ABSTRACT 
 
 
 
 
Title: RISK AND ECONOMIC ESTIMATION OF 
INSPECTION POLICY FOR PERIODICALLY 
TESTED REPAIRABLE COMPONENTS  
 
 Carlos Eduardo Barroeta, M.S., 2005 
Thesis directed by: Professor Mohammad Modarres, Mechanical 
Engineering Department 
 
 
 
This report presents a model to identify the optimal time between surveillance tests 
and overhaul frequency of components whose failures are detected upon inspection. 
The model is based on minimizing the total cost per unit time during the component 
renewal cycle. It considers the component availability assuming that the unit is ?as 
old? after tests and repairs and ?as new? after overhauls. The model takes into 
account costs associated with tests and maintenance, as well as potential losses 
related to unavailability. General conditions and a case study are discussed to 
evaluate the effect of costs, maintenance task durations, and the uncertainty of the 
reliability parameters on the optimal inspection policy of typical tested components. 
This report also discusses the advantage of the cost-based optimization versus the 
traditional approach based on maximal availability. 
  
 
 
 
 
 
 
 
 
 
 
 
RISK AND ECONOMIC ESTIMATION OF INSPECTION POLICY FOR 
PERIODICALLY TESTED REPAIRABLE COMPONENTS    
 
 
 
By 
 
 
Carlos Eduardo Barroeta 
 
 
 
 
 
Thesis submitted to the Faculty of the Graduate School of the  
University of Maryland, College Park, in partial fulfillment 
of the requirements for the degree of 
Master of Science 
2005 
 
 
 
 
 
 
 
 
 
 
Advisory Committee: 
Professor Mohammad Modarres, Chair 
Professor Ali Mosleh 
Professor Aristos Christou 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
? Copyright by 
Carlos Eduardo Barroeta 
2005 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
Acknowledgements 
I really appreciate the guide and advice of Dr. Mohammad Modarres in the 
development of this work, as well as the valuable comments of Dr. Ali Mosleh and 
Dr. Aris Christou. 
I also appreciate the support of my mates in the CTRS:  Jos? Luis, Reza, Mercedes, 
Mohammad and Genebelin. 
I am especially thankful to my wife, Rosa Ana, for her love and company during 
these years. 
 ii 
 
 
Table of Contents 
 
Acknowledgements....................................................................................................... ii 
Table of Contents.........................................................................................................iii 
List of Tables ................................................................................................................ v 
List of Figures.............................................................................................................. vi 
Chapter 1: Introduction................................................................................................. 1 
Chapter 2: Theoretical Background.............................................................................. 4 
2.1 General definitions.............................................................................................. 4 
2.1.1 Non-repairable units..................................................................................... 4 
2.1.2 Repairable units ........................................................................................... 4 
2.2 Availability of repairable items .......................................................................... 5 
2.3 Maintenance and renewal theory ........................................................................ 7 
2.3.1 Ordinary renewal process (ORP)................................................................. 8 
2.3.2 Non-homogeneous Poisson process (NHPP)............................................... 9 
2.3.3 Generalized renewal process (GRP) .......................................................... 11 
2.4 Inspection policies for periodically tested components .................................... 15 
Chapter 3: Analytical Model....................................................................................... 22 
3.1 Time between successive failures..................................................................... 23 
3.2 Cost per unit time and cycle lengths ................................................................. 25 
3.3 Increasing test and repair costs ......................................................................... 27 
3.3.1 Linear function........................................................................................... 28 
3.3.2 Non-linear function 1................................................................................. 28 
3.3.2 Non-linear function 2................................................................................. 28 
3.3 Description of the variables involved in the analytical model.......................... 30 
Chapter 4: Results....................................................................................................... 33 
4.1 Average availability.......................................................................................... 33 
4.2 Cost per unit time.............................................................................................. 35 
4.3 Optimal inspection interval and overhaul frequency........................................ 40 
4.4 Availability versus cost-based optimization ..................................................... 45 
4.5 Sensitivity evaluation........................................................................................ 47 
 iii 
 
 
4.6 Uncertainty in Weibull parameters ................................................................... 51 
Chapter 5: Extensions ................................................................................................. 55 
5.1 Generalized Renewal Process after test cycles ................................................. 55 
5.2 Imperfect surveillance inspections.................................................................... 56 
5.3 Further uncertainty and risk analysis ................................................................ 56 
5.4 Systems of periodically tested components ...................................................... 57 
Chapter 6: Conclusion................................................................................................. 59 
Appendix..................................................................................................................... 62 
Bibliography ............................................................................................................... 68 
 iv 
 
 
List of Tables 
 
Table 1. Comparison between typical stochastic repair processes ............................. 14 
Table 2. Example of parameters for repair cost functions.......................................... 29 
Table 3. Values for the average availability analysis ................................................. 34 
Table 4. Arbitrary values for cost rate function examples.......................................... 36 
Table 5. Input values for safety relief valve example................................................. 40 
Table 6. Availability versus cost-based optimization results for a relief valve.......... 47 
Table 7. Results of sensitivity evaluation for the case study ...................................... 48 
Table 8. Average availability for systems with periodically tested units ................... 58 
 v 
 
 
List of Figures 
 
Figure 1. Categories of stochastic point processes for repairable systems................... 8 
Figure 2. Basic notation for a stochastic point process................................................. 9 
Figure 3. Conditional probability of occurrence of failure......................................... 10 
Figure 4. Approximate point unavailability for periodically tested components ....... 17 
Figure 5. Probability density function considering inspection at T ............................ 19 
Figure 6. Basic notation for the mathematical model................................................. 23 
Figure 7. Conditional probability of occurrence of failure with inspection at T ........ 24 
Figure 8. Behavior of three types of incrementing repair cost functions.................... 29 
Figure 9.  Average availability versus test cycle number for ?  > 1 ............................ 34 
Figure 10. Average availability versus test cycle number for ?  ? 1 ........................... 35 
Figure 11. Cost rate function for different values of N............................................... 36 
Figure 12. Optimal test interval versus overhaul frequency for different ? ............... 37 
Figure 13. Limiting cost rate function for different values of N................................. 38 
Figure 14. T
opt
 versus overhaul frequency for a safety relief valve ............................ 42 
Figure 15. crf(T
opt
) versus overhaul frequency for a safety relief valve ..................... 43 
Figure 16. Cost rate function versus T and N for a safety relief valve ....................... 44 
Figure 17. Average availability versus T for a relief valve with N = 10 .................... 46 
Figure 18. Cost rate function for N = 10 for a safety relief valve............................... 46 
Figure 19. T
opt
 versus N for different incrementing cost models for a relief valve .... 49 
Figure 20. crf vs. T for different N for a relief valve with no unavailability impact .. 50 
Figure 21. Point estimates and 90% bounds for T
opt
 vs. N for a relief valve .............. 52 
Figure 22. Point estimates and 90% bounds for crf(T
opt
) vs. N for a relief valve....... 52 
Figure 23. Frequency chart for the optimal overhaul frequency for a relief valve..... 53 
 
