FAILURE FINDING INTERVAL OPTIMIZATION FOR PERIODICALLY INSPECTED REPAIRABLE SYSTEMS

Transcription

1 ALURE NDNG NERVAL OPMZAON OR PERODCALLY NSPECED REPARABLE SYSEMS by ianqiao ang A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mechanical and ndustrial Engineering University of oronto Copyright by ianqiao ang, 212

2 ALURE NDNG NERVAL OPMZAON OR PERODCALLY NSPECED REPARABLE SYSEMS ianqiao ang Doctor of Philosophy Department of Mechanical and ndustrial Engineering University of oronto 212 ABSRAC he maintenance of equipment has been an important issue for companies for many years or systems with hidden or unrevealed failures (ie, failures are not self-announcing, a common practice is to regularly inspect the system looking for such failures Examples of these systems include protective devices, emergency devices, standby units, underwater devices etc f no periodical inspection is scheduled, and a hidden failure has already occurred, severe consequences may result Research on periodical inspection seeks to establish the optimal inspection interval (ailure inding nterval of systems to maximize availability and/or minimize expected cost Research also focuses on important system parameters such as unavailability Most research in this area considers non-negligible downtime due to repair/replacement but ignores the downtime caused by inspections n many situations, however, inspection time is non-negligible e address this gap by proposing an optimal failure finding interval ( considering both non-negligible inspection time and repair/replacement time A novel feature of this work is the development of models for both age-based and calendar-based inspection policies with ii

3 random/constant inspection time and random/constant repair/replacement time More specifically, we first study instantaneous availability for constant inspection and repair/replacement times e start with the assumption of renewal of the system at each inspection e then consider models with the assumption of renewal only after failure e also develop limiting average availability models for random inspection and repair/replacement times, considering both age-based and calendar-based inspection policies e optimize these availability models to obtain an optimal in order to maximize the system s availability inally, we develop several cost models for both age-based and calendar-based inspection policies with random inspection time and repair/replacement time e formulate the model for constant inspection time and repair/replacement time as a special case e investigate the optimization of cost models for each case to obtain optimal in order to minimize the expected cost he numerical examples and case study presented in the dissertation demonstrate the importance of considering non-negligible downtime due to inspection iii

4 o My ather ( iv

5 ACKNOLEDGEMENS offer my sincerest gratitude to my supervisors, Professor Andrew KS Jardine and Dr Dragan Banjevic for giving me the opportunity to do my PhD with them and for their invaluable guidance, support and encouragement During the nearly ten years since my Master s work with them, they have given me consideration and support which has helped me to overcome all difficulties am fortunate to have them am very thankful to Professor Daoud Ait-Kadi (from Laval University, Professor Baris Balcioglu and Professor Birsen Donmez (from University of oronto who served as members of my examination committee and provided me with invaluable comments and suggestions on my research am grateful to Dr Daming Lin for his friendship and assistance which helped me to succeed wish to express my sincere appreciation to Professor Jianyong Li and Professor Shouguang Sun for their support ithout their kind encouragement and help, would not have finished my PhD Gratitude is extended to a great group of friends and colleagues at C-MORE lab, especially Dr v

6 Elizabeth hompson for her editorial assistance and comments and, Sharareh aghipour for her friendship and help am grateful to the company who provided the data set for my research hanks also go to my colleagues and friends in China for their valuable support and assistance Last, but not least, would like to express my deepest gratitude to my family and my parents who always believed in me and encouraged me during these past years Especially thanks to my dear daughter, whose smile has always been a source of inspiration am fortunate to have them and would like to share my moment of joy with them ianqiao ang ebruary 212 vi

7 PREACE he following papers have been published based on work reported in the thesis: 1 ang, Lin D, Banjevic D, Jardine AKS (211 Errata to "Availability of a periodically inspected system under perfect repair" [J Statist Plann nference 91 (2 77-9] Journal of Statistical Planning nference 141:61 2 ang, Lin D, Banjevic D, Jardine AKS Availability of a periodically inspected system with non-negligible downtimes due to inspection and repair/replacement Journal of Statistical Planning nference (Under 2 nd round of review after minor revision 3 ang, Banjevic D and Jardine AKS (28 Design of a reliability knowledge database model Proceeding of the 28 nternational ndustrial Engineering Research Conference: ang, Banjevic D and Jardine AKS Optimal failure finding interval to maximize availability Reliability Engineering and System Safety (Submitted in April ang, Banjevic D and Jardine AKS Optimal failure finding interval to minimize cost (Pending for submission vii

8 ABLE O CONENS 1 NRODUCON 1 2 LERAURE REVE 7 21 System availability and availability optimization System availability and unavailability models Optimal inspection interval to maximize availability Optimal inspection interval to minimize cost Other research on periodic inspection Concluding Remarks 25 3 AVALABLY MODEL AND OPMZAON ntroduction nstantaneous availability model formulation Age-based instantaneous availability with renewal at each inspection (A Age-based instantaneous availability with renewal after failure (B Calendar-based instantaneous availability with renewal at each inspection (A Calendar-based instantaneous availability with renewal after failure (B Limiting average availability model with age-based inspection policy 42 viii

9 ABLE O CONENS 331 Age-based limiting average availability with renewal at each inspection (A Age-based availability optimization with renewal at each inspection (A Age-based limiting average availability with renewal after failure (B Age-based availability optimization with renewal after failure (B Optimal to maximize availability with renewal after failure (B Optimal for system with lifetime following exponential distribution Limiting average availability model with calendar-based inspection policy Calendar-based limiting average availability with renewal at each inspection (A Calendar-based availability optimization with renewal at each inspection (A Calendar-based limiting average availability with renewal after failure (B Calendar-based limiting average availability formulation (B Extension of calendar-based model assuming downtime longer than Calendar-based availability for system with lifetime following exponential distribution Calendar-based availability optimization with renewal after failure (B Numerical Example 7 36 Concluding Remarks 74 4 OPMAL NSPECON NERVAL O MNMZE COS ntroduction Cost model with age-based inspection policy Age-based cost model with renewal at each inspection (A Age-based cost optimization with renewal at each inspection (A Age-based cost model with renewal after failure (B 9 ix

10 ABLE O CONENS 424 Age-based cost optimization with renewal after failure (B Optimal age-based to minimize cost with renewal after failure (B Optimal for system with lifetime following exponential distribution Cost model with Calendar-based inspection policy Calendar-based cost model with renewal at each inspection (A Calendar-based cost optimization with renewal at each inspection (A Calendar-based cost model under assumption with renewal after failure (B Calendar-based cost model formulation (B Calendar-based cost model for system with lifetime following exponential distribution Calendar-based cost optimization with renewal after failure (B Numerical Example Another cost model with Age-based inspection policy Concluding Remarks CASE SUDES Case study for a thermal power plant ntroduction Optimal inspection interval for safe valve Case study for a mining and refining company Concluding Remarks CONCLUSON AND UURE RESEARCH Conclusion uture Research 142 x

11 ABLE O CONENS 7 REERENCES APPENDCES 155 APPEND A Properties of function g ( 155 APPEND B Properties of function g( a, b 158 APPEND C MALAB Program for numerical examples on availability optimization 16 APPEND D MALAB Program for numerical examples on cost optimization 166 APPEND E MALAB Program for case studies 173 xi

12 LS O ABLES able 21 he overall structure of the research 27 able 31 Optimal with renewal in each inspection (A71 able 32 Overall optimal vs maximum availability 73 able 51 Parameters of Pressure Safety Valves for Boiler steam drum 127 able 52 Parameters of Super heater (SH safety valves 128 able 53 Parameters of Pressure Control Valve (PCV 128 xii

13 LS O GURES igure 11 Classification of literature on inspection interval for hidden failure system 3 igure 31 Calendar-based and age-based inspection policies 31 igure 32 Structure of availability models 33 igure 33 wo different assumptions for age-based inspection policy 34 igure 34 wo different assumptions for calendar-based inspection policy 38 igure 35 Relationship between functiong 1 (, G 2 ( and G( 5 igure 36 he upper bound of optimal 52 igure 37 Cycle length under assumption A with calendar-based inspection policy 53 igure 38 Cycle length under assumption B with calendar-based inspection policy 62 igure 39 Relationship between functiong 1 (, G 2 ( and G( 69 igure 31 Age-based availability vs with 5 (A 7 igure 311 Age-based availability vs with 1 (A 7 igure 312 Age-based availability vs with 2 (A 71 igure 313 Age-based availability vs with 5 (B 72 igure 314 Age-based availability vs with 1 (B 72 igure 315 Age-based availability vs with 2 (B 73 xiii

14 igure 316 Age-based availability vs with 4 (B73 igure 41 wo types of system with hidden failure 78 igure 42 Structure of cost models 79 igure 43 Possible cost function vs 87 igure 44 Relationship between functionc(, G1(, G2( 95 igure 45 Possible relationship between function C(, G1 (, G2 ( and G3 ( 16 igure 46 Possible relations between C(, G1(, G2( 114 igure 47 Age-based cost function for igure 48 Age-based cost function for igure 49 Cost function when C L C C R R 117 igure 41 Cost function under assumption A 119 igure 411 Cost function with a smaller 119 igure 412 Cost function without local when CL C 12 igure 51 Safety valve for the boiler system 128 igure 52 Limiting average availability vs optimal for safety valve 13 igure 53 Cost function vs optimal for safety valve 132 igure 54 Limiting average availability over failure finding interval ( 134 igure 55 nstantaneous availability over time 135 igure56 Limiting average availability for 136 igure A1 Relationship between R( t dt and g ( 156 xiv

15 igure B1 Relationship between R( t dt and g( a, b 159 xv

16 LS O MODELS e list limiting average availability models and cost models in the table Limiting average availability model age-based, assumption A, random inspection time & repair/replacement time Section 331 (38 R ( t dt P44 [1 ( ] A Y R Limiting average availability model age-based, assumption A, constant inspection time, random repair/replacement time Section 331 (39 R( t dt P44 [1 ( ] A Y R Limiting average availability model age-based, assumption A, constant inspection time & repair/replacement time Section 331 (31 R( t dt P44 [1 ( ] A R R Limiting average availability model age-based, assumption B, random inspection time & repair/replacement time Section 333 (313 A, where g ( R( i P47 ( g( i Y xvi

17 LS O MODELS Limiting average availability model age-based, assumption B, constant inspection time, random repair/replacement time Section 333 (314 Limiting average availability model A P47 ( g( age-based, assumption B, constant inspection time & repair/replacement time Section 333 (315 A ( g( Y R P47 Limiting average availability model calendar-based, assumption A, random inspection time & repair/replacement time Section341 (32 E[min{, }] ER( Y E[min{, Y}] E( A ER( Y [1 ] E( P56 Limiting average availability model calendar-based, assumption A, constant inspection time, random repair/replacement time Section341 (321 E[min{, }] ER( Y E[min{, Y}] E( A ER( Y [1 ] E( P57 Limiting average availability model calendar-based, assumption A, constant inspection time & repair/replacement time Section341 (322 A R( R R( t dt R( t dt ( R( R [1 ] ( R P57 Limiting average availability model calendar-based, assumption B, random inspection time & repair/replacement time xvii

18 LS O MODELS Section3431 (327 i A, where g( E[ R( ( (1 g ( j Y ] P64 i j Limiting average availability model calendar-based, assumption B, constant inspection time, random repair/replacement time Section3431 (328 Limiting average availability model i A, where Eg( E[ R( ( (1 Eg ( Y ] P64 i j calendar-based, assumption B, constant inspection time & repair/replacement time Section3431 (329 Cost model A, where (1 g ( g( R[ i( R] P65 age-based, assumption A, random inspection time & repair/replacement time Section421 (41 Cost model C C( Y i ~ CRY ( CL ( t dt P84 ( age-based, assumption A, constant inspection time, random repair/replacement time Section421 (42 C C ( C ( t dt P84 ( R Y L ( C Y Cost model age-based, assumption A, constant inspection time & repair/replacement time Section421 (43 C C ( C ( t dt P84 ( R R L ( C R Cost model age-based, assumption B, random inspection time & repair/replacement time Section423 (45 C g( CRY CL[( g( Y ] C ( ( g( Y P91 xviii

19 LS O MODELS Cost model age-based, assumption B, constant inspection time, random repair/replacement time Section423 (46 C g( CRY CL[( g( Y ] C ( ( g( Y P91 Cost model age-based, assumption B, constant inspection time & repair/replacement time Section423 (47 C g( CRR CL[( g( R ] C ( ( g( R P92 Cost model calendar-based, assumption A, random inspection time & repair/replacement time Section431 (415 C CRY CLD ( C ( ER( Y E( [1 ], where E max{, } ER( Y D ( E max{, Y } ER( P1 Cost model calendar-based, assumption A, constant inspection time, random repair/replacement time Section431 (416 C CRY CLD ( C ( ER( Y ( [1 ], where ER( Y D E Y t dt ( max{, } ( R( P11 Cost model calendar-based, assumption A, constant inspection time & repair/replacement time Section431 (417 R R( R CRR CL[ ( t dt ( t dt] C ( C ( P11 R( R [1 ] ( xix

20 LS O MODELS Cost model calendar-based, assumption B, random inspection time & repair/replacement time Section4331 (421 Cost model C (1 g( CRY CL CL(1 g( C ( P19 (1 g( calendar-based, assumption B, constant inspection time, random repair/replacement time Section4331 (422 Cost model C (1 Eg( CRY CL CL(1 Eg( C ( P19 (1 Eg( calendar-based, assumption B, constant inspection time & repair/replacement time Section4331 (423 C (1 g( CRR CL CL(1 g( C ( P19 (1 g( Cost model age-based, different assumption, constant inspection time & repair/replacement time Section45 (43 C CRR ( ( CA C pa( C ( ( [1 p ( ] ( f ( t tg( t dtdt A R B t B B, where P123 * * A( ( 1 C ( ( B t B B p P f t g t dtdt xx

21 ABBREVAONS : D: CC: HPP: RR: MADM: ailure inding nterval irst ailure Density Common Cause ailure Homogeneous Poisson Process nspection-repair-replacement Multi-Attribute Decision Making xxi

22 1 NRODUCON he maintenance for equipment has been an important issue for companies for many years n recent decades, maintenance and replacement problems have been extensively investigated ithin this body of research, a great deal of work has been done on inspection and maintenance strategies for systems with revealed failures Many excellent surveys summarize the research and practice in this area, including Barlow and Proschan (1965, Pierskalla and Voelker (1976, Sherif and Smith (1981, homas LC (1986, Vandez-lores and eldman (1989, Osaki (22, ang (22, Pham (23, and Nakagawa (25 However, research has paid less attention to inspection strategies for periodically inspected units with hidden or unrevealed failures Hidden failure refers to the case where failure is not self-announcing and remains undiscovered unless an inspection or a test is performed (Cerone, 1993; ortman et al 1994 Such failures are also called unrevealed faults (Phillips, 1979, pending failures (Sherwin, 1979, or latent faults (Shin, 1986 According to Moubray, up to 4% of the failure modes of a complex industrial system fall into the category of hidden failure Up to one third of the tasks generated by a comprehensive, correctly applied maintenance strategy development program are devoted to failure finding (Moubray,

23 1 NRODUCON 2 or systems with hidden or unrevealed failures, an appropriate inspection policy is required to guarantee a satisfactory level of availability nspection policies can be divided into continuous and non-continuous inspections, and the latter may be more appropriate for hidden failures n this type, the inter-checking times, or intervals between successive inspections, may be either identical (periodic inspections or different (sequential or non-periodic inspection (Balow et al, 1963 Research on inspection policies for systems with hidden failures either focuses on single-unit or multiple unit systems igure 11 summarizes the classification of literature on this topic Note that the interval between successive inspections for periodic inspection is called the ailure inding nterval ( n general, systems with hidden failures can be divided into two categories: ype : Protective devices (safety systems or standby unit: the function of these systems is to protect the main (protected system ailure of the system may not cause direct consequences provided it is not needed However, since this type of equipment is often used in emergencies, if a hidden failure has already occurred, severe consequence or even disaster might result if it is suddenly needed Examples of these devices include safety devices, emergency devices, standby units, dormant systems etc (Jardine and sang, 26; Nakagawa 25 ype : Operating devices: their function is the system s main function But when the failures of these systems are hidden and can only be detected during an inspection, a higher percentage of unacceptable items or bad-quality products may result (ang 29 Loss is assumed to incur from the moment of the system s failure until the time when an inspection is performed Examples of this type of equipment include underground pipes, underwater equipment, cutting tools etc

