55, 56

preventive replacement, 105

Prioritization, iii, xviii, 3, 27, 29, 30, 31, 37, 42, 43, 53, 54,

179, 180

Proactive maintenance, 56

probability of survival, 132, 252

R

ranking, 28, 30, 39, 53, 196

RCM. See Reliability Centered Maintenance

recall, 27

relative measurement, 28, 31, 53, 58, 179, 197

Reliability Centered Maintenance, xxiv, 25

replacement, xx, 5, 15, 16, 18, 21, 100, 102, 103, 105, 106,

108, 129, 131, 132, 134, 136, 154, 155, 157, 159, 160,

161, 167, 168, 169, 170, 173, 174, 176, 177, 182, 186,

187, 188, 189, 190, 191, 192, 194, 198, 222, 247, 269,

270, 275, 276, 277, 280, 281, 282, 283

Risk, viii, xiv, 26, 32, 33, 35, 40, 43, 44, 46, 49, 57, 58, 179,

187, 194, 196, 197, 199, 201

risk assessment, xviii, 26, 32, 122

S

scheduled inspection, 26, 80, 105, 108, 110, 111, 117, 118,

128, 129, 141, 142, 143, 144, 146, 151, 168

self-announcing, 5, 67, 78, 79, 175, 180

Soft failure, xv, xvii, 5, 68, 71, 78, 79, 108, 109, 110, 122,

142, 144, 145, 146, 147, 148, 150, 152, 155, 157, 158,

169, 173, 177, 181, 184, 213, 232, 235, 247, 248, 249,

261

SPI. See Safety and Performance Inspection

T

threshold, xix, 5, 103, 105, 109, 117, 118, 122, 123, 125,

126, 127, 175

time-based maintenance, 54, 55, 56

Trend analysis, iii, 3, 4, 23, 61, 62, 73, 81, 83, 90, 98, 180,

183

U

Utilization, xvi, 32, 33, 34, 38, 45, 57, 206, 209

W

Weibull, xv, xvi, xvii, 61, 62, 75, 76, 188, 216, 217, 218

work order, xviii, 19, 20, 21, 22, 64, 67, 68, 69, 213

reliability and maintenance of medical devices · reliability and maintenance of medical devices...

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