 
 vi 
 
 
Chapter 1: Introduction 
While inspection and maintenance strategies have been widely studied for monitored 
components whose failures are immediately detected, less attention has been given to 
periodically tested units.  The latter are often related to emergency or protection systems 
and, therefore, are important elements to be considered in terms of reliability and risk 
assessment. 
The purpose of this work is to present a model to identify the optimal time between 
surveillance tests and overhaul frequency of components whose failures are detected 
upon inspection and are periodically tested to ensure high availability.  This type of 
equipment includes emergency and spare units, as well as hardware components with 
hidden or dormant failures in normal operation. 
Although the average and time-dependent point availability of periodically tested 
components have been studied in the past [1-5], the consideration of economic aspects for 
the study of their optimal inspection interval and overhaul frequency is more recent.  
Adachi and Nishida [6] discussed the optimal inspection policy based on maximizing the 
component availability with ?as new? repairs and preventive maintenance, using random 
test and repair durations. Vaurio [7] analyzed the optimal availability and cost rate 
function (cost per unit of time) of periodically inspected preventively maintained units, 
considering constant duration of repair and maintenance tasks, with ?as old? tests and ?as 
new? repairs.  Bad?a et al. [8] studied the minimal cost per unit time of components with 
less than perfect tests, ?as new? corrective and preventive maintenance, but negligible 
test and repair durations. 
1 
 
 
 
More recently, Martorell et al. [9], and Lapa et al. [10], studied the surveillance test 
policy optimization under constraints, through the use of genetic algorithms.  
The analytical model presented in this report is developed for periodically tested 
components with overhauls (preventive maintenance) after certain number of inspections. 
The model is based on the assessed component availability during the renewal cycle and 
considers ?as old? process after tests and repairs (component aging). It takes into account 
costs associated not only with surveillance tests and maintenance, but also with the 
potential losses related to the unit unavailability.  The model also considers the duration 
of repairs and inspections (often neglected in literature), and allows for uncertainty in the 
parameters of the probability density function (pdf) of the time between failures. 
Summing up, the analytical model in this study is based on the following assumptions: 
1. Component failures are only detected upon inspection. Inspection tasks are perfect 
(probability of detection equals 1). 
2. Inspections are carried out every T units of time (constant interval). Repairs are 
conducted in case of failure detection. 
3. Inspection and repair durations are not negligible but constant. 
4. The component is under aging process and remains ?as old? after surveillance tests 
and repairs. Inspection and repair tasks do not deteriorate the unit. 
5. Periodic overhaul is performed after every N inspection cycles regardless of the unit 
condition.  Component returns to ?as new? after the overhaul. 
6. Component unavailability may cause economic losses with conditional probability 
]Pr[ uL  (probability of losses given unavailable unit).  These losses are independent 
of component age.  
2 
 
 
 
7. Direct inspection and repair costs may increase after every test cycle.  Inflation and 
other financial effects are negligible. 
 
A theoretical background related to the model, including general definitions about 
availability and maintenance of repairable components, is presented in chapter 2.  
Chapter 3 explains the analytical model in detail, and chapter 4 discusses important 
results based on numerical examples.  Possible extensions of the model and scope of this 
work are commented in chapter 5. Finally, chapter 6 presents concluding remarks 
obtained upon the completion of this research work. 
3 
 
 
 
Chapter 2: Theoretical Background 
 
2.1 General definitions 
The study of reliability of components and systems has a growing interest due to the 
increasing complexity of equipment and general awareness about performance, quality 
and especially safety issues.  Reliability is defined to be the probability that a unit or 
system will perform a required function for a given period of time, when used under 
stated operated conditions [11]. It has an important role in the assessment of system 
performance and the potential failures that may lead to adverse consequences in terms of 
equipment, people and environment. 
Reliability and risk analyses are strongly related to maintenance, as they take into account 
the condition, repairs and renewal of units and systems in a moment or period of time. 
When performing reliability studies, it is important to make a distinction between 
repairable and non-repairable items, since they involve different failures characteristics as 
well as methods for predicting their reliability and availability [12]. 
2.1.1 Non-repairable units 
These items are those that are discarded and replaced with new ones when they fail 
(e.g. light bulbs). Their reliability is expressed in terms of the time-to-failure 
probability distribution.  
2.1.2 Repairable units 
These items are in general not replaced after the occurrence of failures; rather, they 
are repaired and put into operation again. Their reliability depends on their renewal 
4 
 
 
 
model and stochastic process different from the models of non-repairable units. 
Valves and pumps (when seen as single items) can be examples of this type of 
units. 
2.2 Availability of repairable items 
Availability is defined as the probability that a repairable element (component or system) 
is performing its required function at a given point in time or over a stated period, when 
operated and maintained in a prescribed manner [11].  Availability depends not only on 
the chance of failure of a given item, but also on its maintainability, which is the 
likelihood that the failed unit will be restored or repaired to a specified condition within a 
period of time when maintenance is performed. 
Like in reliability analysis, the rules of probability theory can be applied to quantify this 
measure.  Accordingly, some important definitions can be established [12]: 
 
a) Instantaneous availability, a(t):  it is the probability that the unit or system is up at 
a time t. 
 
b) Limiting availability: it is defined as the following limit of the instantaneous  
availability, 
a = a(t)                                                 (1) 
??t
lim
 
c)  Average availability:  it is defined for a fixed time interval (mission time), T 
 
       ?(T) = 
?
T
dtta
T
0
)(
1
                                               (2) 
 
5 
 
 
 
d)  Limiting average availability:  
 
      ?
l
 =  
??T
lim
?
T
dtta
T
0
)(
1
                                             (3) 
 
The previous definitions constitute the basis to evaluate the availability of equipment. To 
analyze these measures for a specific unit (component), it is necessary to establish the 
type of unit, whether it is repairable or not and, particularly, the kind of failure it presents. 
Thus, the following classification is typically used in literature: 
 
a) Replaceable components: no repair action is foreseen (non-repairable units) 
b) Repairable components with failures which are immediately detected (revealed faults) 
c) Repairable components with failures which are detected upon demands (faults remain 
unrevealed until next demand occurs) 
d) Repairable units whose failures are detected upon inspection (faults remain unrevealed 
until next inspection is carried out). 
 