24 1 NRODUCON 3 or systems with hidden or unrevealed failures, an appropriate inspection policy is required to regularly inspect the system to guarantee a satisfactory level of availability nspection policies can be divided into continuous and non-continuous inspections, and the latter may be more appropriate for hidden failures or this type, the inter-checking times, or intervals between successive inspections, may be either identical (periodical inspections or different (sequential or non-periodical inspection (Barlow et al, 1963 Research on inspection policies for systems with hidden failures either focus on single-unit system or multiple unit system igure 11 summarizes the classification of literature on this topic n particular, the interval between successive inspections for periodical inspection is called ailure inding nterval ( ailure Revealed failure Hidden failure Periodic inspection Sequential Single unit system Multiple unit system System property Optimization Maximize availability Minimize cost igure 11 Classification of literature on inspection interval for hidden failure system

25 1 NRODUCON 4 Periodic inspection optimization seeks to establish the optimal inspection interval according to the following criteria: maximizing availability, minimizing expected costs, and maintaining a given safety availability level Some research has concentrated on important system parameters such as unavailability, expected profit rate etc Most research in this area considers non-negligible downtime due to repair/replacement but ignores the downtime due to inspection n many situations, however, inspection time is non-negligible and should be considered his thesis aims to fill this gap in the literature by proposing an optimal considering both non-negligible inspection time and non-negligible repair/replacement time he goal is to optimize the to maximize availability and to minimize expected cost e demonstrate the importance of considering non-negligible downtime due to inspection e also formulate instantaneous availability models with non-negligible inspection times for both age-based and calendar-based inspection policies A novel feature of this work is its establishment of a framework to classify methodologies, inspection policies and assumptions for hidden failure systems he three main contributions of this research can be listed briefly as follows: t establishes a framework to summarize research methodology, categories and common assumptions and distinguishes between age-based and calendar-based inspection policies in the case of hidden failures t proposes models for instantaneous availability with non-negligible inspection times for both inspection policies t optimizes cost and availability with non-negligible inspection times for both inspection policies

26 1 NRODUCON 5 he thesis is structured as follows Chapter 2 presents a detailed review of the literature on inspection policies for hidden-failure systems e begin by reviewing inspection strategies focusing on system availability before turning to literature on cost optimization n Chapter 3, we present instantaneous availability models with constant downtime due to inspection and repair/replacement, considering both age-based and calendar-based inspection policies or each inspection policy, we propose instantaneous availability models under two assumptions Under assumption A, the system is always restored to as good as new condition at inspection Under assumption B, the system found working at inspection is returned to operation without further intervention n this chapter, we also study the limiting average availability of a system with both age-based and calendar-based inspection policies e consider availability models under assumption A and assumption B or each case, we construct general models with random inspection time and random repair/replacement time e formulate models with constant inspection time and repair/replacement time as a special case inally, we optimize availability models with the purpose of obtaining an optimal to maximize the limiting average availability n Chapter 4, we develop several cost models for both age-based and calendar-based inspection policies As in the previous chapter, we formulate assumption A with renewal at each inspection and assumption B with renewal only after failure e consider a general case with random downtime due to inspection and repair/replacement and formulate a model for a constant inspection time and repair/replacement time e also discuss cost optimization to obtain an optimal he main contribution of these models is to calculate an optimal to minimize the expected cost incurred by inspection and possible loss

27 1 NRODUCON 6 All above models assume that the system is periodically inspected with interval, and failures are rectified only at the periodic inspection moments As an extension, particularly for protective devices, we develop an age-based cost model with the assumption that failures of protective devices can also be rectified at non-scheduled inspections which are performed after the occurrence of an accident caused by lack of protection n Chapter 5, we use the results of the analysis in Chapters 3 and 4 to present case studies for a mining & refining company and an electric power plant e investigate the current inspection policies for different types of safety devices e then discuss the optimal inspection intervals based on our models n Chapter 6, we provide our conclusions and offer some guidelines for future research

28 2 LERAURE REVE or systems in which failures are only detected at the time of inspection (hidden failures, it is important to determine the optimal time of inspection nspection policies for systems with hidden failures can be divided into periodic and non-periodic inspections f the time between successive inspections is randomly distributed, it constitutes non-periodic or sequential inspection (Balow et al, 1963 or periodic inspections, the interval is constant and is called ailure inding nterval (Jardine and sang, 26 Both periodic and sequential inspections are applicable to single-unit and multiple unit systems Various authors have produced interesting and significant results for a periodic inspection policy; these are reviewed in this chapter Periodic inspection is easier to schedule in practice than a sequential inspection As shown in igure 11, the literature in this area can be divided into two categories: 1 finding an optimal inspection interval either to minimize the cost or maximize the availability; 2 investigating system availability (or unavailability to achieve a certain safety level or each category, most literature deals with a single system, placing less emphasis on multiple unit systems 7

29 2 LERAURE REVE 8 his chapter is structured as follows n Section 21, we review system availability and unavailability models, as well as studies on availability optimization he emphasis is on single unit systems since there is little in the literature on multiple unit systems n Section 22, we present literature on optimal periodical inspection intervals to minimize the cost e begin with research on single unit systems, then go on to consider multiple-unit systems n Section 23, we mention literature on this topic which does not belong in either of the two previous sections Section 24 constitutes concluding remarks n the literature, many different assumptions are made about inspection and repair/replacement time, renewal efficiency, inspection type, preventive maintenance, and so on he most common assumptions can be summarized as follows: he inspection time and repair/replacement time is negligible or non-negligible After inspection, a system will be as good as new (with renewal or as bad as old for a working system e call the former as assumption A and the latter assumption B mperfect repair: hile a failed system will be as good as new after a perfect repair/replacement or as bad as old after a minimum repair, imperfect repair restores the system to somewhere in between (Brown and Proschan, 1983 Perfect inspection or imperfect inspection: Perfect inspection means that the inspection reveals the true state of the system without error Preventive maintenance (PM will restore the system to as good as new An inspection or test may or may not degrade the system

30 2 LERAURE REVE 9 Sequential inspection assumes the interval between successive inspections is not identical he information from previous inspections is used to determine the next inspection schedule A number of researchers have considered this inspection scheme for a single system, including Kaio and Osaki (1984, Keller (1974, Nakagawa and Yasui (198, Chelbi and Ait-Kadi (2, Yang and Kluke (2, Leung (21, Okumura and Okino (23, and Okumura (26 Some studies focus on methods associated with inspection density function; others investigate optimization models based on cost, availability, or number of inspections Sequential inspection schemes for multiple unit system have also attracted attention Naidu and Gopalan (1983 carry out cost benefit analysis of a one-server two-unit standby system hey extend their study to include non-negligible inspection time (1984 Parmigiani (1994 also investigates an inspection schedule for a two-unit standby system to minimize the number of inspections under the constraint of a fixed probability of system failure Sequential inspection strategy for multiple systems is studied by Cui, Loh and ie (24 with negligible inspection time and repair/replacement time Jiang and Jardine (25 propose two new optimization models to find the optimum inspection sequence 21 System availability and optimization Many studies have investigated system availability or unavailability for systems with hidden failures; most focus on the limiting average availability and consider a single unit system As a majority of the literature focuses on steady-state availability, instantaneous availability is seldom investigated n reality, time-dependent availability indices are important A few authors have considered instantaneous availability for continuous monitoring systems, such as Høyland

31 2 LERAURE REVE 1 and Ransand (1994 Sakar and Chaudhuri (1999 also look at instantaneous availability with gamma lifetime and exponential repair times But little research has considered periodic inspections 211 System availability and unavailability models n this section, we begin by reviewing the literature which assumes negligible inspection and repair/replacement times Yang and Klutke (2 introduce limiting average availability A av and long-run inspection rates for periodic inspections with a constant interval hey consider the weakness of a periodic inspection model which ignores information about the remaining life of the system hey give the limiting average availability and inspection rates for other policies, including quantile-based inspections ( QB ( and the hybrid inspection policy Cui, ie and Loh (24 extend this study by adding two new policies, multiple-quantile-based inspection (MQB and time hybrid inspection (HYB, under the same assumptions Kiessler (22 considers the limiting average availability of an inspected system in the case of Markovian degradation n this study, the deterioration process is modeled as a continuous-time Markov chain with finite number of states he models consider an independent, identically distributed sequence of nominal lifetimes that determines the failure criterion A study by Kharoufeh, inkelsten and Mixon (26 extends Kiessler s model by providing reliability and availability measures for a system subject to Markovian wear and degradation

32 2 LERAURE REVE 11 due to random shocks Unlike Kiessler, they propose that a failure occurs when the cumulative degradation (due to wear and shocks reaches or exceeds a deterministic threshold Klutke and Yang (22 examine the limiting average availability of a hidden failure system in which deterioration is driven by shocks and gradual degradation hey develop an expression for the limiting average availability to determine an inspection rate to guarantee a given level of availability his work is an extension of Klutke, ortman and Ayhan (1994, and ortman, Klutke and Ayhan (1996 he former paper considers availability under the assumption that the system degrades according to a continuous deterioration process he latter examines a system that degrades due to random shocks Hismeier et al (1995 present an unavailability model for an ageing standby system heir model assumes that first failure density (D differs according to the distributions of other failures All above models assume negligible inspection time and repair/replacement time under perfect repair Assumption B is considered where a failed system is replaced by a new device and the system found working is left undisturbed aking a different approach, some recent work assumes non-negligible repair/replacement time, ie, negligible inspection time but non-negligible repair time Sarkar and Sarkar (2 examine both instantaneous availability and limiting average availability with constant repair/replacement time under perfect repair heir study assumes perfect inspection and negligible inspection time hey consider two different assumptions Under assumption A, an unfailed system is as good as new upon inspection; a failed system is

33 2 LERAURE REVE 12 perfectly repaired with immediate restoration Under assumption B, an unfailed system is as bad as old; a failed system is perfectly repaired with restoration at the next inspection time A similar study by Cui and ie (25 accepts assumption A by Sarkar and Sarkar (2, but assumption B differs slightly, in that a failed system is restored at the time of completion of repair/replacement Sarkar and Sarkar consider constant repair time; Cui and ie extend the repair time to include a random case However, they both assume that the inspection time is negligible Note that the inspection policies for the above studies are different n Sarkar and Sarkar (2, the system is inspected every time units he repair/replacement time is included within the interval Cui and ie (25 assume the next inspection is scheduled to commence time units after the completion of the last inspection or repair e use the concept of calendar-based and age-based inspection policies to represent the above two cases and introduce them in the next chapter Dialynas and Michos (1992 investigate time-dependent unavailability indices with a fixed repair time in a calendar-based inspection policy and a preventive maintenance setting hey assume the system will be restored upon the completion of repair or maintenance f there is no repair/maintenance in the cycle, the system will remain as bad as old without intervention n their study, they evaluate exact point, interval, and average unavailability he above models all assume perfect repair Sarkar and Sarkar (21 study availability models under perfect repair or perfect upgrade n particular, the system undergoes perfect upgrade or perfect repair depending on whether is it is working or failed at the time of

34 2 LERAURE REVE 13 inspection Both instantaneous availability and limiting average availability are presented with random repair or random upgrade time t is not necessary to assume that the repair or upgrade time is greater than the inspection interval, since the inspection is suspended if the repair or upgrade is not complete As in Sarkar and Sarkar (2, the repaired or upgraded unit is restored at the next scheduled inspection Biswas, Sarka, and Sarkar (23 introduce availability function and limiting average availability under imperfect repair he system is maintained through several imperfect repairs before being replaced with a new system or before undergoing a perfect repair (Barlow and Proschan, 1975 As in Sarkar and Sarkar (2, they assume a calendar-based inspection policy with interval ; the repaired system is restored at the next scheduled inspection heir model considers both constant repair time and random repair time However, with the assumption of the restore policy, the random repair time becomes the nonnegative integral multiple of All models reviewed above make a general assumption of negligible inspection time n some situations, however, the downtime due to repair/replacement and the downtime due to inspection are not negligible Sim (1986 considers a case when inspection duration is non-negligible but only addresses steady state unavailability/availability for a system in the case of exponential failure-time distributions 212 Optimal inspection interval to maximize availability Cha and Mi (27 look at optimal lower and upper bounds for the burn-in time to maximize the system availability A calendar-based inspection policy is assumed for assumption A with

35 2 LERAURE REVE 14 renewal at each inspection and assumption B with renewal only after repair hey make an additional assumption by embedding assumption B with preventive maintenance (PM activity Both inspection time and repair/replacement time are negligible Vaurio (1995 determines an inspection interval to maximize the steady-state availability under imperfect inspection with error n this study, the inspection time is negligible, and a test is required before restoration either after the repair time for a failed system or after inspection for a working system he system will be as good as new after the test Only Jardine and sang (26, Jiang and Jardine (26, Pak (26, and Barroeta and Modarres (25 consider both non-negligible inspection time and repair/replacement time in their optimization studies However, they discuss only the limiting average availability Jardine and sang present an approach to obtain optimal inspection interval that maximizes availability under perfect repair (Jardine and sang, 26 heir study considers an age-based inspection policy under assumption A with renewal at each inspection Similarly, Jiang and Jardine (26 propose an approach to optimization based on maximizing expected availability heir study is an extension of Moubray s model to a more general case Barroeta and Modarres investigate an optimal inspection interval to minimize expected cost based on the method of Jarine and sang (26 (refer to Section 22 Pak (26 considers the expected availability ( for an age-based inspection policy n his study, as in Hilsmeier (1995, he assumes the D is different in other instances Some attempts have been made to study optimal inspection policies for multi-unit systems with hidden failures Uematsu and Kowada (1981 look at the optimal inspection period, minimizing the steady unavailability for a two-unit system he failure of the operating unit is

36 2 LERAURE REVE 15 self-announcing but hidden for a standby unit he stochastic behavior is modeled by a semi-regenerative process hey present point unavailability and steady state unavailability as well Kangvansaichol, Pittayapat, Eua-arporn (2 estimate the optimum routine test interval for a multi-unit system model Pascual et al (21 investigate the optimal inspection interval for multiple-unit systems with dependent failures under partial inspections, in which not all potential failure modes are observable hey assume that all system components are inspected every time units, and the system is renewed by a full inspection after N partial inspections heir objective is to maximize the overall availability of the system while respecting a safety availability constraint 22 Optimal inspection interval to minimize cost A great deal of research on periodic inspection for hidden failure addresses the issue of cost n this work, the optimum inspection interval and maintenance policy are obtained by minimizing the expected cost in a given time period or the expected long-run average cost per unit time (Sarkar and Sarkar, 21 A number of periodic inspection models have been established according to cost criteria based on different assumptions, such as per cost categories, inspection effects, quality or effectiveness of repair etc he assumptions are as follows: (1 nspections are performed at interval, and inspections do not degrade the system; (2 he failure time of the system has a general distribution (t, and its failure is detected only at the next inspection;

37 2 LERAURE REVE 16 (3 he time required for inspection is negligible; (4 Repair is conducted immediately in case of failure detection, and repair makes the system as good as new (perfect repair Barlow et al (1963 have created a basic model, a purely inspection model for determining the optimal inspection interval to minimize the expected cost he total cost per inspection cycle is given by C x c N t c, (21 ( 1[ ( 1] 2 t where Nt ( is the number of checks in [, t ], t is the time to discovery if the system fails at time t Countering assumption (4 above, they assume that the cycle is terminated upon detection of failure; in other words, no replacement or repair takes place Accordingly, in their model, only two costs are included: a fixed cost C 1 for each inspection; and a loss cost C 2 per unit time resulting form the time elapsed between system failure and the next inspection he optimal inspection policy is the one that minimizes E[ C( t ] As a result, Barlow et al obtain the cost model for periodic inspection with an interval of x : C( x c ( k 1 c [( k 1 x t] ( Based on Barlow s model, some researchers have investigated algorithm to determine an optimum inspection time, such as Luss (1976, Schultz (1985 and Alfares (1999 hile the above models only consider inspection cost and loss cost, other research takes the repair/replacement cost into account Nakagawa (25 considers an age-based inspection policy for a single standby unit with some extension of the basic model His model assumes that both inspection and repair will renew the system and that repair time has a general