As mentioned before, this report focuses on repairable components whose failures are 
revealed upon inspection and have to be periodically tested in order to detect possible 
faults (type d). This kind of units include items used in emergency conditions, 
components in spare or storage, and also normal operated units with hidden (dormant) 
faults only detectable by inspection. 
6 
 
 
 
2.3 Maintenance and renewal theory 
Maintenance is defined as the combination of all technical and corresponding supervision 
and administrative actions, intended to retain or restore an entity to a state in which it can 
perform its required function [13]. 
Two general types of maintenance can be distinguished: reactive and proactive 
maintenance.  The former is performed in response to unplanned or unscheduled 
downtime of the unit, usually as a result of a failure, whether it be internal (inherent) or 
external (e.g. operator-induced).  On the other hand, proactive maintenance is performed 
prior to failures and may be either preventive or predictive [11].  Preventive maintenance 
is a scheduled downtime, usually periodical, in which well designed set of tasks, such as 
repair, replacement, cleaning, adjustment, etc. are performed.  Predictive maintenance 
estimates, through diagnostic tools and measurements, when a part is near failure and 
should be repaired or replaced (i.e. maintenance based on condition).  Candidates for 
predictive maintenance are normal operating equipment whose condition can be 
monitored over time.  Unlike regular preventive maintenance, maintenance based on 
diagnostic (predictive tasks) is not necessarily periodical and represents a cost-effective 
alternative for monitored items. 
As discussed before, reliability and availability are related to maintenance.  Particularly, 
the study of repairable components and systems strongly depends on the model of repair 
or renewal involved in the maintenance process. 
A repairable item is one which undergoes repair and can be returned to operation by a 
method other than replacement of the entire item. The following sections discuss 
7 
 
 
 
different stochastic models considered for the analysis of repairable units and systems 
(see Figure 1). 
 
 
 
 
Categories of repair
Perfect repair
(AGAN)
Normal repair
Minimal repair
(ABAO)
Homogeneous
Poisson Process
Other life 
distributions
Generalized 
renewal
Non-homogeneous
Poisson Process
Superimposed 
renewal
 
 
    Figure 1. Categories of stochastic point processes for repairable systems 
 
 
2.3.1 Ordinary renewal process (ORP) 
This model assumes that, following a repair, the unit returns to an ?as good as new? 
(AGAN) condition.  In this process, the interarrival times, x
i
, between successive 
failures (see Figure 2) are considered independently and identically distributed 
random variables. It is a generalization of a Homogeneous Poisson Process (HPP). 
This model represents an ideal situation; it is only appropriate for replaceable items 
and hence has very limited applications in the analysis of repairable components 
and systems. 
Variations of the ORP can also be defined.  The modified renewal process, where 
the first interarrival time differs from the others, and the superimposed renewal 
process (union of many independent ORPs) are examples of these possible 
variations [14].  
 
 
8 
 
 
 
   
  
 
 
 
 
 
 
 
 
t
t
4
t
3
t
2
t
1
x
1
x
3
x
2
x
4
x
t: arrival times x: interarrival times
 
Figure 2. Basic notation for a stochastic point process 
 
 
2.3.2 Non-homogeneous Poisson process (NHPP) 
This model is also called ?minimal repair? and it assumes that the unit returns to an 
?as bad as old? (ABAO) condition after a repair. So that, after the restoration the 
item is assumed to be operative but as old as it was before the failure.  The NHPP 
differs from the HPP in that the rate of occurrence of failures varies with time rather 
the being constant [14].  Unlike the previous model, in this process the interarrival 
times are neither independent nor identically distributed. 
The NHPP is a stochastic point process in which the probability of occurrence of n 
failures in any interval [t
1
, t
2
] has a Poisson distribution with: 
 
?
=
2
1
)(
t
t
dttmean ?                                                     (4) 
 
9 
 
 
 
where ? (t) is the rate of occurrence of failures (ROCOF) defined as the inverse of 
the expected interarrival times, 1/E[x
i
]. 
One of the most common forms of ROCOF used in reliability analysis of repairable 
components and systems is the Power Law Model: 
 
1
)(
?
?
?
?
?
?
?
=
?
??
?
?
t
t                                                 (5) 
 
This form comes from the assumption that the interarrival times between successive 
failures follow a conditional Weibull probability density function, with parameters 
?  and ? .  This model implies that the arrival of the ith failure is conditional on the 
cumulative operating time up to the (i ? 1)th failure.  Figure 3 shows a schematic of 
this conditionality [15]. 
The Weibull distribution is typically used due to its flexibility and applicability to 
various failure processes, however, solutions to Gamma and Log-normal 
distributions are also possible. 
 
 
 
 
f(t )
t
tt
1
P(Time ? t | Time > t
1
)
 
 
 
Figure 3. Conditional probability of occurrence of failure 
10 
 
 
 
2.3.3 Generalized renewal process (GRP) 
A repairable system may end up in one of the five possible states after a repair: 
 
a. As good as new 
b. As bad as old 
c. Better than old, but worse than new 
d. Better than new 
e. Worse than old 
 
The two models described before, ordinary renewal process and NHPP, account for 
the first two states respectively.  However, the last three repair states have received 
less attention since they involve more complex mathematical models. 
In 1986 Kijima and Sumita [16] proposed a probabilistic model for all the after-
repair states called Generalized Renewal Process (GRP). According to this 
approach, the ordinary renewal process and the NHPP are considered specific cases 
of the generalized model. 
The GRP theory of repairable items introduces the concept of virtual age (A
n
).  This 
value represents the calculated age of the element immediately after the nth repair 
occurs.  For A
n
 = y the system has a time to the (n + 1)th failure, x
n+1
, which is 
distributed according to the following cumulative distribution function (cdf): 
 
)(1
)()(
)(
yF
yFyxF
yAxF
n
?
?+
==                                    (6) 
 
where F(x) is the cdf of the time to the first failure (TTFF) distribution of a new 
component or system. 
 