38 2 LERAURE REVE 17 distribution Gt ( n addition to inspection cost C 1 and the loss cost factor C 2, the repair cost C is included in the cost model where C2 C C1 he study discusses two models: (1 an optimum interval * to minimize the expected cost; (2 the largest such that the probability (standby unit fails to function when called upon is not greater than a pre-specified value ε Because of analytical difficulties, Nakagawa only derives solutions of optimum interval * and for a simplified case where the failure time is exponential and the repair time is negligible Vaurio (1999 studies the availability and cost functions for age-based inspected preventively maintained units he objective is to minimize the total cost rate by selecting both optimal inspection interval and maintenance interval M he cases of M 1 and M are discussed in the paper; these constitute assumption A with renewal at each inspection and B with renewal after repair/replacement respectively Vaurio s model proposes another cost incurred by possible accident due to the failure of standby safety systems his model assumes non-negligible repair time and maintenance duration However, only the limiting unavailability is studied Jardine and sang (26 study the cost model for an age-based inspection policy with non-negligible inspection time he cost includes inspection cost and failure cost Pak (26 has a similar study with Jardine and sang (26 Baker (1989 takes a different approach, proposing a simple profit maximization model for exponential lifetime system Based on this model, Chung (1992 discusses an approximate analytical result Vaurios (1994 gives a more accurate and explicit equation for the optimal

39 2 LERAURE REVE 18 inspection interval Ben-Daya et al (1998 extend their work by introducing a more general model of optimal intervals to maximize the expected profit A special feature of all the models discussed above is that the system is renewed on detection of failure, either by replacement or perfect repair But in some cases, it is practical to perform minimal repair rather than to make a replacement or perform a complete overhaul n this case, a minimal repair rather than a perfect repair is assumed Policies of periodic replacement with minimal repair for normal failures have been widely discussed by many researchers, such as Barlow et al (1967, Boland and Proschan (1982, Nakagawa (1981, 26, ang (22, ang and Pham (26, etc Some investigate periodic inspections with minimal repairs for systems with hidden failures A general assumption is that the system is replaced at multiples of some period and minimal repairs are carried out when failures are discovered at inspections Vaurio (1997 analyzes the optimal inspection interval of standby units, incorporating a preventive maintenance policy As in Dialynas and Michos (1993, his model assumes that the maintenance interval M is a multiple of the inspection interval Unavailability and cost rate equations are developed for selecting optimal inspection and maintenance intervals, under different sets of assumptions concerning the renewal efficiency of inspection and repairs He assumes negligible time for inspection, repair and replacement Yeh (1995 studies a similar system under minimal repair and determines a maintenance policy that minimizes long-run expected cost per unit time while ensuring that the time-dependent availability exceeds a given lower bound Her model considers cost of

40 2 LERAURE REVE 19 inspection, repair and replacement, as well as a fixed penalty cost that depends on the lower bound on availability Another approach is used by Mohandas et al (1992 hey propose a model to maximize the profit per unit assuming negligible inspection time hey extend their study with the assumption of non-negligible inspection time n this study, the cost of minimal repair is the non-decreasing function of age hey investigate the optimal replacement time and the corresponding optimal number of inspections using a dynamic programming formulation and numerical search Barroeta and Modarres (25 examine cost rate function with an age-based inspection policy under minimal repair hey consider non-negligible inspection time and repair time he model takes into account costs associated with inspections and maintenance, as well as losses hey further assume that the repair cost is upgraded with the age of the system heir study does not consider risk limits Lienhardt et al (27 perform a study considering risk limit hey investigate the problem of selecting failure finding maintenance strategy for a repairable aircraft system hey develop an optimization model based on the Markov model, considering the cost rate as an objective function, and using the risk of corrective maintenance as the constraint function he model is applied to a system with an exponentially distributed lifetime More recently, ahgipour and Banjevic (211 investigate the optimal inspection interval for a multi-unit repairable system to minimize expected cost over a finite and infinite time horizon Similar to the study of Nakagawa (1987 and Murthy and Nguyen (1981, they assume that

41 2 LERAURE REVE 2 failed components may be minimally repaired or replaced with age dependent probabilities he failures of each component follow a non-homogeneous Poisson process (NHPP (Ross, 27 he cost of inspection, replacement, repair and penalty are considered in the cost rate function ang (28 develops an optimization model to determine inspection intervals for a process with two types of inspections and repairs to minimize the expected cost, based on the delay time concept Ahmadi and Kumar (211 develop a cost rate function model to identify the optimum interval and frequency of inspection and restoration of an aircraft s repairable components Similar to Vaurio (1999, in addition to the cost of inspection, repair/replacement and loss, they consider the cost parameter of accidents his model assumes non-negligible inspection time and repair/replacement time with an age-based inspection policy under minimal repair Sometimes inspections do not reveal the true status of the system, or the information obtained through inspection is not reliable or example, an inspection may incorrectly identify the system as up when in fact it is down or vice versa his is called an imperfect inspection homas et al (1987 introduce a discrete Markov decision process model and investigate the optimal inspection policy for maximizing the expected time until a catastrophe occurs, taking into account both the inspection and repair times hey consider the case of imperfect inspection Luss and Kander (1976 also formulate inspection policies with a non-negligible inspection time under perfect repair Costs are introduced for checking, loss and repairing or replacing

42 2 LERAURE REVE 21 hey allow imperfect inspection by introducing two probabilities and is a false detection of failure and is the probability of erroneously indicating a failure at inspection he models are solved by differentiation and by dynamic programming Badía et al (21 examine a model minimizing cost per unit time under imperfect inspection with two types of error and of efficiency of inspection he model includes cost due to inspections, type errors, downtime loss and corrective maintenance hey give the optimum policy for exponential and Pareto failure-time distributions An extension of Badía et al s inspection model is presented by Berrade (212 Using a procedure with two different inspection intervals, 1 and 2, Badía et al consider a single inspection interval every units of time his two-phase inspection policy adapts itself to changes in the system reliability hen the unit reaches M 1 and no failure has been detected, an inspection is carried out every 2 time units from that point on n this way, different inspection frequencies may detect both early failures and failures due to deterioration Note that the above models with imperfect inspection assume perfect repair Yeh (1995 proposes an inspection-repair-replacement (RR policy for a standby system under minimal repair his goal is to obtain an optimal RR policy so that the availability of the system is high enough at any time, and the long-run expected cost per unit item is minimized Unlike most researchers above, Badía et al (22 develop an inspection policy along with maintenance procedures for a single unit system whose failures are either revealed or unrevealed (with probability p hey assume imperfect inspection and negligible inspection

43 2 LERAURE REVE 22 and repair times nspection and preventive maintenance occur at periodic times N for unrevealed failures and corrective maintenance occurs for revealed failures hey also discuss optimum inspection time (* to minimize expected cost hile the above inspection polices consider an infinite time span, Nakagawa and Mizutani (27 summarize maintenance policies for a finite interval n the same vein, aghipour et al (21 propose a model to find the optimal periodic inspection interval on a finite time horizon for a multi-component repairable system hey assume that components of the system are subject to hard failures and soft failures with minimal repairs and consider a policy associated with soft failures which can only be detected and fixed at inspection Hard failures are ignored in the model by making the assumption of instantaneous minimal repairs he objective is to find the optimal scheduled inspection interval that minimizes the expected cost incurred over a cycle of length (= n he inspection models reviewed thus far consider single unit systems However, some research deals with inspection policies for multi-unit systems As this is not our focus, we provide only a short review of the relevant literature Nakagawa et al (1976 summarize optimum preventive maintenance policies for two-unit standby redundant systems Shima and Nakagawa (1984 investigate an optimal inspection policy for a protective device which prevents shocks to the operating system heir model assumes that the main operating system will fail as a result of any shock which occurs randomly (exponentially distributed with rate α if it is not protected by the protective device f the protective device is working, ie, it has not failed by itself, it prevents the failure of the

44 2 LERAURE REVE 23 operating system caused by a shock with probability 1 p,( p 1 Both operating system and protective device will fail as a result of the shock with probability p Associated costs are C 1 for the failure of both operating system and device, and C for the failure of the device 2 and inspection cost C where C 3 1 C2 C3 hey present a general model for minimizing expected cost rate per unit time and derive optimal inspection interval (* for the case that t ( is exponential to and Nakagawa (1995 investigate an optimal inspection policy for a storage system with high reliability he system has two kinds of units whereby unit 1 is renewed at inspections, and unit 2 remains unchanged A system is inspected at periodic times N ( N 1, 2 K N (N=1,2 K and is overhauled at failure or at time ( K 1 when the system reliability is lower than a pre-specified value q he cost model includes cost for inspection, replacement, and loss An optimal inspection time is discussed to minimize the average cost for the case of eibull failure times to and Nakagawa (2 extend their study for a two-unit storage system with degradation, whereby unit 1 is renewed at inspections, and unit 2 degrades with time and at each inspection A system is inspected at periodic times N ( N 1,2 and is overhauled at time N t when the system reliability is lower than a pre-specified value q he optimal inspection times are derived for a given to maximize N t and minimize the average cost associated with inspection and overhaul Mok and Seong (1996 introduce an optimal inspection policy for multi-unit safety systems to minimize the expected cost under imperfect inspection More references are reviewed in Nakagawa (25

45 2 LERAURE REVE Other research for periodic inspection Criteria of the optimization problem in periodic inspections include cost, availability, reliability, and other system performance indexes Some inspection optimization studies do not consider minimizing cost or maximizing availability Bulter (1979 deals with the problem of determining an optimal inspection policy for maximizing the expected life time of the system Siqueira (24 develops a probabilistic model to optimize the frequency of maintenance tasks for power system equipment Motta and Colosimo (22 propose an optimization model considering the multiple failure probability of the protective device and the protected system Many believe that in the case of safety-critical systems, safety analysis should take a proactive rather than a reactive approach hus, safety study goes beyond accident investigation to include proactive measures to prevent the occurrence of dangerous events (Bukowski, 21 Zhang et al (28 present a method to analyze performance indexes of safety-critical systems, incorporating imperfect diagnostics and imperfect periodic inspection and repair into a Markov model Cepin (22 determines the optimal scheduling to improve the safety of equipment outages in nuclear power plants by minimizing the mean value of the selected time-dependent risk measure Other studies in this area include Yang et al (2, Hokstad et al (1995, Harunuzzaman and Aldemir (1996 and Ahmadi et al (21 hile the majority of the literature reviewed so far assumes the independent failure of components in a multi-component system, Yu et al (27 consider reliability optimization of a redundant dependency Vaurio (23 presents common cause failure probabilities in standby

46 2 LERAURE REVE 25 safety systems fault tree analysis with time-dependency He outlines an approach to incorporating assessment uncertainties in estimations of multiple failure rates Recently, some literature focuses on multi-criteria or multi-objective optimization models (riantaphyllou, 2 erreria et al (29 develop a multi-criteria decision model to determine inspection intervals for condition monitoring based on delay time analysis (see Christer, 22 hey formulate inspection models dealing with two criteria simultaneously, namely, cost and downtime Anmeida and Bohouis (1996 propose a decision model for planning maintenance strategy for a standby system based on Multi-Attribute Utility heory (MAU ang and Pham (211 design a dependent competing risk model for a hidden failure deteriorating system subject to shocks hey present models for the cost rate and asymptotic unavailability of the hidden failure system, as well as a multi-objective optimization based on the Genetic Algorithm (GA 24 Concluding Remarks his chapter reviews periodic inspection policies for hidden failure systems Many researchers focus on optimization models to establish the optimal inspection interval to maximize availability or minimize expected costs n this chapter, we review the general assumptions associated with inspection time and repair/replacement time, renewal efficiency, inspection type, preventive maintenance, and so on e also review the literature on the various assumption combinations

47 2 LERAURE REVE 26 Other studies concentrate on important systems parameters such as availability and unavailability Many study limiting average availability; few consider instantaneous availability Most research on period inspections for hidden failures assumes that the times for inspection are negligible However, in some cases, the inspection time cannot be ignored f we treat the downtime due to inspection as negligible, the result might be very different, especially when is small herefore, in our study, we consider availability models and cost models with non-negligible inspection times, making our work more generalizable and more realistic e distinguish between two inspection policies for hidden failure systems, namely, age-based and calendar based policies, and we create a framework for the various categories of periodic inspection hile age-based and calendar-based inspection policies are well addressed in the literature on systems with revealed failures, they are less commonly found in the literature on systems with hidden failures n Chapters 3 and 4, we will investigate the optimal inspection interval to maximize availability and minimize expected cost for a single unit system with non-negligible downtime due to inspection and repair/replacement e will also study system instantaneous availability with same assumption able 21 summarizes the overall structure of the research

48 2 LERAURE REVE 27 able 21 he overall structure of the research Age-based Calendar-based Remarks Availability Model nstantaneous availability A(t Assumption A, B* Assumption A, B Constant downtimes** nspection nterval Optimization o Maximize Availability A o Minimize Cost Assumption A, B Assumption A, B Assumption A, B Assumption A, B Random downtimes Random downtimes Note: *A assumes renewal at each inspection; B assumes renewal after failure **Downtimes refer to downtime due to inspection, and down time due to repair/replacement

49 3 AVALABLY MODEL AND OPMZAON 31 ntroduction n this chapter, we investigate system availability models and optimal failure finding intervals to maximize the availability of a single unit system e consider a repairable system subject to hidden failures under periodic inspections at equal intervals e assume that both downtime due to inspection and downtime due to repair/replacement are both non-negligible e make the following assumptions in our research: 1 nspection is performed at fixed intervals nspections do not deteriorate the system 2 he failure time of the system has a general distribution (t, and the failure is detected only at inspection 3 Both inspection and repair are perfect n other words, an inspection always correctly detects a failure if it has occurred, and a repair/replacement can always restore the system to as good as new condition 4 hen the system is found failed at an inspection, repair or replacement starts immediately after inspection 28

50 3 AVALABLY MODEL AND OPMZAON 29 5 henever an inspection or a repair/replacement task is performed on the system, its functions stop and it does not age his research uses the following notations: : fixed inspection interval length L ( : cycle time : constant downtime due to inspection R : constant downtime due to repair/replacement, if failure found during inspection : total constant downtime due to inspection and repair/replacement, the sum of and R i : random variable, downtime due to inspection, with mean Y i : random variable, downtime due to repair/replacement if a failure is found during an inspection, with mean Y Y : downtime due to repair/replacement in the first interval of a cycle ux: ( density function of random inspection time vy: ( density function of random repair/replacement time : time to failure of the system f( t : density function of the time to failure of the system R (t : reliability (survival function of the system t (: cumulative distribution function of the system 29

51 3 AVALABLY MODEL AND OPMZAON 3 : mean lifetime of the system, R( t dt A (t : instantaneous availability of the system at time t A : limiting average availability of the system, ie, 1 A lim t t t A( u du (t : up (=1 or down (= state of the system at time t ~ 1 : age of a unit at the first failure * 1 : calendar time at the first failure 1 : calendar time at the inspection when the first failure is found e consider two types of inspection policies in the research he first type, which is called calendar-based inspection policy, schedules inspections at fixed calendar intervals, say, for example, every Monday (once a week or this inspection policy, the downtime due to inspection and repair/replacement is included in the interval he second type, the age-based inspection policy, schedules inspections at fixed age intervals he inspection time and necessary repair/replacement time are not included in the interval ig 31 shows the difference between the two inspection policies n the special case where the downtime due to inspection and the downtime due to repair/replacement are negligible, ie,, the calendar-based and the age-based inspection policy become exactly the same 3

52 3 AVALABLY MODEL AND OPMZAON 31 ailure (1a Calendar-based inspection policy ailure (1b Age-based inspection policy ig 31 Calendar-based and age-based inspection policies Calendar-based and age-based inspection policies have both advantages and disadvantages or example, a calendar-based inspection policy is easier to schedule in practice than an age-based inspection policy since the latter requires keeping track of the previous inspection time However, it is less efficient than an age-based inspection policy since it may schedule an unnecessary inspection just before a renewal e study both policies under two different assumptions: Assumption A: At an inspection, the system is always restored to as good as new condition Assumption B: At an inspection, the system found working is returned to operation without intervention n section 32, we present instantaneous availability models with constant downtime due to inspection and repair/replacement, considering both age-based and calendar-based inspection policies or each inspection policy, we propose instantaneous availability models under assumption A and B 31

53 3 AVALABLY MODEL AND OPMZAON 32 n section 33, we study the limiting average availability of a system with an age-based inspection policy e study availability models under assumptions A and B or each case, we construct general models with random inspection time and random repair/replacement time e then formulate models with constant inspection time and repair/replacement time as a special case inally, we optimize availability models for each case with the purpose of obtaining an optimal to maximize the limiting average availability n section 34, we develop limiting average availability models for a system with a calendar-based inspection policy As in section 33, we propose availability models under assumption A and B or each case, we formulate availability models with both random downtime and constant downtime e also investigate the optimization of availability models to find the optimal to maximize the availability All models calculate the availability in terms of failure finding interval and obtain an optimal interval * to maximize availability n section 35, we present numerical examples to show different cases of availability models and their optimization he structure of availability models in Chapter 3 is summarized in igure 32 32