11 
 
 
 
The summation: 
?
=
=
n
i
in
xS
1
                                                   (7) 
 
with S
0 
= 0, is called the real age of the element.  The model assumes that the nth 
repair only compensates for the damage accumulated during the time between the 
(n - 1)th and the nth failure.  With this assumption, the virtual age of the component 
or system after the nth repair is: 
 
   A
n
 = A
n-1 
+ qx
n
 = qS
n
                                           (8) 
 
where q is the repair effectiveness (or rejuvenation) parameter and A
0
 = 0. 
According to this model, the result of assuming a value of q = 0 leads to an ordinary 
renewal process (as good as new), while the assumption of q = 1 corresponds to a 
non-homogeneous Poisson process (as bad as old).  The values of q that fall in the 
interval 0 < q < 1 represent the after-repair states in which the condition of the 
element is better than old but worse than new, whereas the cases where q > 1 
correspond to a condition worse than old.  Similarly, cases with q < 0 would suggest 
a component or system restored to a state better than new.  Therefore, physically 
speaking, q can be seen as an index for representing the effectiveness and quality of 
repairs [15]. 
Even though the q value of the GRP model constitutes a realistic approach to 
simulate the quality of maintenance, it is important to point out that the model 
assumes an identical q for every repair in the item life. A constant q may not be the  
 
12 
 
 
 
 
case for some equipment and maintenance process, but it is a reasonable approach 
for most repairable components and systems. 
The three models described above have advantages and limitations.  In general, the 
more realistic is the model, the more complex are the mathematical expression 
involved.  Table 1 summarizes the main strengths and weakness of each repair 
approach. 
As mentioned in the table, the NHPP model has been proved to provide good results 
even for realistic situations with better-than-old but worse-than-new repairs [17]. 
Based on this, and given its conservative nature and manageable mathematical 
expressions, the NHPP (ABAO repair model) was selected for this particular work.  
The specific analytical modeling is discussed in chapter 3. 
13 
 
 
 
 
 
          Table 1. Comparison between typical stochastic repair processes 
 
Repair model Strengths Weakness 
Ordinary renewal  
process (AGAN) 
- Represents a first, simple 
approach to model repairable 
components. 
- It is appropriate for 
modeling replaceable units. 
- In general, its mathematical 
expressions are simpler than 
the ones of the other models. 
 
- It is generally not 
appropriate for systems, since 
replacements typically apply 
for a single part and not for 
the entire system. 
- Assumes that the interarrival 
times between failures are 
independent and identically 
distributed. 
Non-homogeneous 
Poisson process 
(ABAO)  
- It is a useful and quite 
simple model to represent 
equipment under aging 
(deterioration). 
- Involves relatively simple 
mathematical expressions.  
- It is a conservative approach 
and in most cases provides 
results very similar to those 
of more complex models like 
GRP with 0.1 < q < 1 [17]. 
- Is not adequate to simulate 
repair actions that restore the 
unit to conditions better than 
new or worse than old. 
Generalized renewal 
process (GRP) 
- It is a realistic general 
model which cover all the 
possible restoration 
conditions, from better than 
new to worse than old. 
- Involves an additional 
parameter (q) and more 
complex mathematical 
equations. 
- Assumes constant 
rejuvenation parameter (q) for 
all the repairs. 
 
14 
 
 
 
2.4 Inspection policies for periodically tested components 
Until this point, repair models, reliability and availability definitions have been discussed 
without making any special distinction between components and systems.  Components 
are single units or elements that represent the minimum level where information 
available. On the other hand, systems are arrangements of two or more components 
usually working simultaneously. 
As mentioned before, this work is oriented to components (single units) which are 
periodically tested and, in case of failure, remain failed until next inspection.  From now 
on, emphasis will be given to components under this assumption, and particular 
expressions for availability and cost functions will be discussed. 
The availability study of periodically tested components (PTC) in most cases turns into 
difficult analyses, especially if the repair and inspection times are treated as random 
variables. This problem has been studied by different approaches [1-5]. 
Particularly, Hilsmeier, Aldemir and Vesely [5] presented general expressions for 
calculating the point unavailability of aging standby components according to a standard 
extension of the classic renewal equation.  In particular, for the case of tests performed 
every T hours, considering inspection and repair times negligible with respect to T, ?as 
old test? and ?as new repairs?, the following equations were presented: 
 
?
?
=
?+=?=
n
k
t
nT
tkTqTknRkTqdttftatq
1
~
),(])[()(')'()(1)(      nT < t < (n+1)T     (9) 
 
where: 
?
?=
t
dtttR
0
]')'(exp[)( ?                                              (10) 
15 
 
 
 
 
and 
?
?
?
??=
kTt
Tkn
dtttkTq
)(
]')'(exp[1),( ?                                       (11) 
where: 
~
f : is the first failure density function, 
n: is the cycle number,  k = 0, 1, 2, ....n 
T: is the fixed time between inspections, and 
? (t): is the time-dependent failure rate. 
 
According to these expressions, the availability of PTC goes to dampening oscillations, 
and eventually settles to the so called ?saw-tooth? periodic behavior, where the settling 
time increases when the interval between inspections is decreased. 
Since the availability of PTC is a periodic function of time, the study of the average 
availability over the test interval becomes especially interesting. For this situation, the 
effect of inspection and repair times are usually considered in practical cases. 
For constant rates, assuming ? T << 1, and renewal after each cycle, the following 
expression is often presented in textbooks (see Figure 4) [12], [18]: 
 
Average availability per cycle = ?(T)
?
??
????
T
tT
t
2
1                    (12) 
 
where: 
T: is the inspection interval, 
t
t
: is the constant time to test (t
t
 << T), 
? : is the failure rate, and 
?: is the corresponding repair rate (mean time to repair = 1/? << T) 
 
 
16 
 
 
 
 
Approximate
pointwise
unavailability
1-a(T)
T
t
t
1
1/?
Time
? T?
 
 
 
 
 
 
 
Figure 4. Approximate point unavailability for periodically tested components 
 
 
Clearly, the availability of the unit is strongly affected by the time between surveillance 
tests (T). Therefore, the study of the optimal inspection interval is particularly interesting 
for maximizing the average or mission availability of periodically tested components. 
From the previous equation, if ?(T) is differentiated with respect to T, and the result is set 
equal to zero: 
0
2
)(?
2
=+?=
T
t
dT
Td
t
?
                                          (13) 
 
The optimal inspection interval, for this simple case, can be established as: 
 
?
t
opt
t
T
2
=                                                      (14) 
 
Other authors have studied the availability maximization in periodically tested 
components as a function of the time between inspections, considering different 
assumptions in more realistic cases [3], [6], [19]. 
17 
 
 
 
 
In particular, Jardine [19] presented an approach based on the expected uptime per cycle 
(average availability), considering the following: 
 
- The component is periodically tested every T units of time, and it will be repaired if 
found to be failed. 
- The unit is considered renewed after inspection and repairs (as good as new). 
- The lengths of time needed to inspect (T
i
) and repair (T
r
) are known and constant. 
 
Thus, the average availability is expressed as:   
 
?(T)=
length cycle expected
uptime expected
                                             (15) 
 
According to the theory of expectations, the expected uptime is the operative time of a 
good cycle, T, multiplied by its probability, R(T), plus the mean time to failure given that 
inspection takes place at T
, 
multiplied by [1-R(T)].  
To determine the mean time to failure given the periodic inspection, the mean of the 
shaded portion of Figure 5 is considered. Thus: 
 
MTTF
shaded
 = 
)(
)(
TF
dtttf
T
?
??
                                                 (16) 
 
The mean of the shaded region is similar to the mean of the entire distribution, but 
considers that the unshaded portion is an impossible region for failures [19]. 
18 
 
 
 
 
  Figure 5. Probability density function considering inspection at T 
 
 
Likewise, the expected cycle length is the duration of a good cycle, (T + T
i
), multiplied 
by its probability, R(T), plus the length of a failure cycle, (T +
 
T
i
 + T
r
) multiplied by its 
probability [1-R(T)].  
Accordingly, the average availability in the renewal cycle becomes: 
 
?(T) =
)](1[
)()(
TRTTT
dtttfTRT
ri
T
?++
+?
?
??
                                             (17) 
 
This expression can be used to calculate the average availability and the optimal 
inspection interval that maximizes the uptime for different probability density functions, 
f(t).  The equation assumes that after tests the unit is in the ?as new? state, which may be 
as a result of minor modifications being made during the surveillance inspections. 
In practice, this assumption can be reasonable and it will be the case if the failure 
distribution of the component is the exponential distribution.  If the ?as new? assumption 
is not the case and the time to failure distribution has an increasing rate, the expression 
for the average availability becomes more complex.  
19 
 
 
 
Until now, the optimal time between inspections in periodically tested components has 
been discussed considering the maximum availability without taking into account the 
economical considerations related to periodic inspection and repairs, as well as the cost of 
unavailability. 
Nowadays, besides maximizing the operative time of units, operators and inspectors have 
to face economical limitations which may affect the frequency of surveillance tests. Thus, 
to identify the optimal inspection interval, it becomes necessary to consider not only the 
maximum operative time (availability) but also the direct cost of test and repairs as well 
as the potential losses associated to unavailability.  Summing up, it is important to 
consider the following: 
 
The cost of unavailability due to random failures ? 
? 
? 
? 
The downtime cost due to surveillance tests and possible repairs 
The direct cost of periodic inspections  
The direct cost of repairs  
 
For downtime losses (unavailability cost), distinctions have to be made between 
periodically tested components that are in standby and those in normal operation.  For 
modeling this aspect a probability factor of actual losses given unavailable unit, ]uLPr[ , 
can be introduced.   
Likewise, for the test interval optimization, the cost rate function (cost per unit time) is 
considered in literature [7], [8].   
As a general approach, the following expressions could be used to study the optimal time 
between inspections based on cost minimization: 
20 
 
 
 
 
Cost per unit time = 
][
][)(
LE
DECppTFCrCi ??+?+
                      (18) 
where 
T: inspection interval 
Ci: direct cost of inspection 
Cr: direct cost of repair 
Cp: cost of lost production 
p: probability of losses given unavailable component 
E[D]: expected downtime 
E[L]: expected cycle length 
 
The expected downtime and the expected cycle length are defined as: 
 
Expected downtime = E[D] = E[L] ? E[U]                              (19) 
 
Expected cycle length = E[L] = [Ti+T]R(T) + [T+Ti+Tr]F(T) = Ti + T + Tr?F(T)    (20) 
 
Expect. uptime = E[U] =  )()(
)(
1
)(
0
TFdttft
TF
TR
T
??+?
?
T =    (21)                     
?
?+?
T
dttftTRT
0
)()(
 
where T
i
 is the time to inspect and T
r
 is the mean time to repair. 
 
 
Equations 18 to 21 apply to estimate the optimal inspection interval for PTC with ?as 
good as new? restoration after every test cycle.  More general expressions will be 
presented in next chapter. 
 
 
21 
 
 
 
Chapter 3: Analytical Model 
This chapter presents the mathematical model developed to calculate the availability and 
cost per unit time of periodically tested components.  The model uses the minimal repair 
approach (non-homogeneous Poisson Process) after every test cycle and is based on the 
following assumptions: 
 
1. Component failures are only detected upon inspection; in case of failure, the unit 
remains failed until the next scheduled test. Inspection tasks are perfect (probability 
of fault detection equals 1). 
2. Inspections are carried out every T units of time (constant interval). Repairs are 
conducted in case of fault or failure detection.  
3. Inspection and repair durations are not negligible but constant (deterministic values). 
4. The component is under aging process and it remains ?as bad as old? after 
surveillance tests and repairs. Inspection and repair tasks do not deteriorate the unit. 
5. Preventive periodic overhaul is performed after every N inspection cycles regardless 
of the unit condition.  Component returns to ?as good as new? after the overhaul. This 
maintenance action defines the renewal cycle. 
6. Component unavailability may cause economic losses with conditional probability 
]Pr[ uL  (probability of losses given unavailable item).  These losses are considered in 
terms of cost per unit of downtime (C
p
). This cost is independent of component age. 
7. Direct inspection and repair costs increase after every test cycle.  Inflation and other 
financial effects are negligible. 
 
22 
 
 
 
The analytical model used in this work is an extension of the approach introduced in the 
last pages of previous chapter.  The model extends the notion of expected availability and 
expected cycle length to ?as bad as old? test and repairs.  It uses the concept of cost rate 
function (cost per unit time) in the renewal cycle as a basis for identifying the optimal 
inspection interval and overhaul frequency, N (i.e. values that minimize the cost rate 
function). Figure 6 shows a schematic of the basic notation. 
 
 
N?TiT2T ??????
Test cycle
Renewal cycle
0
Return to ?as new? condition
T
Overhaul
ABAO test and
possible repair
time
??????
 
 
 
 
 
 
 
 
Figure 6. Basic notation for the mathematical model 
 
3.1 Time between successive failures 
This model assumes that the interarrival times between successive failures follow a 
conditional Weibull probability distribution, where the arrival of the ith failure is 
conditional on the cumulative operating time up to the (i ? 1)th failure. This 
conditionality comes from the fact that the component retains an ?as bad as old? state 
after repairs. 
23 
 
 
 
The rate of occurrence of failures (ROCOF) under this assumption corresponds to a 
power law expression, with parameters ?  and ? : 
 
1
)(
?
?
?
?
?
?
?
=
?
??
?
?
t
t                                                    (22) 
 
For the case of components inspected at time T (see Figure 7), the following conditional 
probability is defined: 
 
)(
)(
1
)(
)(1)(1
)(
)()(
)(
TR
tR
TR
TRtR
TR
TFtF
TtimettimeP ?=
+??
=
?
=>?            (23) 
 
 
 
f(t )
t
t
T
P(Time ? t | Time > T)
 
 
 
 
 
 
Figure 7. Conditional probability of occurrence of failure with inspection at T 
 
 
In equation 23, the functions F and R are the probability of failure and reliability (1 - F) 
at the respective times.   Then, considering the Weibull probability distribution where 
R(x) = exp(x/? )
?
, equation 23 becomes: 
 
24 
 
 
 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?=
?
??
??
ii
i
tt
tF
1
exp1)(                                       (24) 
 
and the conditional Weibull density function, dF(t
i
)/dt
i
: 
 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
? ???
????
?
iii
i
ttt
tf
1
1
exp)(                                   (25) 
 
The previous equation constitutes the basis of the model in terms of probability of failure. 
  