54 3 AVALABLY MODEL AND OPMZAON 33 Availability Models 32 nstantaneous availability model 33 Limiting average availability with age-based policy 34 Limiting average availability with calendar-based policy 321 Age-based model with renewal at each inspection (A 322 Age-based model with renewal after failure (B 323 Calendar-based model with renewal at each inspection (A 331 Age-based availability with renewal at each inspection (A 332 Age-based availability optimization with renewal at each inspection (A 333 Age-based availability with renewal after failure (B 341 Calendar-based availability with renewal at each inspection (A 342 Calendar-based availability optimization with renewal at each inspection (A 343 Calendar-based availability with renewal after failure (B 324 Calendar-based 334 Age-based availability 344 Calendar-based model with renewal optimization with renewal availability optimization with after failure (B after failure (B renewal after failure (B (BAvailability optimization igure 32 Structure of availability models 32 nstantaneous availability model formulation n this section, we study instantaneous availability of the hidden failure system e assume that the failure can only be detected upon inspection e construct instantaneous availability models, first for an age-based inspection policy then for a calendar-based inspection policy or each inspection policy, we consider assumption A with renewal at each inspection and assumption B with renewal after failure 321 Age-based instantaneous availability with renewal at each inspection (A igure 33 compares assumptions A and B for an age-based inspection policy 33

55 3 AVALABLY MODEL AND OPMZAON 34 nspection ailure R Renew Renew * 1 1 Renew Assumption with renewal at each inspection (A nspection ailure R Assumption with renewal only after failure (B * 1 1 Renew igure 33 wo different assumptions for age-based inspection policy Under assumption A, the system is renewed at each inspection e have the following results for the instantaneous availability of the system he instantaneous availability of the system with an age-based inspection policy is given recursively by: R( t, t, t A ( t (31 R( R( t, t R( A( t [1 R( ] A( t, t Proof Obviously, A( t R( t when t, and A ( t when t or t, we have 34

56 3 AVALABLY MODEL AND OPMZAON 35 A( t P( ( t 1 P( ( 1 P( ( t 1 ( 1 P( ( P( ( t 1 ( R( R( t (1 R( R( R( t, since the system found working at time (ie ( 1 is renewed at time, and the system found failed at time (ie ( needs duration to repair/replace Hence, it will be down in the interval (,, ie, ( t, for t or t, we have A( t P( ( 1 P( ( t 1 ( 1 P( ( P( ( t 1 ( R( A( t (1 R( A( t, since the system found working at time is renewed at time, and the system found failed at time is renewed at time he above completes the proof of Equation (31 Our model reduces to the case studied by Cui and ie (25 when heir paper gives only instantaneous availability e consider both instantaneous availability and limiting average availability 322 Age-based instantaneous availability with renewal after failure (B Under assumption B, the system remains unchanged if it is found working at inspection e obtain the following results for the instantaneous availability and the limiting average availability of the system he instantaneous availability of the system with an age-based inspection policy is given 35

57 3 AVALABLY MODEL AND OPMZAON 36 recursively by: At ( mt ( A( t t p, t n( t( i i i1 mt ( i i i1 R( t n( t A( t t p, t n( t(, (32 where m( t ( t /(, n( t t /(, ( i 1(, and t i p i R(( i 1 R( i * Proof Since the system is down in time interval,, we have [ 1 1 A( t P( ( t 1 P( ( t 1, t P( ( t 1 * 1 * 1 P( ( t 1, t P( * 1 * 1 t P( ( t 1, t P( ( t 1, 1 1 t 1 t (33 rom the definition of n (t, we have n t( t ( n( t 1( ( Note that before the first failure, the system is always up in time interval n( t(, n( t( ] and always [ down in time interval n( t(,( n( t 1( because this is an inspection period ( herefore, the first term on the right-hand side of Equation (33 can be expressed by: P( ( t 1 t P( t * * 1 1 * P( 1 t, t n( t( * 1 P( 1 t, t n( t(, t n( t( P( 1 t n( t, t n( t(, t n( t( R( t n( t, t n( t( 36

58 3 AVALABLY MODEL AND OPMZAON 37 Note that all the inspections up to 1 are performed at t i, i 1,2,, m( t Given that the first failure is found at t i, the system is renewed at t i Removing the first cycle [, t i ] and counting from t i, the situation is exactly the same as the original situation at time his means P ( ( t 1 1 ti A( t ti, for t ti Based on this determination, the second term on the right-hand side of Equation (33 can be expressed as P( ( t 1, t mt ( i1 mt ( i1 1 P( ( t 1 t P( t 1 i 1 A( t t P( t t * i i 1 i i mt ( A( t t P(( i 1 i i1 mt ( A( t t p i1 i i 1 i he above completes the proof of Equation (32 hen, our model reduces to the case considered by Cui and ie (25 Our model gives the same result for the instantaneous availability as in their paper t also gives the formula for the limiting average availability 323 Calendar-based instantaneous availability with renewal at each inspection (A n the previous section, we discussed system availability when an age-based inspection policy is applied n this section, we study the instantaneous availability of a system with a 37

59 3 AVALABLY MODEL AND OPMZAON 38 calendar-based inspection policy under assumptions A and B igure 34 shows these two assumptions with a calendar-based inspection policy nspection ailure R Renew Renew * 1 1 Renew Assumption with renewal at each inspection (A nspection ailure R Renew Assumption with renewal only after failure (B * 1 1 igure 34 wo different assumptions for calendar-based inspection policy Under assumption A, the system is renewed at each inspection t is reasonable to assume that e obtain the following results for the instantaneous availability of the system he instantaneous availability of the system with a calendar-based inspection policy is given recursively by: R( t,, A( t R( t k R( t k A( k, A( k R( t k [1 A( k], t k t k k k t k t ( k 1 (34 where k 1,2, Proof A( t R( t when t, and A ( t when k t k 38

60 3 AVALABLY MODEL AND OPMZAON 39 or k t k, we have A( t P( ( t 1 P( ( t 1 ( k 1 P( ( k 1 P( ( t 1 ( k P( ( k R( t k A( k [1 A( k] R( t k A( k, since the system found working at time k (ie ( k 1 is renewed at time k, and the system found failed at time k (ie ( k needs duration to repair/replace Hence it will be down in the interval (k, k, ie, ( t, for k t k or k t ( k 1, we have A( t P( ( t 1 ( k 1 P( ( k 1 P( ( t 1 ( k P( ( k R( t k A( k R( t k [1 A( k ], since the system found working at time k is renewed at time k, and the system found failed at time k is renewed at time k he above completes the proof of Equation (34 Note that when, it reduces to the case discussed by Sarkar and Sarkar (2 324 Calendar-based instantaneous availability with renewal after failure (B Under assumption B, the system remains unchanged if found working at inspection e obtain the following results for the instantaneous availability and the limiting average availability of the system he instantaneous availability of the system with a calendar-based inspection policy is given recursively by: 39

61 3 AVALABLY MODEL AND OPMZAON 4 4 k t k p i t B k t R k t k p i t B k t R k t k t t R t A k i i k i i 1 (, ( (, ( (,, ( ( (35 where (,, ( ( (, ( (, ( 1 k i i k i i R t t k t k B t R t k B t i q k t k R t k B t i q k t k (36, ( (, ( ( 1 1 i i i i i i R R q R R p,,, (, 1,2, (, 1,2, i i i i i i i i and 1,2, k Proof ( ( t R t A when, t and ( t A when k t k Since the system is down in time interval, [ 1 * 1, we have 1, ( ( ( 1 ( ( 1, ( ( 1, ( ( 1, ( ( 1 ( ( ( 1 * 1 * * 1 * 1 t t P t P t t P t t P t t P t t P t P A t (37 Note that before the first failure, assuming the first failure is after ( 1 k, the system is always

62 3 AVALABLY MODEL AND OPMZAON 41 up in time interval [ k,( k 1 ] n this case, the first term on the right-hand side of Equation (37 can be expressed by: * * * ~ P( ( t 1 1 t P( 1 t 1 P( 1 t P( 1 t k R( t k for k t ( k 1 Note that all inspections in a calendar-based policy are performed at i, i 1,2, Given that the first failure is found at i, the system is renewed at i he renewed cycle starting from time i will be inspected at ( i 1, ie, the first inspection interval for the renewed cycle will be counting from the renewed time i n fact, all the renewed cycles will have the first inspection interval counting from the renewed time Let B (t be the availability at time t under a calendar-based inspection policy with the first inspection interval (counting either from the installation time or the renewed time and all subsequent inspection intervals until a failure is found his gives us P ( t 1 i B( t i ( 1 or k t k, the second term on the right-hand side of Equation (37 can be expressed as 41

63 3 AVALABLY MODEL AND OPMZAON ( ( ~ 1( (( ( ~ ( ( 1 (( ( ( ( ( 1 ( ( 1, ( ( * 1 * k i i k i k i k i p i t B i i P i t B P t B i i P i t B P t B i P i t P t t P Similarly for k t k 1 (, the second term on the right-hand side of Equation (37 can be expressed as ( 1, ( ( 1 1 k i p i i t B t t P he above proves Equation (35 Applying the above approach used for (t A to (t B, we can similarly derive Equation (36 Note that ( ( t B t A when Our model reduces to the case studied by Sarkar and Sarkar (2 33 Limiting average availability model with age-based inspection policy n this section, we investigate availability optimization of a hidden failure system with an age-based inspection policy irst, we formulate limiting average availability models under assumption A with renewal at each inspection e then develop availability models under assumption B with renewal only after failure Under each assumption, we present the general model with random inspection time and repair/replacement time e also develop availability

64 3 AVALABLY MODEL AND OPMZAON 43 models with constant downtime due to inspection and repair/replacement as special cases inally, under each assumption, we discuss the optimization of availability models to obtain optimal inspection intervals to maximize the availability 331 Age-based limiting average availability with renewal at each inspection (A Since the system is always renewed at each inspection independently of whether or not the system is found working, the limiting average availability of the system can be obtained by the property of a renewal process: Expected availability in one cycle A Expected time in one cycle irst we establish the limiting average availability model for random downtime due to inspection and repair/replacement Assume the inspection time is i, which is randomly distributed with mean, and the repair/replacement time is also a random variable Y i, which is distributed with mean of Y e obtain the expected cycle time as follows: Possible cycle Cycle length L1 Y L2 Y 43

65 3 AVALABLY MODEL AND OPMZAON 44 L( ( ( ( Y ( Y ( he expected cycle time will be: EL( E EY ( ( Y he expected uptime in one cycle is E[min{, }] R( t dt he limiting average availability model with random inspection time and repair/replacement time is as follows: R ( t dt (38 [1 ( ] A Y R he cost model for constant inspection time, say,, and random repair/replacement time is represented by (39: R ( t dt (39 [1 ( ] A Y R f both the inspection time and repair/replacement time are constant, denoted by and R, equation (39 can be rewritten as: R( t dt (31 [1 ( ] A R R 332 Age-based availability optimization with renewal at each inspection (A rom equation (38, we have: 44

66 3 AVALABLY MODEL AND OPMZAON 45 2 [ Y [1 R( ] [ R( R( t dt R( Y R( ( Y f ( R( t dt] A Let G( R( R( t dt R( R( ( f ( R( t dt Y Y G( is continuous, and G( G( R( R( t dt R( R( ( tf ( t dt R( R( ( R Y e have: lim G( ( tf ( t dt Given this, must exist such that G ( herefore, there must be a maximum for A e may apply similar discussion in cases when the down time due to inspection and/or repair/replacement is constant simply replacing and Y by and R respectively 333 Age-based limiting average availability with renewal after failure (B Under assumption B, we assume the system remains unchanged if found working at inspection; and renewed after the repaired/replacement time when a failure is detected e derive availability models with random down time due to inspection and repair/replacement e then study the availability of constant inspection time and repair/replacement time as a special case irst we investigate the expected cycle time Conditioning on, Y, 1,, we have: 45

67 3 AVALABLY MODEL AND OPMZAON 46 Possible cycle Cycle length Probability Y N L1 Y p 1 R( 1 N 2 L2 2 ( 1 Y p R( R(2 2 1 Y i-1 Y +C i N i Li i i1 Y p R (( i 1 R ( i i i1 hen the cycle length (given, Y, 1,, is: i L(, Y,, [( i Y ] p [( i Y ][ R(( i 1 R( i] 1 i1 i i1 i1 i1 i1 i1 i i i [( i i1[ R(( i 1 R( i] Y ( i i1 pi Y i1 i1 i1 i1 he expected cycle time is: i EL(, Y, 1, E [( i i1 pi ] EY i( E i1 pi Y i1 i1 i1 ( i1 ip i Y ( i R( i Y Since the uptime during the cycle is R ( t dt, the limiting average availability of the system for 46

68 3 AVALABLY MODEL AND OPMZAON 47 random downtime due to inspection and repair/replacement is: A ( R( i i Y (311 Let g ( R( i (312 i Equation (311 can be written as: A ( g( Y (313 he general properties of g ( are presented in Appendix A Similarly, we may develop models for the scenario of constant inspection time and repair/replacement time as special cases, as given in equation (314 and (315 he availability model for constant inspection time and random repair/replacement time is: A ( g( Y (314 he availability model for constant inspection time and repair/replacement time R is: A ( g( (315 R 47

69 3 AVALABLY MODEL AND OPMZAON Age-based availability optimization with renewal after failure (B n this section, we investigate the optimal to maximize liming average availability under assumption B e start with a general approach for availability optimization, and then discuss optimal for a system whose failure follow exponential distribution as a special case 3341 Optimal to maximize availability with renewal after failure rom equation (313, A ( g( Maximizing A is the same as minimizing ( g( Y Recall (312, where g( R( i i Let G( ( g( rom Appendix A, we know that max{1, } g ( 1 So, max{1, }( G( ( (1 Let G1 ( max{1, }(, 2 ( ( (1 G or (, G 1 1 (, G (, G1( lim G1( lim( ; G1( lim G1( lim( 48

70 3 AVALABLY MODEL AND OPMZAON 49 or, G ( 1 is decreasing in ; or, G ( 1 is increasing in ; or, min G1 ( or G (, 2 G ( lim G ( ; G2( lim G2( 2 2 G2 ( Note that 1 2 or or or, G ( is decreasing in ; 2, G ( is increasing in ; 2, 2 min G ( 2 G( is bounded by G ( and G ( igure 35 shows the relationship between the 1 functions So min G ( exists, and 2 min G ( 2 (316 Accordingly, maximum A exists such that: max A 2 Y Y, or, max A 2 ( Y Y 49

71 3 AVALABLY MODEL AND OPMZAON 5 G 2 ( 2 min G2 ( G( min G1 ( G 1 ( igure 35 Relationship between function G, G ( and G( 1( 2 o obtain optimal, [( g( ] need to be zero ie, g( ( g( Y hen, we should find such that g( 1 [ln g ( ], which will give an optimal g( e may use a similar application in cases when the down time due to inspection and/or repair/replacement is constant or those cases, we simply replace and Y by and R respectively or constant inspection time and random repair/replacement time, max A exists, giving us: max A 2 ( Y Y or both constant inspection time and repair/replacement time, max A exists; max A 2 ( R R 5

72 3 AVALABLY MODEL AND OPMZAON Optimal for system with lifetime following exponential distribution n this section, we discuss the optimal of a system the lifetime of which follows an exponential distribution with failure rate Note that R( t exp( t, so from (312, we obtain, 1 g( R( i 1 exp( i Under assumption B with renewal only after failure, from equation (313 we get, A ( g( Y Maximize A is same as minimizing ( g( Let ( ( ( g( 1 exp( o get the minimum value of (, the numerator of ( should be zero, ie, (1 e ( e ( e 1 ( e 1 ( By solving the above equation we may obtain an optimal to minimize ( or, equivalently, to maximize A 51

73 3 AVALABLY MODEL AND OPMZAON 52 e may determine the upper limit of the optimal in the following fashion 1 1 e x x e x x 2 2 x 2 x 2 1, 1, so 1 e ( hen, ( 2 2 hus, the optimal * is smaller than igure 36 shows the relations of the demonstration e 1 1 ( * igure 36 he upper bound of optimal A similar conclusion can be drawn for cases with constant inspection time and repair/replacement time 34 Limiting average availability model with calendar-based inspection policy n this section, we study the availability optimization of a hidden failure system with a calendar-based inspection policy As in section 33, we develop limiting average availability models under assumption A with renewal at each inspection and under assumption B with renewal only after failure Please refer to igure 34 52