3.2 Cost per unit time and cycle lengths 
For a component periodically tested every T units of time, the failure probability and 
reliability in the ith test cycle are: 
 
dttfTF
iT
Ti
i
)()(
)1(
?
?
=                                                 (26) 
 
dttfTR
iT
Ti
i
)(1)(
)1(
?
?
?=                                              (27) 
 
where f(t) is the conditional Weibull pdf presented above. 
 
Likewise, the cost per unit time for the ith test cycle, considering inspection every time T, 
is defined as fallows: 
 
Cost per unit time
i
 =
][LE
Cost
i
i
= 
][
][)(
LE
DECppTFCrCi
i
iiii
??+?+
             (28) 
25 
 
 
 
 
where: 
 
T: inspection interval 
Ci
i
: direct cost of inspection in the ith test cycle 
Cr
i
: direct cost of repair in the ith test cycle 
Cp: cost of lost production 
p: probability of losses given unavailable component, ]Pr[ uL  
 
and 
 
Expected downtime
i
 = E[D]
i
 = E[L]
i
 ? E[U]
i
                               (29) 
Expected cycle length
i
 = E[L]
i
 = T + Ti + Tr?F
i
(T)                           (30) 
Expected uptime
i
 = E[U]
i
 =  )()(])1([
)(
1
)(
)1(
TFdttfTit
TF
TR
i
iT
Tii
i
????+?
?
?
T      (31)      
 
being Ti and Tr the average time to test and repair (constant deterministic values). 
 
Here, it is important to explain that the expected uptime (equation 31) is defined as the 
test interval, T, multiplied by its probability, R
i
(T); plus the mean of the pdf between two 
successive tests multiplied by its probability, F
i
(T). As commented before, this 
expression is an extension of the concepts explained in chapter 2.   
In accordance with the assumptions, in this model the probability of failure, reliability, 
and cost rate function (cost per unit time) are different for each inspection cycle.  
Likewise, the values of the expected uptime and cycle length vary after every test interval 
since the component is under aging and remains ?as bad as old? after every test/repair. 
After N test cycles the component is subject to overhaul, and returns to the ?as good as 
new? condition. Thus, the renewal cycle length is defined as follows: 
 
26 
 
 
 
?
=
=
N
i
itotal
LEL
1
][                                                (32) 
 
Similarly, the total cost in the renewal cycle is: 
 
?
=
+=
N
i
itotal
CostCoC
1
                                              (33) 
where Co is the cost of overhaul. 
 
Therefore, the cost per unit time or cost rate function (crf) in the renewal cycle becomes: 
 
total
total
L
Cost
NTcrf =),(                                               (34) 
 
Equation 34 constitutes the basis for this study, since the purpose of this work is to 
identify an inspection policy that minimizes the costs during the renewal cycle.  Notice 
that the crf is a function of two variables, T and N.  So that, for every overhaul frequency, 
N, there will be a test interval, T, that minimizes the cost per unit time (crf).   
 
3.3 Increasing test and repair costs 
As indicated in the model assumptions, the approach presented in this report considers 
incrementing costs of surveillance tests and repairs (corrective actions). In accordance 
with the ?as bad as old? premise, as the unit becomes older, it may be more expensive to 
perform inspections and possible subsequent repairs.  This effect is taken into account by 
considering costs that vary as function of the test cycle number (i).   
27 
 
 
 
Although it is possible to affirm that in real situations the direct cost of tests and repairs 
will increase with the component age, the specific function that would represent the 
increment is uncertain and will depend on the specific type of component.   
Given this uncertainty, three different incrementing-cost functions are considered in this 
work: a linear relation, and two non-linear functions: 
3.3.1 Linear function 
According to this model the costs of inspection and repair in the ith test cycle are: 
 
 
imCiCi
i
?+=
0
                                                (35) 
 
 
ibCrCr
i
?+=
0
                                                (36) 
 
 
3.3.2 Non-linear function 1 
In this case the costs of inspection and repair in the ith test cycle are: 
 
 
i
i
mCiCi +=
0
                                                (37) 
 
 
i
i
bCrCr +=
0
                                                (38) 
 
3.3.2 Non-linear function 2 
For this case the costs of test and repair in the ith test cycle are assumed to be: 
 
 
m
i
iCiCi +=
0
                                                (39) 
 
 
b
i
iCrCr +=
0
                                                (40) 
 
 
28 
 
 
 
where Ci
0
, Cr
0
, m and b are arbitrary constants, and i is the test cycle number. 
 
 
To give an idea of the behavior of these incrementing models, the three repair cost 
functions are plotted in Figure 8.  For the example, the following values are 
assumed:  
 
Table 2. Example of parameters for repair cost functions 
 
Incrementing repair cost parameters 
Linear Non-linear 1 Non-linear 2 
Cr
0
 = 1000 Cr
0
 = 1000 Cr
0
 = 1000 
b = 100 b = 2 b = 3 
 
 
 
0
500
1000
1500
2000
2500
12345678910
Inspection cycle number
C
o
s
t
 of r
e
pa
i
r
 
($
)
Cr (linear)
Cr (non-linear 1)
Cr (non-linear 2)
 
 
Figure 8. Behavior of three types of incrementing repair cost functions 
 
 
 
29 
 
 
 
3.3 Description of the variables involved in the analytical model 
As mentioned before, the cost rate function (crf) is a function of two variables, T and N.  
Yet, equation 34 involves many different cost values and task durations that will depend 
on the specific component to be evaluated.  The following paragraphs summarize each of 
these values: 
 
? : The scale parameter of the conditional Weibull distribution. This parameter is similar 
in magnitude to the mean of the distribution, so that, it can be seen as a representation of 
the mean time between successive failures. The value of ?  is always greater than zero and 
depends on the specific unit to be studied.   
 
? : The shape parameter of the conditional Weibull density function. Like ? , this value is 
an attribute of the component under study. Values of ?  smaller than 1 are used to 
represent units with decreasing failures rates (e.g. infant mortality). ?  = 1 corresponds to 
constant failure rate, and values of ?  greater than 1 represent components with increasing 
hazard functions. 
The model presented in this work, is basically oriented to the latter case, that is, to 
components under aging process whose failure rate increases with time. Typically, the 
value of ?  ranges from 1 to 5 for units under wear out (deterioration). 
In next chapters, this work explores the effect of ?  and ?  in the optimal test interval of 
periodically tested units, and considers variability (uncertainty) in these two parameters. 
 
Ci: The direct cost of a planned surveillance inspection, expressed in dollars or other 
money unit.  This cost is assumed to include both materials and labor. As explained 
30 
 
 
 
before, this value may increase after each test cycle according to the equations presented 
in the previous section.  Nevertheless, this cost is a non-random value. 
 
Cr: The direct cost of the repair tasks (corrective actions) that may follow a surveillance 
test, expressed in dollars or other money unit.  This cost is assumed to include both 
materials and labor. As explained before, this value may increase after each test cycle 
according to the equations presented in the previous section.  Nevertheless, like the 
inspection one, this cost is a non-random variable. The magnitude of this value is 
generally greater than the test cost. 
 
p: The conditional probability of losses given unavailable component, ]Pr[ uL . For the 
case of normally operated units, this is the chance that a component absence actually 
impacts the production of the system or facility.  For standby or spare items, this 
probability represents the chance of simultaneous failures, that is, the chance that the 
component is required when being unavailable. This value varies from 0 to 1. 
 
Cp: The cost (economic impact) of downtime. It is sometimes the profit margin 
associated with the system or plant where the component is located.  It is expressed in 
terms of money per unit of time.  This value is possible to establish in most production 
facilities and represents the amount of money that operators loose when a given system 
becomes unavailable. 
 
Co:  The total cost of preventive overhaul, expressed in dollars or other currency.  It 
involves the preventive maintenance tasks, materials and labor required to restore the 
component to an ?as good as new? condition.  It may be a total replacement of the unit.  
31 
 
 
 
Since it is a well planned maintenance task, it is assumed to be short in duration, and to 
be performed in operational windows, without any production impact due to unit 
downtime.  This is a deterministic value in the analytical model, and is greater than the 
cost of corrective (unplanned) repairs. 
 
Ti: The duration of the surveillance test, expressed in units of time.  For this particular 
model, this is assumed to be a constant (deterministic) value.  It can be estimated as the 
average test duration according to the experience and maintenance history of the 
component. 
 
Tr: The duration of the repair task that may follow an inspection, expressed in units of 
time.  For this particular model, this is assumed to be a constant (deterministic) value.  
However, it can be estimated as the average or mean time to repair according to the 
experience and maintenance history of the component. This duration is usually larger 
than the average time to inspect. 
 
32 
 
 
 
Chapter 4: Results 
The analytical model described in the previous chapter was computed by using 
Mathcad? Professional version 8. With this tool, it is possible to solve the equations 
involved in the model, which in most cases require numerical calculation.  The following 
sections discuss the results obtained. 
 
4.1 Average availability 
An important measure in the study of repairable components is the average availability in 
a period of time.  This value represents the probability that the unit is operative for a 
specific mission time or interval.  For the periodically tested components studied in this 
work, the average uptime (availability) in a given test cycle is particularly interesting. 
The probability of being operative in the test cycle is calculated as the expected uptime 
divided by the expected test cycle length. It depends on several factors: inspection 
interval, chance of failures, duration of the test and possible repairs.   
In particular, the chance of failures within the inspection interval is determined by the 
probability function of the time between failures; thus, by the conditional Weibull density 
function with parameters ?  and ?  (equation 25). 
The scale parameter, ? , gives us an idea of the mean time between failures [11]. The 
parameter ?  of this equation defines the type of failure rate: whether it is decreasing, 
constant, or increasing with time. The latter corresponds to components under aging 
processes, that is, to units whose failure rate augments with the age of the item. 
The effect of the shape parameter, ? , on the test cycle average availability is depicted in 
Figures 9 and 10.  These charts are based on arbitrary values of T, Ti, Tr and ?  (Table 3) 
33 
 
 
 
and show the uptime probability per test cycle in a component with ?as bad as old? 
inspections/repairs (a component that retains its old condition after every test cycle). 
 
Table 3. Values for the average availability analysis 
 
Inspection interval Test duration Repair duration Weibull scale parameter
T = 325 days Ti = 2 days Tr = 8 days ?  = 20000 days 
 
0.9800
0.9850
0.9900
0.9950
1.0000
12345678910
Test cycle number
A
v
a
i
la
bi
li
t
y
?  = 1.2
?  = 2
?  = 2.5
 
Figure 9.  Average availability versus test cycle number for ?  > 1 
 
Figure 9 illustrates the availability behavior of a unit with increasing failure rate (?  > 1).  
As anticipated, the chart shows how the average uptime decreases as the unit becomes 
older, and how the aging process is more significant for larger values of the shape 
parameter.  ?  greater than one represents components under aging, which is the case of 
most maintained units, and the focus of this work. 
 
 
34 
 
 
 
0.9100
0.9300
0.9500
0.9700
0.9900
12345678910
Test cycle number
A
v
a
i
la
bi
li
t
y
?  = 1
?  = 0.7
?  = 0.5
 
 
Figure 10. Average availability versus test cycle number for ?  ? 1 
 
Figure 10 depicts the average availability for ?  equals 1 and smaller values (decreasing 
failure rate).  The former case constitutes the particular situation where the Weibull 
distribution becomes the exponential pdf (constant failure rate). Under this circumstance, 
even with the ?as bad as old? assumption, the component does not exhibit any aging 
(availability is the same in every cycle) due to the non-memory condition of the 
exponential distribution. 
For values of ?  smaller than one, the average uptime in the test cycles increases with the 
age of the unit.  This reflects the behavior of items whose failure rate decreases with time. 
An example of this situation is the so called ?infant mortality?, a characteristic observed 
in some components in very early stages of their lifespan. 
 