74 3 AVALABLY MODEL AND OPMZAON 53 Under both assumptions, we first present general availability models with random inspection time and repair/replacement time e then establish availability models with constant downtime due to inspection and repair/replacement as special cases e explore availability optimization, seeking to determine an optimal to maximize limiting average availability 341 Calendar-based limiting average availability with renewals at each inspection (A he limiting average availability of the system can be obtained by the property of a renewal process as follows: A Expected availability in one cycle Expected time in one cycle Under assumption A, the system will be renewed after inspection time, while a failed system will be renewed after failure e define one cycle as the time between two inspections at which failures have been detected (igure 37 Note that the inspection and repair/replacement times in the first interval belongs to this cycle, while the times for the next repair/replacement and inspection detecting failure fall into the next cycle nspection ailure R Renew Renew Renew One cycle R Next cycle igure 37 Cycle length under assumption A with Calendar-based inspection policy 53

75 3 AVALABLY MODEL AND OPMZAON 54 irst, we consider the scenario whereby the inspection time and repair/replacement time are random variables i and Y i, distributed with means of and Y respectively e assume that i and Y i are bounded with constant x m and y m respectively t is practical to assume x y, to ensure the total time for inspection and repair/replacement is smaller than m m interval irst we investigate the expected cycle time Conditioning on, Y, 1,, we have: Possible cycle Event Probability Y N= Y p 1 ( Y 1 Y Y 1 2 N=2 Y 1 1 N=3 Y p R( Y 2 ( p R( Y 3 R( ( Y 1 1 i-1 i-1 N i ( i 2 Y 1 1 i2 i2 i 1 i 1 p R( Y i i2 j1 [ R( ] ( i1 j hen the cycle length given, Y, 1, is: (,, 1, i i1 L Y i p Recall that, 1, i, have the same distribution, so that the expected cycle time will be: 54

76 3 AVALABLY MODEL AND OPMZAON 55 EL(, Y, 1, E( i pi E( p1 E( ipi i1 i2 E[ ( Y ] E{ ir( Y [ R( ] ( } i2 j i1 i2 j1 E[ ( Y ] ie{ R( Y [ R( ] ( } i2 j i1 i2 j1 i2 i2 E[ ( Y ] ier( Y [ ER( ] E( (317 Let C ER( Y, D ER( Equation (317 will be: 1 i 2 i2 EL(, Y,, (1 C C(1 D id (1 C C(1 D id i2 i2 1 1 C C D id D C C D i1 i (1 (1 [ ] (1 (1 [ ] 2 i1 i (1 D 1D C C (1 C C (1 1D 1D ER( Y [1 ] (318 E( where E( ( x u( x dx, x m ym xm E[ R( Y ] R( x y u( x v( y dxdy he uptime using conditioning on, Y, 1, is U (, Y,, min{, Y } min{, } ( Y L min{, } ( Y,,, i2 i i 1 1 i1 i1 where the contribution in each interval is considered separately 55

77 3 AVALABLY MODEL AND OPMZAON 56 he expected uptime in one cycle, U (, will be U ( EU (, Y,, E[min{, Y }] E[min{, } P( Y ] L E[min{, } P( Y,, ] i2 i2 i i 1 1 i1 i1 E min{, Y } E min{, } ER( Y 1 1 E ER Y ER i1 min{, } ( [ ( ] Let A E Y B E min{, }, min{, }, C ER( Y, D ER( n this case, BC 1 D i1 i1 i U ( A BC BC D A BC D A BC D A i2 i1 i hen, the expected uptime is U E Y E[min{, }] ER( Y ( [min{, }] E( (319 herefore, the age-based limiting availability model for random inspection time and repair/replacement time appears as follows: E[min{, }] ER( Y E[min{, Y}] E( A, ER( Y [1 ] E( y x xy m m where E min{, Y} tf ( t u( x v( y dtdxdy, (32 E min{, } m tf ( t u( x dtdx x x f the inspection time is not random but a constant, from (32, the limiting availability 56

78 3 AVALABLY MODEL AND OPMZAON 57 model is given as: E[min{, }] ER( Y E[min{, Y}] E( A, ER( Y [1 ] E( y y m where E min{, Y} tf ( t v( y dtdy, (321 E min{, } R( t dt Suppose we take time for inspection; if a failure is detected, an additional time of R is necessary for repair/replacement his scenario is a special case of (321 he limiting average availability model for constant inspection time and repair/replacement time is given by: A R( R R( t dt R( t dt ( R( R [1 ] ( R (322 Equation (322 can be written as:, R ( R( t dt R( ( R R t dt A [ ( R( ] R or (323 R (1 R( R( t dt R( ( R R t dt A [1 R( R( ] R 57

79 3 AVALABLY MODEL AND OPMZAON 58 or the case of constant downtime due to inspection and repair/replacement, we may also obtain the limiting average availability model from the instantaneous availability (34 as follows: Recall that R herefore, from equation (34, we have: R( t,, A( t R( t k R( t k A( k, A( k R( t k [1 A( k], t k t k k k t k t ( k 1 where k 1,2, Letting t ( k 1 and applying equation (34, we have A(( k 1 [ R( R( ] A( k R(, k 1,2,, which is a first-order nonhomogeneous difference equation Based on heorem 113 in Elaydi (25, lim A( k exists since R( R( 1 k Letting k in equation (34, we have R( lim A( k k 1 R( R( Note that the above formula can be also derived by the ergodicity of the Markov chain with state space of {failure found, failure not found} and transition probability of the system moving from the current inspection interval k to the state in the next inspection interval ( k 1 or any u (,, and applying equation (34 we obtain 58

80 3 AVALABLY MODEL AND OPMZAON 59, u A( u k R( u A( k, u R( u A( k R( u [1 A( k], u Letting k, we have, u lim A( u k R( u, u k R( u (1 R( u, u Hence, the limiting average availability is: A 1 ( [ ( (1 ( ] R u du R u R u du 1 R( u du (1 R( u du 1 R( u du (1 R( u du (324 Note that equation (323 can also be written as: R (1 R( R( t dt R( ( R t dt A [1 R( R( ] 1 R( t dt (1 R( t dt herefore, equation (323 and (324 are identical 59

81 3 AVALABLY MODEL AND OPMZAON Calendar-based availability optimization with renewal at each inspection (A or simplicity, we investigate the optimal for constant downtime due to inspection and repair/replacement rom equation (324, the limiting average availability is: 1 A R( t dt (1 R( t dt, where R( 1 R( R(, 1 R( 1 1 R( R( 1 Note that A R( t dt R( t dt R( t dt 1 R( t dt R( t dt R( t dt Let G1 R( t dt, G2 f ( [1 R( R( [1 R( ] R( R( 2 ] [ f ( f ( ] f ( [1 R( [1 R( ] f ( R( 2 ] R( hus, is a decreasing function of Note that R( t dt is also decreasing in since R (t is a decreasing function herefore, G 1 is a decreasing function G R( R( t dt 2 or G 2, 2 Letting ( ( ( R R t dt, 6

82 3 AVALABLY MODEL AND OPMZAON 61 ( R( f ( R( f ( hen, ( e have lim ( lim( R( R( t dt hus, there must be one such that ( G2 Hence, we have G2 for and for G1 G2 hen, if for, the maximum of A will be at Otherwise, there will be a maximum of A for 343 Calendar-based limiting average availability with renewal after failure (B Under assumption B, we assume a system remains unchanged if found working at inspection and is renewed only after repair/replacement when a failure is detected One cycle is the time between two inspections in which failures have been detected (igure 38 Note that the inspection time and repair/replacement time in the first interval belongs to this cycle, while the time for the next inspection (in which a failure is detected and repair/replacement belongs to the next cycle he definition of one cycle is the same as in section 341 under assumption A (igure 37 Recall that under assumption B, the system is only renewed after failure, and under assumption A, renewal happens at each inspection 61

83 3 AVALABLY MODEL AND OPMZAON 62 nspection ailure R Renew R Renew One cycle Next cycle igure 38 Cycle length under assumption B with Calendar-based inspection policy e formulate limiting average availability models with both random and constant downtime due to inspection and repair/replacement e also present availability models for a system whose lifetime follows an exponential distribution as a special case inally, we discuss the optimal to maximize availability 3431 Calendar-based limiting average availability formulation (B Suppose inspection time and repair/replacement time are random variables i and Y i, distributed with means of and Y respectively i and Y i are bounded with constant x m and y m respectively Similar to assumption A with renewal at each inspection, it is assumed y x to ensure that inspection and repair/replacement can be finished in interval m m Conditioning on, Y, 1,, we have: Event Probability N= q1 Y ( Y 62

84 3 AVALABLY MODEL AND OPMZAON 63 1 Y 1 Y N=2 N=3 q R( Y R( ( Y 2 j1 j q R( ( Y R( ( Y 3 j1 j1 j1 j1 Y 1 i-1 1 i-1 N i qi R( i 1 R( i,, i ( j1 Y, i 1 i j1 he cycle time conditioning on, Y, 1, is L(, Y, 1, i q iq R( i i i i1 i1 i he expected cycle time will be EL(, Y, 1, E[ R( ( j Y ] i i j1 E[ R( ( j Y i j i E[ R( ( j Y ] i j i (325 Let i j (326 i j g( E[ R( ( Y ] he expected cycle time can be written as (1 g( he expected uptime in one cycle is 63

85 3 AVALABLY MODEL AND OPMZAON 64 E( tf ( t dt R( t dt he calendar-based availability model with random inspection time and random repair/replacement time under assumption B is the following: A (327 (1 g ( e may also formulate the cost model when inspection time is a constant, as a special case of (327 e assume that ym Let g( R( ia b hen a, b Y, we have Eg( E[ R( i( Y ], i1 i1 which is a special case of g ( he limiting average availability model with constant inspection time and random repair/replacement time will be the following: A (328 (1 Eg ( f both the inspection and repair/replacement times are constant, say and R, we may obtain the limiting average availability model as a special case of (328 n this case, g( R( ia b, a, b R he calendar-based availability model with constant i1 inspection time and constant repair/replacement time is given by 64

86 3 AVALABLY MODEL AND OPMZAON 65 A (1 g ( ( Extension of calendar-based model assuming downtime longer than or calendar-based inspection policy, we assume as shown in previous availability models with a constant inspection time and repair/replacement time However, it is also of interest to consider the situation of n particular, assuming that ( k 1 k, k 1,2,, and the system remains unchanged if found working at inspection, the limiting average availability of the system is: A i1 R( t dt ( i k 1 q i ( k 1 i R( i (33, i where qi R( i1 R( i, i i( ( k 1 i 1,2, 3433 Calendar-based availability for system with lifetime following exponential distribution n section 3431, we present availability models for a hidden-failure system with an arbitrary distributed lifetime n this section, we discuss availability models for a system whose failures follow an exponential distribution, ie R( t exp( t Suppose the downtimes for inspection and repair/replacement are random rom (326, 65

87 3 AVALABLY MODEL AND OPMZAON 66 i g( E[ R( ( Y ] E exp[ ( ( Y ] j j i j i j i e ( ( 1 ( (331 i1 Y [exp( ( ] Y( i e where Y ( and ( are the Laplace transform of Y and respectively; these exist at because Y and are bounded ( Y ym, xm hen, equation (327 can be written as follows: 1 A (332 (1 g( or the case with random repair/replacement time but constant inspection time, the availability model is a special case of (332, given by: 1 A, where (1 Eg( ( e Eg( ( ( Y (333 1 e f repair/replacement time is also a constant R, from (333, the limiting average availability model can be written as follows: 1/ A, where (1 g ( ( R e (334 g ( ( 1 e t is convenient to assume repair/replacement time follows (, distribution since Y i is bounded by Yi ym f Z follows a standard B distribution, 66

88 3 AVALABLY MODEL AND OPMZAON 67 f( z z (1 z (, 1 1, z 1 Y ymz, then ymz ym y y hus, ( y P( ymz y P( Z Z( y y, m m y y ( (1 y f ( y Z ( y y y y ym ym y ( ym y 1 m m m (, m (,, y ym (337 hen, for a system with an exponential distribution, z (1 z Y ( Ee Ee e dz (, Y yz yz A (, distribution can be used for bounded inspection time and repair/replacement time Similarly, we may use (, distributed inspection time since i is bounded by i xm 344 Calendar-based availability optimization with renewal after failure (B or simplicity, we consider the case of constant inspection time and random repair/replacement time rom (328, A (1 Eg ( Maximizing A is the same as minimizing (1 Eg( Let G( (1 Eg( rom Appendix B, we know that ( a b br( a b max{, } g ( a a 67

89 3 AVALABLY MODEL AND OPMZAON 68 Since a, b Y, EY E[ Y R( Y ] Eg( EY E[ Y R( Y ] hen, max{, ( } G( [1 ] Note that E[ Y R( Y ] EY and Rt ( 1, we have max{, ( Y Y } G( (1 ( y Let G1 ( max{, }, G or (, G ( Y (1 R (, Y G (, Y ( (1 G 1 Y ; G1( lim G1( Y G1 ( ( Y or Y,, G 2 1 ( ( is decreasing in or Y, G ( 1 is increasing in when Y, min G1 ( Y G2 ( ( Y or G 2 (, 1 2 ( or (, G 2 ( is decreasing or (, G 2 ( is increasing Y Y 68

90 3 AVALABLY MODEL AND OPMZAON 69 or ( Y, G ( is minimal 2 ( Y MinG2 ( ( ( Y [1 ] ( ( ( ( Y Y Y 2 ( Y Y herefore, 2 Y min G( ( Y (336 Max A exists, and max A 2 ( Y Y or the scenario of constant inspection time and repair/replacement time, we have: max A 2 ( R R igure 39 shows the relationship between G 1 ( and G 2 ( G ( is bounded by G 1( and G ( 2 G 2 ( ( R 2 R G( G 1 ( igure 39 Relationship between function G1(, G2( and G ( 69

91 3 AVALABLY MODEL AND OPMZAON 7 35 Numerical example Assume that the failure of a hidden failure system follows a eibull distribution with 8 hours he inspection time is 8 hours he repair/replacement time for a failed system, R is 8 hours e investigate the limiting average availability model of the system under assumption A with renewal at each inspection and compare the results with those of assumption B with renewal after failure, considering both age-based and calendar-based inspection polices Availability function under assumption A with renewal at each inspection rom equation (31, the limiting average availability with an age-based inspection policy is A R R R( t dt igures show the limiting average availability as a function [1 ( ] of the based on different values, eg, equals 5, 1, and 2 igure 31 Age-based availability vs igure 311 Age-based availability vs with 5 (A with 1 (A * * Amax 8619, 171 hours Amax9557, 356 hours 7

92 3 AVALABLY MODEL AND OPMZAON 71 igure 312 Age-based availability vs with 2 (A A hours * max 9869, 911 According to equation (323, the calendar-based limiting average availability model is R (1 R( R( t dt R( ( R R t dt A [1 R( R( ] R e omit the figures showing a calendar-based inspection policy since they are very similar to those of age-based inspection policy able 31 summarizes the optimal under assumption A for age-based and calendar-based inspection policies able 31 Optimal with renewal in each inspection (A Age-based Calendar-based A * max A * max = h h = h h = h h 71

93 3 AVALABLY MODEL AND OPMZAON 72 Both policies have very close maximum limiting average availability ( A max for the same value due to short inspection time and repair/replacement time he optimal inspection interval ( * for the case of 1 is smaller than for 1 he optimal for calendar-based inspection policies are slightly longer than for age-based policies Availability function under assumption B with renewal only after failure igures show the over limiting average availability based on different values, eg, equals 5, 1, and 2 e omit the figures showing a calendar-based inspection policy since they are very similar to those of age-based inspection policy he shape parameters are shown in igures igure 313 Age-based availability vs igure 314 Age-based availability vs with 5 (B with 1 (B * * Amax 9672, 458 hours Amax9557, 34 hours 72

94 3 AVALABLY MODEL AND OPMZAON 73 igure 315 Age-based availability vs igure 316 Age-based Availability vs with 2 (B with 4 (B A hours * max 9531, 322 A hours * max 9536, 326 Note that the optimal is almost same if the shape parameter is greater than 1 igure 32 shows availability over the when is 4 hile the function is not unimodal, the optimal changes little in comparison to those with a smaller value, say, 2 he overall optimal and maximum limiting average availability are summarized in able 32 n the table, we compare the optimal under assumptions A and B using both inspection policies able 32 Overall Optimal vs maximum availability Assumption A Assumption B =8h Age-based Calendar-based Age-based Calendar-based A * max A * max A * max A * max = h h h h = h h h h = h h h h = h h 73

95 3 AVALABLY MODEL AND OPMZAON 74 or 1, the optimal under assumption A is much shorter than under assumption B he maximum limiting average availability ( A max is higher with renewal after failure (assumption B than at each inspection (assumption A Under assumption B, when is close to or greater than 1, the maximum limiting average availability ( to the value for either policy Both policies have a very similar A max and optimal ( * are not sensitive A max, in the range of 95 he optimal is almost the same, even if the shape parameter is changed; it remains similar up to 1 with the same value or both inspection policies, the optimal for calendar-based inspection policy (around hours is slightly longer than for age-based policies ( hours for the same value t is interesting to see how the results change with a fixed but a different scale parameter hen changes from 8 hours to 6 hours, the optimal is 277 hours, with the maximum limiting average availability of 946 inally, for the case of 1, the optimal is more sensitive to than to 36 Concluding remarks Availability optimization is an important topic for hidden-failure systems his chapter presents availability models for periodically inspected repairable systems under different assumptions Both age-based and calendar-based inspection policies are considered under assumptions A (renewal at each inspection and B (renewal after failure or each case, we consider both random downtime and constant downtime due to inspection and repair/replacement n addition, 74

96 3 AVALABLY MODEL AND OPMZAON 75 we study instantaneous availability models for both inspection policies with constant inspection time and repair/replacement time e show that if an age-based inspection policy is applied to the system, the optimal inspection interval always exists to maximize system availability, both in assumption A with renewal at each inspection and in assumption B with renewal after failure e provide the range of the maximum availability under assumption B f a calendar-based inspection is applied to the system, it is reasonable to assume the downtime due to inspection and repair/replacement is smaller than the inspection interval, since the downtime is relatively short compared to the inspection interval Obviously, the assumption is very common in real applications However, in this chapter, we also show the limiting average availability model for the case in which constant downtime overpass inspection interval under assumption B as a theoretical extension ith regards to a calendar-based inspection policy, we show that an optimal exists to maximize system availability under assumption B with renewal after failure e present the range of maximum availability for this case Under assumption A with renewal at each inspection, an optimal inspection interval may or may not exist, depending on the distribution of time to failure of the system n particular, if there is no optimal, the maximum availability will be at Otherwise, there is an optimal to maximize availability for e note that optimal and maximum limiting average availability with age-based and calendar-based policies are often very similar, as downtimes for inspection and repair/replacement are relatively small he calendar-based inspection policy is recommended 75

97 3 AVALABLY MODEL AND OPMZAON 76 since it is easier to schedule than the age-based inspection policy n this chapter, we also explore availability of system with particular distribution or example, we consider the case that failure of the system follows exponential distribution e also investigate the availability model with random inspection time and repair/replacement time following Beta distribution Moreover, we extend the availability model to include instantaneous availability with constant inspection time and replacement time e consider both age-based and calendar-based inspection policy Our work includes the results obtained by Sarkar and Sarkar (2 and Cui and ie (25 as special cases of our models Note that consideration of non-negligible inspection downtime is of importance n many real applications, the inspection time cannot be ignored, especially in the case when the repair/replacement time is close to the downtime due to inspection e will demonstrate it in case studies in Chapter 5 n the next chapter, we will discuss cost optimization assuming non-negligible downtimes due to inspection and repair/replacement 76

98 4 OPMAL NSPECON NERVAL O MNMZE COS 41 ntroduction n this chapter, we investigate the optimal to minimize the expected cost for a single unit system n general, systems with hidden failures can be classified into two categories: ype : Protective device (safety system: its function is to protect the main (protected system function eg safety valve ype : Operating device: its function is the main function but system failures are hidden and only can be detected during an inspection, eg underground equipment he consequences of the failures of the two types of system are different ailure of a type system will cause direct loss even if the failure is not detected f a type system fails, there may not be a direct loss right after the failure, but an accident cost might occur if the protected system fails due to lack of protection Considering both types of systems in this study, we investigate optimal failure finding interval to minimize the long run average cost induced by inspection, repair/replacement, production loss, or possible accidents 77

99 4 OPMAL NSPECON NERVAL O MNMZE COS 78 a ype (Protective device ailure b ype (Operating device ailure igure 41 wo types of system with hidden failure irst, we develop several cost models with an age-based inspection policy in section 42 e start with the assumption of renewal at each inspection of the hidden failure system, assumption A hen, we formulate models based on assumption B, which assumes that the system remains unchanged if found working at inspection and is renewed by repair/replacement only when failure is detected or both cases, we construct general models for random inspection time and repair/replacement time e also formulate models for constant inspection time and repair/replacement time as special cases inally, we optimize cost models for each case with the purpose of obtaining the optimal to minimize the expected cost Second, we develop cost models with a calendar-based inspection policy in section 43 As in our study of an age-based inspection policy, we start with assumption A with renewal at each inspection and assumption B with renewal after failure or both cases, we formulate cost models in general, before moving to a special case Again, we investigate the optimization of cost models to explore the optimal to minimize the expected cost All models assist us in 78

100 4 OPMAL NSPECON NERVAL O MNMZE COS 79 calculating the excepted cost in terms of the failure finding interval, and in obtaining an optimal to minimize cost All above models assume that the system is periodically inspected with interval and failures are rectified only at periodic inspection f an accident happens due to lack of protection, we fix the protected system but not the protective devices As a result, if the protected system needs to be protected before the next inspection time, there may be another accident his assumption is common in cost optimizations for single-unit systems due to its simplicity (Vaurio, 1995 n practice, however, we may also fix the protective device immediately after an accident e develop a model based on this assumption in section 45 with the age-based inspection policy he structure of the cost models derived in Chapter 4 is shown in igure 42 Numerical examples are presented in section 46 Cost Models 42 Age-based inspection policy 43 Calendar-based inspection policy 44 Another model with age-based inspection Policy 421 Age-based cost model with renewal at each inspection (A 422 Age-based cost optimization with renewal at each inspection (A 423 Age-based cost model with renewal after failure (B 424 Age-based cost optimization with renewal after failure (B 431 Calendar-based cost model with renewal at each inspection (A 432 Calendar-based cost optimization with renewal at each inspection (A 433 Calendar-based cost model with renewal after failure (B 434 Calendar-based cost optimization with renewal after failure (B igure 42 Structure of cost models 79

101 4 OPMAL NSPECON NERVAL O MNMZE COS 8 e use the following notations: : fixed inspection interval length L ( : cycle time : constant downtime due to inspection R : constant downtime due to repair/replacement if a failure is found during inspection i : random variable, downtime due to inspection, with mean Y i : random variable, downtime due to repair/replacement if a failure is found during an inspection, with mean Y Y : downtime due to repair/replacement in the first interval of a cycle ux:density ( function of random inspection time vy:density ( function of random repair/replacement time : time to failure of the system f( t : probability density function associated with the system lifetime R (t : reliability (survival function of the system t (: cumulative distribution function of the system : mean lifetime of the system, R( t dt C ( : expected average cost per unit time C : cost of inspection per unit time C ( C : cost of repair/replacement per unit time for a failed system R C L : cost of loss due to system failure per unit time 8

102 4 OPMAL NSPECON NERVAL O MNMZE COS 81 C A( R L C, and C : cost of an accident (protective system has already failed when called upon K : frequency of true demands calling for the system to function 42 Cost model with age-based inspection policy n this section, we study cost model optimization of a hidden failure system with an age-based inspection policy e assume that if an accident happens due to the lack of protection, we fix the main system immediately with negligible downtime, and the protective device is inspected at the next inspection time (refer to igure 33 in Chapter 3 irst, we construct cost models under assumption A with renewal at each inspection e then develop cost models under assumption B with renewal at repair/replacement whenever failure is detected or both cases, we discuss the optimization of the cost model to obtain the optimal interval to minimize the expected cost 421 Age-based cost model with renewal at each inspection (A he cost function model of the system can be obtained by the property of a renewal process as follows: C( Expected cost in one cycle Expected time in one cycle he expected cost in one cycle includes the following: 1 nspection cost his is the product of the mean cost of inspection per unit time ( C and inspection time in one cycle 81

103 4 OPMAL NSPECON NERVAL O MNMZE COS 82 2 Repair/replacement cost for a failed system his is the product of the mean cost of repair/replacement per unit time ( C and repair/replacement time in one cycle R 3 Cost of downtime e define C L as the loss cost factor which is the mean loss cost of the system per unit he loss cost depends on the product of C L and downtime during the cycle he loss cost might be zero if the failure of the system itself does not cause any loss Normally, most type equipment belongs here, eg failure of a protective device does not incur a productive loss he only consequence is that the protected system is not protected until the next scheduled inspection 4 Cost of accident An accident cost is incurred if the (protective system has already failed when called upon C A is the mean cost of an accident Normally, C A is much higher than C, C R and C L, since severe consequences or even a disaster may eventually occur Parameter K reflects the average frequency of true demands on the protective system to function he demands may reflect different type of protection, eg, for a safety valve, there is a demand to open, a demand to close etc Provided the (protective system does not fail, no accident cost is incurred However, if the (protective system fails during the cycle, the expected accident cost will be CA K multiplied by the downtime of the (protective system in the cycle e define ~ C L C L C A K to facilitate model formulation Assume the inspection time is randomly distributed with mean, and the repair/replacement time is also a random variable with mean of Y herefore, we have: 82

104 4 OPMAL NSPECON NERVAL O MNMZE COS 83 Possisble cycle Cycle length Cost L1 C Y L2 Y C CRY CL( rom section 331 in Chapter 3, we know the following: he expected cycle time, EL( is ( he cost in one cycle is C ( [ C CRY CL( ] ( he expected cost in one cycle is Y EC( C E C E[ Y ( ] C E[( ( ] R L C C ( C E[( ( ] R Y L Since E( ( E ( P( [ ( t f ( t dt] ( ( t f ( t dt ( ( = ( t f ( t dt ( t dt, the expected cost in one cycle is C C ( C ( t dt R Y L Recall that C equals C C K herefore, the cost model is as follows: L L A 83

105 4 OPMAL NSPECON NERVAL O MNMZE COS ( ( ~ ( ( dt t C C C C Y L Y R (41 n particular, if the inspection time is not random but a constant, say,, the cost model for constant inspection time and random repair/replacement time is represented by (42: ( ( ~ ( ( dt t C C C C Y L Y R (42 Obviously, if both the inspection time and repair/replacement time are constant, as denoted by and R, (41 can be rewritten as ( ( ~ ( ( dt t C C C C R L R R (43 As cost factors L C and A C may be associated with failure modes of the system, for different failure modes, they can be different 422 Age-based cost optimization with renewal at each inspection (A rom (4,1, ( ( ~ ( ( dt t C C C C Y L Y R hen, both ( and dt t ( tend to zero, we have ( ( ( lim ( lim R Y L C C C t dt C C C

106 4 OPMAL NSPECON NERVAL O MNMZE COS Note that ( ( t dt R t dt hus, equation (41 can be rewritten as ( ( ~ ~ ( ( dt t R C C C C C Y L L Y R hen, ( 1, we have lim ( R t dt hen, ( ( ( lim ( lim R Y L L L L A Y C C C C R t dt C C C C C K Under the assumption of renewals at each inspection, ( C will approach C when tends to zero and approaches K C C A L when tends to infinity hen tends to zero, all failures will be prevented so that only inspection cost is incurred hen tends to infinity, there is no cost for inspection or repair/replacement so that ( C approaches K C C A L e further investigate the case with an optimal local Note that ] ( [ ] ( ][1 ( ~ ( [ ] ( ][ ( ~ ( [ ( 2 f dt t C C C C f C C Y Y L Y R Y L Y R hen, ( 1 Recall our assumption of ( lim f, 2 ( ] ( ~ ~ ( [ ] ( [ ( ~ lim ( lim Y L L Y R Y L dt t R C C C C C C 2 ( ( 1 ( ( ( ( ( lim ( L L Y R Y L Y C C C C C R t dt =

107 4 OPMAL NSPECON NERVAL O MNMZE COS 86 hen, ( and ( t dt tend to zero hen, C( C lim lim f ( ( C R Y 2 ( (1 f ( C f ( ( C C f ( C lim R R Y 2 ( Y Y f ( ( CR C C Case : f C C, ie C CL CAK L n this case, C( C(, is never the optimal solution may be the optimal solution, depending on the parameters and the distribution of the system igure 43 (A shows the possible cost functions for the case of ~ C C L 1 f f ( ( C C C, Y R ie f ( C C(, then lim ( CR C Y he cost function decreases right after and there is at least one local optimal ( to minimize the cost he solid line in igure 43 (A indicates the case with an optimal C 2 f f (, and if there exists any such that C( C, ie ( C C R Y RY ( L L ( [ Y ( ] C C C C R t dt C, or, ( C C C R( t dt ( C C (, then similar to case 1, there is at least a local L L R Y optimal with the minimum cost smaller than C L 86

108 4 OPMAL NSPECON NERVAL O MNMZE COS 87 C Otherwise, if f (, and if for all, C( C, ie, ( C C R Y ( CL C CL R( t dt ( C ( R C Y, then is the optimal solution, with C( C as the minimum cost he dashed line in igure 43 (A indicates this case C( C( C L +C A K C C C L +C A K A B igure 43 Possible cost functions vs t is difficult to obtain further conclusion for the general case; we study the optimal for a system whose failures follow an exponential distribution in more details Here, f ( 1 f 2 f C ( C C R Y C ( C C R Y, there is at least one local optimal to minimize the expected cost, is the optimal solution Proof: f C( C for all, ie, 87

109 4 OPMAL NSPECON NERVAL O MNMZE COS 88 t CL ( CL C CL e dt ( C ( R C Y ( CL C (1 e ( CR C Y ( C C ( C [ L ( ] ( ( [ L ( ](1 t L C CR C Y CL C CR C Y e CL f ( CR C Y, the above equation is obviously true CL f ( CR C Y, note that both sides of the equation are zero for aking CL derivatives on both sides, we have CL C [ ( CR C Y ] f ( Note that f( CL f CL C [ ( CR C Y ], then CL CL C [ ( CR C Y ] f ( CL CL hus, ( C L C [ ( CR C Y ] ( CL C [ ( CR C Y ] Or, C CL C CL ( CR C Y ( C C R Y herefore, if C ( C C R Y, C( C C( for all he minimum cost approaches C when tends to zero or example, if there are many failures ( is high which are expensive ( C is relatively significantly greater than R C, we should apply inspections as often as possible 88

110 4 OPMAL NSPECON NERVAL O MNMZE COS 89 Case : f C C, ie C CL CAK L n this case, C( C(, is never the optimal solution may be the optimal solution, depending on the parameters and failure distribution of the system igure 43 (B shows the possible cost functions for the case of C CL 1 f for all, C( CL, ie C CRY ( CL CL R( t dt C ( ( L Y, or ( C C ( C C ( C R( t dt, then the minimum cost approaches L R L Y L CL CAK when tends to infinity he possible cost function is shown as the dashed lines in igure 43 (B 2 Otherwise, if there exists any such that ( C C ( C C ( C R( t dt, then there is at least a local optimal L R L Y L to minimize the expected cost he solid line in igure 43 (B indicates this case Proposition: Suppose failure of the system follows exponential distribution, is the optimal solution if and only if CL ( C C ( C C L R L Y Proof: f C( CL for all, then ( C C ( C C ( C R( t dt, or, L R L Y L CL ( C CL ( CR CL Y ( (1 e CL CL ( C CL [( C R CL Y ] ( ( C CL [ ( CR CL Y ] ( 89

111 4 OPMAL NSPECON NERVAL O MNMZE COS 9 CL Or equivalently, ( C CL ( CR CL Y since ( 1 ie CL ( C C ( C C L R L Y CL So, if C CL and, there is a local optimal to minimize ( C C ( C C L R L Y the expected cost Otherwise, if CL, the minimum cost approaches ( C C ( C C L R L Y CL CAK when tends to infinity or example, if the cost for downtime and accident is significantly smaller than that for inspection and/or repair, there is no need for inspection e may apply a similar discussion in cases when the downtime due to inspection and/or repair/replacement is constant hey can be regarded as special cases of random downtime n such special cases, we simply replace and Y by and R respectively 423 Age-based cost model with renewal after failure (B Under assumption B, the system remains unchanged if found working at inspection and is renewed only at repair/replacement when a failure is detected e develop the cost model for random inspection time and repair/replacement time, then move to the special case of constant downtime due to inspection and repair/replacement Recall g ( i R( i ; from Chapter 3, we know that the expected cycle time is 9

112 4 OPMAL NSPECON NERVAL O MNMZE COS 91 EL( ( R( i ( g( Y Y i he expected uptime during the cycle is R ( t dt, then the expected downtime is D ( ( g( Y (44 he expected total cost in one cycle includes the following costs: 1 he expected inspection cost is E( C p C ip i i i i1 i1 2 he expected repair/replacement cost is E( CRY CR Y 3 he expected loss cost and possible accident cost is ( C C K E[ D ( ] L A According to (44, the expected cost for loss and accident will be C [( g( ] L Y he cost model with random inspection and repair/replacement time is: C g( CRY CL[( g( Y ] C ( ( g( Y (45 he cost models for the scenario of constant inspection time and random repair/replacement time represent a special case, given by equation (46 he age-based cost model with a constant inspection time of is: ~ C g( CRY CL[( g( Y ] C( (46 ( g( Y he cost model for constant inspection time and repair/replacement time R is: 91

113 4 OPMAL NSPECON NERVAL O MNMZE COS 92 C C( ~ g( CRR CL[( ( g( R g( R ] (47 Note that in most cases, for type (safety system, the value of C L might be zero Normally we use C ~ to represent the loss and accident cost factor, which in this case will be CL CAK L 424 Age-based cost optimization with renewal after failure (B n this section, we investigate the optimal inspection interval to minimize the cost under assumption B e start with a general approach to cost optimization; we then discuss the optimal for a system with failures following an exponential distribution as a special case 4241 Optimal age-based to minimize cost with renewal after failure (B rom equation (45, C C( C( C L ~ g( CRY CL[( g( Y ] ( g( Y C g( CRY CL (48 ( g( Y rom appendix A, lim g (, and g( C g( CRR CLY lim C( lim[ C ] C C g( g( g( g( hen, g ( 1 as L L Y Y hen, 92

114 4 OPMAL NSPECON NERVAL O MNMZE COS 93 C g( CRR CLY lim C( lim[ CL ] CL g( g( g( g( Y Y hus, when tends to zero, the cost will be C C C K ; and when, C ( tends L A to C, or C C K L L A Proposition 1: is the optimal solution if and only if CL C CRY Proof: rom (48, we note that if for all, C( CL, then is the optimal solution ie, CL CRY C g( CRY CL, or g ( e proved in appendix A that g ( 1, so C CL CRY if 1, or equivalently, CL C CRY, then C( CL for all he C minimum cost will be C L when tends to infinity CL CRY Otherwise, if 1, there exists at least one local optimal to minimize the C expected cost Proposition 2: f CL C CRY, then C C C C CL min C( C G (, where is the at L R Y L R Y L 1 Y Y 2 which G ( has the minimum value 2 C g( Proof: Let G1 ( ( g( Y, and ( CL CRY G2 ( g( Y 93

115 4 OPMAL NSPECON NERVAL O MNMZE COS 94 hen C( CL G1( G2( Letting G( ( g(, (49 ~ then from equation (316 in section 3341, we know that that G ( is a bounded function with a minimum value between min G ( 2 herefore, G ( has a maximum value which satisfies 2 C C C C max G ( L R Y L R Y 2 2 Y Y Accordingly, we may estimate the range of minimum value of C ( as C C C C CL min C( C G (, L R Y L R Y L 1 Y Y 2 where is the at which G ( has the minimum value 2 igure 44 shows the possible relationship of G, G (, and C( functions he bold solid 1( 2 line shows the case of a local optimal he normal solid line indicates the case when is the optimal solution 94

116 4 OPMAL NSPECON NERVAL O MNMZE COS 95 C ( C L C( C G 1 ( G 2 ( igure 44 Relationship between C(, G, G ( 1( 2 e may apply a similar discussion to cases when the downtime due to inspection and/or repair/replacement is constant hey can be regarded as special cases of random downtime, and we simply replace and Y by and R respectively 4242 Optimal for system with lifetime following exponential distribution n following section, we discuss the optimal of a system with failures following an exponential distribution with failure rate e have R( t exp( t, ( g i 1 exp( R( i 1 exp( exp( 1 CL CRY rom proposition 1 in section 4241, if 1, ie, CL / CRY C, or C C ( C C, then there is a local optimal to minimize the expected cost Otherwise, L R Y the minimum cost will be C ~ L when tends to infinity 95

117 4 OPMAL NSPECON NERVAL O MNMZE COS 96 Suppose now C ( C C, from equation (48, L R Y C( C L C C L L C g( CRY CL ( g( Y C exp( / (exp( 1 CRY CL ( exp( / (exp( 1 Y exp( ( C CRY CL / CRY CL / (41 exp( ( Y Y o obtain the optimal, let A C C C /, B C / C R Y L L R Y Equation (41 can be written as C( C L exp( A B exp( ( Y Y o get the minimum value of C(, the numerator of C( should be zero, ie, e A[ e ( ] ( e A B[ e ( e ] Y Y Y A[ e ( ] ( e A B[ ( 1] Y Y Y Ae ( A e A( e A B( B Y Y Y Y A e A B( B Y Y Replacing A and B when necessary, we have: ( C C C / e A ( C / C ( ( C / C R Y L Y L R Y Y L R Y e A ( C / C ( C ( C / C L R Y L R Y e A B B C B, (411 where A C C C /, B C / C R Y L L R Y 96

118 4 OPMAL NSPECON NERVAL O MNMZE COS 97 By solving equation (411, we may obtain an optimal to minimize the expected cost he general solution can be applied to constant inspection time and repair/replacement time by replacing and with and respectively Y R 43 Cost model with calendar-based inspection policy n this section, we study cost models optimizing a hidden failure system using a calendar-based inspection policy As in section 42, we first develop cost models under assumption A with renewal at each inspection e then formulate cost models under assumption B with renewal only at repair/replacement whenever failure is detected Please refer to igure 34 in Chapter 3 to see the two different assumptions as they appear in a calendar-based inspection policy or both assumptions, we develop general cost models for random inspection time and repair/replacement time e then discuss cost models with constant downtime due to inspection and repair/replacement as special cases inally, we investigate optimal intervals to minimize the expected cost 431 Calendar-based cost model with renewals at each inspection (A Under assumption A, the system found working at inspection will be renewed after inspection time, while a failed system will be renewed after repair/replacement e define one cycle as the time between two inspections at which failures have been detected (refer to igure 37 in Chapter 3 Note that the inspection time and the repair/replacement time in the first interval belong to this cycle, while the times for the next repair/replacement and inspection detecting 97

119 4 OPMAL NSPECON NERVAL O MNMZE COS 98 failure belong to the next cycle As in section 421, the expected cost in one cycle includes the following: inspection cost, repair/replacement cost, cost for downtime and accident cost irst, we consider a scenario where inspection time and repair/replacement time are random variables i and Y i, distributed with means of and Y respectively e assume that i and Y i are bounded with a constant x m and y m respectively e also assume y x m m As shown in equation (318 in Chapter 3, the expected cycle time will be: EL Y ER( Y E( (,, 1, [1 ] (412 o derive the cost model, we first find the expected cost per cycle Conditioning on, Y,,, we have: 1 Possible cycle Probability Cost Y R+C 1 Y 1 ( C CRY C ( Y N= p1 Y N=2 p2 R( Y ( 1 L C ( C Y 1 R C ( L Y 1 2 N=3 p3 R( Y R( ( C C Y i1 R i1 C ( L

120 4 OPMAL NSPECON NERVAL O MNMZE COS 99 1 Y 1 i-1 i-1 N i ( i 2 p R( Y i i2 j1 R( ( i1 j i1 C C Y i1 R i1 C ( L i1 i1 he expected total cost in one cycle includes the following costs: 1 he expected inspection cost conditioning on, Y, 1, is: E (, Y, C E[ ( Y ] C ie( p, 1 j1 i i2 j1 i1 i1 C E[ ( Y ] C ie{ R( Y [ R( ] ( } j1 k i1 i2 j1 k 1 i2 i1 i2 C [ E( Y ier( Y [ ER( ] E( ] i2 Recall equation (318 in Chapter 3; the second term of the above equation is [1 E( ] ER( Y E( he expected cost in one cycle is given by: 1 E( E (, Y, 1, C [ ER( Y ER( Y ] E( C ER( Y E( [1 ] (413 2 he expected repair/replacement cost is E( CRY CRY 3 he expected loss and accident cost will be C ( C C K times the expected downtime L L L 99

121 4 OPMAL NSPECON NERVAL O MNMZE COS 1 Let the expected downtime in one cycle be D ( ; thus, D ( EL (, Y,, E max{, Y } E max{, } P( Y D E max{,,} P( Y,, i2 i2 i i 1 1 i1 i1 E max{, Y } E max{, } ER( Y 1 1 E ER Y ER i1 max{, } ( [ ( ] As in section 341, let A E max{, Y }, B E max{, }, C ER( Y, D ER( BC 1 D i1 D ( A BC BC D A i2 herefore, the expected downtime is D E Y E max{, } ER( Y ER( ( max{, } (414 rom equations (413-(415, the calendar-based cost model with random inspection time and repair/replacement time under assumption A is given by: C CRY CLD ( C ( ER( Y E( [1 ], E max{, } ER( Y where D ( E max{, Y }, ER( ym xm xy, E max{, Y } ( x y t f ( t u( x v( y dtdxdy (415 x m x E max{, } ( x t f ( t u( x dtdx 1

122 4 OPMAL NSPECON NERVAL O MNMZE COS 11 f the inspection time is not random but a constant, the cost model for constant inspection time and random repair/replacement time is as follows: C CRY CLD ( C ( ER( Y ( [1 ], ER( Y where D( E max{, Y } ( t dt, R( (416 ym y E max{, Y } ( y t f ( t v( y dtdy Suppose we take time for inspection; if a failure is detected, an additional time of R is necessary for repair/replacement his scenario is a special case of (415 and leads to the following cost model: R R( R CRR CL[ ( t dt ( t dt] C ( C ( (417 R( R [1 ] ( 11

123 4 OPMAL NSPECON NERVAL O MNMZE COS Calendar-based cost optimization with renewal at each inspection (A or simplicity, we discuss the optimal with constant inspection time and repair/replacement time rom cost model (417, we have R R( R ( t dt ( t dt C ( C R R C( CL (418 R( R R( R [1 ] [1 ] ( ( hen, R( tends to one, R R R C ( ( C L RR C t dt R lim C ( R ( ( ( 1 ( ( ( 1 R R R R R R (1 ( R R R ( R L ( C C C t dt ( (1 ( R R hen, the first two terms of equation (418 tend to zero, we have R R( R ( t dt ( t dt ( lim C( CLlim CL R( R (1 ( n what follows, we further investigate the cost function of equation (418 to discuss the property of optimal interval Recall R ; then the second term of equation (418 will be: C R R CRR ( CRR (1 R( R( R (1 [ ( R( ] [1 R( R( ] ( 12

124 4 OPMAL NSPECON NERVAL O MNMZE COS 13 Recall 1 R( 1, and 1 R( R( R(, 1 R( R( equation (418 can be written as: C (1 ( ( ( ( C R R t dt R t dt C( CL [1 R( R( ] C CRR C L L R R C (1 ( t dt C ( t dt (419 C ( ( C C t dt C t dt C R R (1 L L R R Let ( C CR C R L ( t dt C L ( t dt C RR G1, G2 ( (1, G3 ( 1 G ( 1 is a decreasing function hen, ( C CRR G1 hen, G 1 ( tends to zero 2 Property of G ( : 2 e prove in section 342 that is a decreasing function of, and 1 is an increasing function of CL ( t dt Let G2 (, then G2 ( C L ( ( 2 t dt 13

125 4 OPMAL NSPECON NERVAL O MNMZE COS 14 Let ( ( ( k t dt Note that k ( ( f ( ( f ( ; k ( is a non-decreasing function of hen, k ( hus, k ( for all herefore, G ( 2 ; G2 ( is non-negative increasing in hus, G ( is increasing in e see that 2 G2( lim G2(, lim G2 ( CL 3 Property of G ( : 3 CL ( t dt C RR G3( CL( CL ( t dt C RR Let G3 ( e have 2 Let k ( be the numerator of the above equation Note that k ( CL[ ( f ( ( ] CL f (, thus, k ( is non-decreasing in hen, R k( C ( C ( t dt C L R L R R Since R ( ( ( t dt, ( R R R k when hen k ( for all herefore, G ( 3 is increasing in CL ( t dt C RR e further notice that G3 ( R 14

126 4 OPMAL NSPECON NERVAL O MNMZE COS 15 C ( t dt C G3( lim G3( lim L R R R( CL( CL CRR lim ( 1 R( R( hus, C ( is the summation of G (, i 1,2,3 hen takes the possible minimum value of i C (1 ( R CRR ( R CL ( t dt, C ( inally, C ( will approach to ( (1 ( R R R C L when tends to infinity Case : C( C L n this case, is never the optimal solution may be the optimal solution, depending on the parameters and the distribution of the system igure 43 (A shows the possible cost functions for the case of C( C L f for some, and if we have 3 i1 Gi (, and C( C(, then there exists at least one optimal with the minimum cost he bold solid line in igure 45 (A indicates a possible case with an optimal Otherwise, if C( C( for all, then is the optimal solution and the minimum cost will be C ( he regular solid lines in igure 45 (A show this possibility 15

127 4 OPMAL NSPECON NERVAL O MNMZE COS 16 Case : C( C L n this case, is never the optimal solution f for some, if 3 i1 Gi (, and C, then there exists at least one optimal with the minimum cost he bold solid ( CL line in igure 45(B shows this possibility Otherwise, the minimum cost will tend to CL CAK when tends to infinity he regular solid lines in igure 45(B show this instance C C C L C( G 1 ( G 2 ( C L C( G 2 ( G 3 ( A G 3 ( B G 1 ( igure 45 Possible relations between C(, G, G ( and G ( 3 1( Calendar-based cost model with renewal after failure (B Under assumption B, the system remains unchanged if found working at inspection and renewed only at repair/replacement when failure is detected (refer to igure 34 e develop cost models with both random and constant downtime due to inspection and repair/replacement e then formulate cost models for a system with an exponential distribution as a special case inally, we discuss the optimization of the cost model to minimize the expected cost per cycle 16

128 4 OPMAL NSPECON NERVAL O MNMZE COS Calendar-based cost model formulation Suppose inspection time and repair/replacement time are random variables i and Y i, distributed with means of and Y respectively i and Y i are bounded with constant x m and y m respectively As in assumption A with renewal at each inspection, we assume y x to ensure inspection and repair/replacement can be finished in interval m m As shown in equation (325 in Chapter 3, the expected cycle time will be: EL(, Y, 1, (1 g(, where g( E[ R( ( j Y ] i j i (42 Conditioning on, Y, 1,, we have: Possible cycle Probability Cost Y R+C 1 Y 1 N= q1 Y ( C CRY q R( Y N=2 2 2 R( ( Y j1 j1 C ( Y L C ( C Y 1 R C ( Y L 1 1 Y N=3 2 q R( ( Y 3 j j1 3 R( ( Y j1 j1 2 C C Y j R j C [ Y L 2 j1 ( ] j 2 17

129 4 OPMAL NSPECON NERVAL O MNMZE COS 18 Y 1 1 i-1 i-1 N i qi R( i 1 R( i,, ( Y, i i 1 i j1 j1 i1 C C Y j R j C [ Y L i1 j1 ( ] j i1 he expected total cost in one cycle includes the following costs: 1 he expected inspection cost given, Y, 1, is 1 i1 i1 E (, Y,, ( C q C q E C R( i j i j i i1 j i1 j i C {1 E[ R( ( Y ]} j i j i C (1 g( 2 he expected repair/replacement cost is E( CRY CRY 3 he expected cost of downtime and accident will be the product of C ~ L and the expected downtime D ( Recall that the function of the expected downtime in one cycle is D ( (1 g( he calendar-based cost model for random inspection time and repair/replacement time with renewal after failure is the following: 18

130 4 OPMAL NSPECON NERVAL O MNMZE COS 19 C (1 g( CRY CL[ (1 g( ] C ( (1 g( Or, C (1 g( CRY CL CL(1 g( C ( (421 (1 g( e may also formulate the cost model when inspection time is a constant, as a special case of (421 e assume that ym he cost function model of a system with constant inspection time and random repair/replacement time is given by: C (1 Eg( CRY CL CL(1 Eg( C (, (1 Eg( where Eg( E[ R( i( Y ] i1 (422 f both the inspection time and repair/replacement time are constant, say, and R, the cost model will be: where i1 C (1 g( CRR CL CL(1 g( C (, (1 g( g( R( i( R (423 19

131 4 OPMAL NSPECON NERVAL O MNMZE COS Calendar-based cost model for system with lifetime following exponential distribution n this section, we discuss cost models for a system whose failure follows an exponential distribution, ie R( t exp( t Suppose the downtime due to inspection and repair/replacement are both random According to i e ( Y( equation (331 in Chapter 3, we have g( E[ R( ( j Y ] 1 e (, i j where Y ( and ( are the Laplace transforms of Y and respectively; these exist at because Y and are bounded ( Y ym, xm herefore, equation (422 can be written as follows: C (1 g( CRY CL / CL(1 g( C ( (1 g( (424 or the case of random repair/replacement time but constant inspection time, the cost model is a special case of (424, given by where C (1 Eg( CRY CL / CL(1 Eg( C (, (1 Eg( ( e Eg( ( ( Y 1 e (425 f the time for repair/replacement is not random but a constant, say, R, the cost model is given by 11

132 4 OPMAL NSPECON NERVAL O MNMZE COS 111 where C (1 g( CRR CL / CL(1 g( C (, (1 g( ( R e g ( ( 1 e (426 Or we may rewrite equation (426 as CL C R R C C( CL ( R e [1 ] ( 1 e ( CL e 1 ( C R R ( C ( R C e 1 e L (427 As in section 3433, it is convenient to use a (, distribution since i and bounded by i xmand Yi ymrespectively herefore, from (335, Y i are f( y y ( y y y 1 1 m 1 m (,, y ym, and f( x x ( x y x 1 1 m 1 m (,, x xm hen for a system with exponential distribution, ( Y 1 e ymz z (1 z (, x (1 mz z z dz, and ( e dz (, 434 Calendar-based cost optimization with renewal after failure (B or simplicity, we consider the case of constant inspection time and repair/replacement time rom (423, 111

133 4 OPMAL NSPECON NERVAL O MNMZE COS 112 (1 ( (1 ( ( C g C R R C L C L g C C R R C L C CL (428 (1 g( (1 g( hen R, we have C( C CRR CL CL (1 g( C CRR CL lim C( lim( CL CL (1 Eg( Proposition 1: is the optimal solution if and only if CL C CRR Proof: C CRR CL f C( CL, hen, (1 g( C CRR CL CL C CRR C (1 g( CRR CL g( 1 g ( C rom appendix B, g ( So, if CL C CRR, or, CL C CRR, then C( CL for all he minimum cost will be C L when tends to infinity Otherwise, if C C C, there exists optimal with a minimum cost which is L R R less than C L, depending on the system distribution and cost parameters Suppose now C C C rom the above proof, it is obvious that if L R R CL C CRR g ( 1, then the optimal C * 1 112

134 4 OPMAL NSPECON NERVAL O MNMZE COS 113 e further define: ( C G1 CL, 2 ( CL CRR G (1 g( hen G( G1( G2( t is obvious that G ( is a decreasing function of, and 1 lim G1 ( CL or G (, section 344 (derivation of equation (336 shows that 2 2 R min G( ( R, where G( (1 g( n this instance, CLCRR CLCRR G2 ( has a maximum value between max G 2 2( ( R R Proposition 2: Let C C C hen L R R C C C C C C min G( min{ C, C } C L R R L R R L L L 2 L R ( R, where is the value at which G2( reaches maximum Proof: rom G( G1( G2(, min G1 ( max G2 ( G( max G1 ( min G2( CL CRR hus, CL min G( CL rom min G( G1 ( G2 (, we have the other part R igure 46 shows possible examples of the relationship of G, G (, and C( functions for 1( 2 CL C CRR he bold line indicates the case where a local optimal exists with a minimum cost less thanc he normal solid line shows the case when as the optimal solution L 113

135 4 OPMAL NSPECON NERVAL O MNMZE COS 114 C G 1 ( C L C( G 2 ( igure 46 Possible relations between C(, G, G ( 1( 2 f the failure of the system follows an exponential distribution, the cost model is as shown in equation (427: ( CL e 1 ( C R R ( C ( R C( C e 1 e L As shown in proposition 1, if C L C CRR, or, C ( L C CRR, is the optimal solution Otherwise, if CL ( C CRR, there exists optimal with a minimum cost less than C L Letting CL / C CRR CL C CRR g ( 1, or g ( 1, then the optimal C C * is less than 1 44 Numerical example Assume a safety valve has been installed to protect the main system n order to prevent the failure of the safety valve, inspections are needed on a regular basis e investigate the cost model of the system under assumption A with renewal at each inspection and assumption B with 114

136 4 OPMAL NSPECON NERVAL O MNMZE COS 115 renewal after failure, considering both age-based and calendar-based inspection polices Cost function under assumption B with renewal after failure Case : Assume that failure of the device follows a eibull distribution with a shape parameter and a scale parameter of 8 hours nspection time is 8 hours and R is 1 hours he cost of inspection ( C is $1/h, and the cost of repair/replacement for a failed system C is $3/h he cost of loss ( C is $5/h, and the cost of an accident ( C is $1 he ( R L A frequency of true demands for the protective device to function K is 5 Note that CL C CRR where 1 (1, an optimal exists with a minimum expected cost for both inspection policies rom equation (47, the cost model with an age-based inspection policy is C g( CRR CL[( g( R ] C (, where g ( R( i ( g( i R According to equation (323, the calendar-based limiting average availability model is C (1 g( CRR CL CL(1 g( C (, where (1 g( g( R( i( R i1 igure 47 shows the cost as a function of the failure finding interval with an age-based inspection policy based on equals 3 Note that the expected average cost per unit time approaches $55/h ( C when is large enough he optimal is 352 hours with a minimum L 115

137 4 OPMAL NSPECON NERVAL O MNMZE COS 116 cost of $2841/h e omit the figure of calendar-based inspection policy since the minimum cost is very close to that of an age-based he optimal cost is $2849/h with a slightly longer of 36 hours Both policies have a very similar optimal expected cost or the case of 1, the optimal is not sensitive to value igure 47 Age-based cost function for =3 igure 48 Age-based cost function for =1 e now investigate the cost function model of the device where 1 ailure of the device follows an exponential distribution with 1/ 125 igure 48 shows the expected cost versus the Note that the minimum cost is slightly less than that derived when 3 and the optimal is slightly longer he optimal cost is $2695/h with an optimal of 37 hours for an age-based policy he minimum cost with a calendar-based policy is $2696/h with a slightly longer of 379 hours t is of interest to investigate the cost function with different values of parameters such as C A, C L, C R and K etc to see changes in the optimal t is quite different when 116

138 4 OPMAL NSPECON NERVAL O MNMZE COS 117 CL C CRR According to our conclusion in previous sections, there is no local optimal to minimize the cost; the expected cost will approach to CL CAK when tends to infinity Case : Suppose the following new parameters: 15h and C L is zero; CA $5 / h ; CR $8 / h; and K is 1 hen 3, we have 1 (1 134h Note that CL C CRR igure 49 shows the cost function with an age-based inspection policy when 3 e omit the calendar-based cost function due to its similarity to the age-based one here is no local optimal to minimize the cost he optimal cost will be $5/h ( CL CAK when tends to infinity igure 49 Cost function when CL C CRR Cost function under assumption A with renewal at each inspection Case : Assume failure of the device follows a eibull distribution Suppose the following 117

139 4 OPMAL NSPECON NERVAL O MNMZE COS 118 parameters as in Case under assumption B: 8 hours, C $1 / h, C $3 / h, R CL $5 / h, C $1 / h, K 5 hen 3, we have A 1 (1 134h rom equation (43, the cost model with an age-based inspection policy is C C( ~ C ( C R R ( R L ( t dt Note that C CL, and f ( According to Section 422, if f ( ( C R C C Y, there is at least one local optimal ( to minimize the expected cost igure 41 shows the cost function versus he optimal interval is 993 hours with a minimal cost of $17/h According to equation (417, the calendar-based limiting average availability model is R R( R CRR CL[ ( t dt ( t dt] C ( C ( R( R [1 ] ( Note that the figure for a calendar-based policy is very close to that of an age-based he optimal cost is $14/h with slightly longer of 125 hours 118

140 4 OPMAL NSPECON NERVAL O MNMZE COS 119 igure 41 Cost function under assumption A igure 411 Cost function with a smaller t is interesting to see how the results change with fixed a but a different scale parameter igure 411 shows the cost function when changes from 8 hours to 1 hours or the case of 1, the optimal is more sensitive to than to Case : Suppose the following new parameters: 1, 2, C $1 / h, C L is zero, CA $5 / h, C $5 / h; and K is 1 R igure 412 shows the cost function with an age-based inspection policy e omit the calendar-based cost function due to its similarity to the age-based one Note that CL C, and CL ( C C ( C C L R L Y he minimum cost approaches to $5/h when tends to infinity e do not apply inspections for this case 119

141 4 OPMAL NSPECON NERVAL O MNMZE COS 12 igure 412 Cost function without local when CL C 45 Another cost model with an age-based inspection policy n the previous section, we assume inspections are performed at the end of each interval n some circumstances, for example, if an accident happens due to lack of protection, we may not wait for the regular inspection time, but replace/repair the protective device immediately n such instances, the failures of protective device can also be rectified at non-scheduled inspections he cycle will stop at the end of each interval or after an accident n this section, we discuss a cost model for a protective device based on this new assumption e assume the protected system follows an exponential distribution; this obviously simplifies our study but the assumption is reasonable in real applications n addition to the notations defined in section 41, we include the following: g (t : probability density function of protected system C : cost of an accident A 12

142 4 OPMAL NSPECON NERVAL O MNMZE COS 121 * C : time when the protective device is called upon to function * 1 : time at first failure of protective device Suppose the protected system is A, and the protective device is B * 1 C * 1 R R C CRR * * 1, C * C * 1 R R C CRR * * * 1, C 1 * * 1 C R * * < 1 C * C R CA CRR * * * * * * he expected cycle length ( (1 P( P( E[ ( ] 1 C R 1 C 1 C Let ( (1 P( ( E[ ( ] * * * * * 1 C R C 1 C * * A( ( 1 C ( ( B t B B p P f t g t dtdt (429 E[ ( ] E[ E( ( ] * * * * * * * * C 1 C C 1 C 1 C E[ tg( t dt] * * P( 1 C P( tb * * 1 C f ( t tg( t dtdt B tb B 121

143 4 OPMAL NSPECON NERVAL O MNMZE COS 122 hus, the expected cycle length is EL ( [1 p ( ] ( f ( t tg( t dtdt A R B t B B he cost function model of the system is: Expected cost in one cycle EC C ( Expected cycle time EL he expected cost in one cycle includes the following costs: 1 nspection cost hen the cycle is stopped by an accident due to lack of protection, this implies the failure of the protective device herefore, the inspection time for detecting failure is skipped and only repair/replacement time is significant As a result, the inspection cost will be zero if an accident happens 2 Repair/replacement cost for a failed unit 3 Accident cost his occurs if the protective device has already failed when called upon to function he accident cost is denoted by C A he expected cost in one cycle is the following: EC C (1 P( C P( C P( * * * * * 1 C R R 1 A 1 C C [1 p ( ] C ( C p ( A R R A A C C ( ( C C p ( R R A A he cost model is determined as follows: 122

144 4 OPMAL NSPECON NERVAL O MNMZE COS 123 C CRR ( ( CA C pa( C ( ( [1 p ( ] ( f ( t tg( t dtdt A R B t B B (43 Suppose the protected system follows an exponential distribution with failure rate n this case, we have: t A( ( ( ( B B t B B t B B p f t g t dtdt f t e dtdt tb B ( ( t f t ( ( B e e dtb e f t B dtb e f t B dtb rom (43, f ( t tg ( t dtdt t ( B t B f t te dtdt B B t B B, t B t t tb e e Note that te dt [ te ] t t e dt t B Be e B tb be written as, the above equation can e e 1 e ( ( ( ( ( ( tb t B tb tb f t B tbe e dtb t Be f tb dtb e f t B dtb f t B dtb n this section, we discuss a cost model for a protective device based on a new assumption where the failures of protective device can be rectified at non-scheduled inspections However, we only propose an initial model which can be extended in more details in the future 46 Concluding remarks his chapter studies optimal failure finding intervals to minimize costs induced by inspection, repair/replacement, production loss, or accidents e consider non-negligible inspection time 123

145 4 OPMAL NSPECON NERVAL O MNMZE COS 124 and repair/replacement time n many real applications, the inspection time cannot be ignored or example, if a protective device is found failed in the inspection, one common sense solution is to replace it with a new one rather than waiting to fix it, as this could save time n this case, in comparison to downtime due to repair/replacement, the downtime due to inspection cannot be ignored his chapter proposes several cost function models for a periodically inspected repairable system under different assumptions t considers both age-based and calendar-based inspection policies under assumptions A (renewal at each inspection and B (renewal after failure or each case, we consider both random and constant downtime due to inspection and repair/replacement Other than the two-unit cost model in section 45, all models assume that failures of the system are only detected at regular inspections and are rectified accordingly n the models under assumption A, the expected cost tends to C (= C L C A K, which is the L sum of the production loss and the accident cost e show that for both age-based and calendar-based inspection policies, the local optimal inspection interval may or may not exist, depending on cost parameters and the distribution of time to failure of the system n particular, C with regards to an age-based inspection, if C CL and f ( ( C C R Y, an local optimal exist f we only renew the system after failure rather than at each inspection (assumption B, the cost will also approach to CL CAK when tends to infinity or both age-based and calendar-based inspection policies, if the value of C is greater than C CRR, an L 124

146 4 OPMAL NSPECON NERVAL O MNMZE COS 125 optimal can be found regardless of the system distribution Otherwise, is the optimal solution so that we should not apply inspections e provide the range of the maximum availability under assumption B Note that cost functions for age-based and calendar-based policies are often very similar as their downtimes for inspection and repair/replacement are both relatively small n the future, we may investigate the upper bound of inspection and repair/replacement time, above which cost functions for the two inspection policies are different his chapter also explores systems with a particular distribution or example, we consider a system whose failures follow an exponential distribution e also investigate the cost models with random inspection time and repair/replacement time with a Beta distribution he proposed model is extended to assuming failures of protective device can also be rectified upon an accident (of the protected system due to lack of being protected, rather than at scheduled inspections only n this case, we propose a cost model with age-based inspection policy, and we assume that the failure of protected system (being protected follows an exponential distribution; this is realistic in many applications 125

147 5 CASE SUDES n this chapter, we propose two case studies to investigate the optimal inspection interval for safety devices he models used to analyze the safety devices are based on the results of Chapter 3 and Chapter 4 n section 51, we analyze the optimal for high reliable pressure safety valves in a power plant to maximize availability and/or minimize costs n section 52, we show a case study for safety valves of a mining & refining company e analyze availability functions of the safety valves; provide an optimal to maximize availability 51 Case study for a thermal power plant he study was developed using the historical data of safety valves of EPPZJK, a large thermal power plant in China he plant has eight generator units with a productive capacity of 3 M each per year he annual electrical power of the plant reaches 15 million Mh e analyze pressure safety valves for a boiler steam drum; these valves are very important for the plant s operation he company has an inspection policy based on regulations Our 126

148 5 CASE SUDES 127 conclusions support the current policy, but we indicate the changes in availability and cost in terms of his is useful for the company to make further decisions 511 ntroduction he boiler is one of the most important pieces of equipment in the thermal power plant, and it is part of the power supply system Safety valves are installed to protect the boiler heir main function is to ensure the pressure inside the boiler henever the pressure reaches a critical value (called opening pressure, a safety valve must act (open, relieving the air pressure automatically hen the pressure inside the boiler falls to a specified safe value (re-seating pressure, the safety valve must close to prevent leakage of the air n EPPZJK, safety valves are installed on the boiler of each power unit or example, No 2 power unit has 11 safety valves Among these, six safety valves are more important, because they protect the steam drum and super heater: three protect the steam drum, two protect the super heater (SH, and there is one Pressure Control Valve (PCV he following tables show their parameters able 51 Parameters of Pressure Safety Valves for Boiler steam drum DNumber Model Open pressure MPa re-seating pressure MPa Exhaust volume Kg/h 1 HE ,9 2 HE ,745 3 HE ,122 Accounting for Max furnace evaporation 784% 127

149 5 CASE SUDES 128 able 52 Parameters of Super heater (SH safety valves DNumber Model Open pressure MPa Re-seating pressure MPa Exhaust volume Kg/h 4 HC ,577 5 HC ,917 Accounting for Max furnace evaporation 2239% able 53 Parameters of Pressure Control Valve (PCV DNumber 11 Model EOL121N7 BRA5P1 Open pressure MPa Re-seating pressure MPa Exhaust volume Kg/h Accounting for Max furnace evaporation ,765 11% he PCV is the first screen on the system ts open pressure is lower than that of the steam drum s safety valves e analyze the PCVs since they are the main safety devices, accounting for maximum evaporation of 784% All three safety valves for steam drum are identical except for different preset parameters Numbers 2 and 3 are redundant igure 51 Safety valve for the boiler system of EPPZJK A failure of the safety valve is defined as its failure to open to relieve the pressure or to reset properly he main failure mode is air leakage through the valve port Note that the failure of the 128