4.2 Cost per unit time 
The cost rate function, crf(T, N), represents the cost per unit of time in the renewal cycle.  
To find the optimal time between inspections, T
opt
, it is necessary to find the value of T 
35 
 
 
 
that minimizes the cost rate function.  Since crf is a function of two variables, an optimal 
T can be found for every value of N.  The overhaul frequency, N, is often established a 
priori, so that, the optimal test interval is usually found once the value of N is fixed by the 
maintenance planner. Nevertheless, we will see later that an optimal combination of 
overhaul frequency and test interval can also be determined.   
Figure 11 shows the cost rate function versus T for three different values of N.  For this 
chart, the following arbitrary values were assumed: 
 
Table 4. Arbitrary values for cost rate function examples 
 
Cost values Time between  
failures $ $/day 
Durations (days) 
?  (days) ?  
Ci
0
* Cr
0
* Co p?Cp Ti Tr 
20000 1.2 500 1000 20000 (0.7)?(20000) 2 8 
*with linear increment according to equations 35 and 36 (m = 50, b = 100) 
 
 
 
 
 
 
 
 
 
 
 
600
100
crf1 T()
crf5 T()
crf10 T()
8001 T
0 100 200 300 400 500 600 700 800
100
200
300
400
500
600
(d ays)
(d
o
l
l
a
rs
/
d
a
y
)
N = 1
N = 5
N = 10
C
o
st
 rate f
unct
i
o
n
(d
o
l
l
a
rs
/
d
a
y
)
(d
o
l
l
a
rs
/
d
a
y
)
C
o
st
 rate f
unct
i
o
n
 
Figure 11. Cost rate function for different values of N  
36 
 
 
 
According to Figure 11, the minimum of the curve for N = 1 is reached at T  510 days, 
while for N = 5 and N = 10 the minimum corresponds to T 
?
?  360 and T  320 days 
respectively.  These three values constitute the optimal test interval for these three cases 
of overhaul frequency.  
?
It is important to notice that T
opt
 decreases as N becomes larger. This behavior is 
consistent with the inputs used for this example.  For N = 1 (overhaul at every test) it is 
economically feasible to have relatively large inspection intervals.  On the other hand, 
since the component is under aging (?  > 1), if the overhaul is performed every five or ten 
inspection cycles, it becomes necessary to decrease the test interval to reduce the 
expected cost of unavailability and hence the total cost per unit time. 
Based on the inputs of Table 4, the behavior of the optimal inspection interval versus N is 
plotted in Figure 12.  Here it is important to observe that the value of T
opt
 decreases as N 
increases, but in all cases tends to an asymptotic value for very large N. 
 
0.0
100.0
200.0
300.0
400.0
500.0
600.0
123456789101121314151617181920
N
T
o
p
t (d
a
y
s
)
?  = 20000
?  = 10000
?  = 5000
 
Figure 12. Optimal test interval versus overhaul frequency for different ?  
37 
 
 
 
Another important aspect to be discussed is the limiting value of the cost rate function.  
To observe this limiting behavior, the three curves of Figure 11 are plotted in Figure 13 
for very large values of the test interval, T. 
 
 
 
 
  
crf
 
crf
 
 
 
 
1.
20000
0
1 T()
crf5 T()
10 T()
30000001 T
0 5 10
5
1 10
6
5 10
6
2 10
6
2.5 10
6
3 10
6
0
2000
4000
6000
8000
1 10
4
1.2 10
4
1.4 10
4
1.6 10
4
1.8 10
4
2 10
4
(d ays)
(
d
o
lla
r
s
/d
a
y
)
p.Cp
N = 1
N = 5
N = 10
cr
f
(
d
o
l
lars/d
ay)
(
d
o
lla
r
s
/d
a
y
)
cr
f
(
d
o
l
lars/d
ay)
 
Figure 13. Limiting cost rate function for different values of N 
 
 
 
From Figure 13, two observations are important.  First, it can be seen that the cost rate 
function has only one minimum, and second, it is important to point out that the limiting 
value of the function corresponds to p?Cp regardless of the overhaul frequency N.  This 
fact confirms the importance of the cost of unavailability in this analytical model. 
The following set of equations verifies the limiting behavior depicted in Figure 13: 
Knowing that R(?) = 0, and F(?) = 1, for a given test cycle we have, 
 
Cost
i
 = Ci ][DECppCr
iii
??++                                    (41) 
38 
 
 
 
E[L]
i
 = Ti + Tr + T                                              (42) 
 
E[U]
i
 = MRL
i
                                                  (43) 
 
where MRL
i
 is the mean residual life after the (i-1)th test cycle. 
 
E[D]
i
 = E[L]
i
 ? E[U]
i
  = Ti + Tr + T - MRL
i
                        (44) 
 
So, for the renewal cycle the total cost is: 
 
Cost = Co + +  + p?Cp           (45) 
?
=
N
i
i
C
1
?
=
N
i
i
Cr
1
?
?
?
?
?
?
?++
?
=
N
i
i
MRLTTrTiN
1
)(
 
and the renewal cycle length is, 
L = N(Ti + Tr + T)                                           (46) 
 
Thus, with Ti + Tr + T T, the cost rate function for the renewal cycles is: ?
 
crf = 
TN
CrCCo
N
i
N
i
ii
?
?
?
?
?
?
?
?
++
??
==
1
11
+ p?Cp ?
?
?
?
?
?
?
??
?
=
N
i
i
MRLTN
1
TN ?
1
            (47) 
 
which for T?  ? , yields 
Limiting crf = p?Cp                                             (48) 
 
39 
 
 
 
4.3 Optimal inspection interval and overhaul frequency 
In the previous sections general results for arbitrary data were discussed.  This section 
presents particular results based on a case study. 
Safety relief valves are typical industrial components that have to be tested and calibrated 
every certain period of time.  These units are normally installed in systems or operating 
plants; they are used under emergency or abnormal situations to relief overpressures and 
avoid failure of pressure vessels or process pipelines.  Relief valves should operate 
automatically in case of overpressure, so that, their availability should be guaranteed by 
periodic inspection and appropriate calibration of the pressure set-point. 
As a practical example, a typical safety relief valve was considered for this study.  The 
following data were used: 
 
Table 5. Input values for safety relief valve example 
 
Cost values Time between 
failures 
$ $/month 
Durations  
(months) 
?  
months 
?  Ci* Cr* Co p?Cp Ti Tr 
3500 1.5 500+50?i 5000+500?i 2x10
4
 0.4?8x10
5
 0.05 0.25 
*with linear increment after every test cycle according to equations 35 and 36; i = cycle number 
 
The reliability parameters, ?  and ? , were selected according to failure data of nuclear 
facilities [12].  The Weibull scale parameter (3500 months) gives an idea of the mean 
time to failure of the unit.  The shape parameter (1.5) indicates that the component has an 
increasing failure rate (aging process). 
40 
 
 
 
Direct costs of inspection and repair were assumed based on experience and engineering 
judgment.  Here, a linear cost increment according to equations 35 and 36 is taken into 
account. 
Similarly, the cost of overhaul (preventive maintenance) and cost of unavailability were 
selected according to experience and typical data of industrial plants.  Cost of overhaul 
involves material and labor, while cost of unavailability expresses the total losses, in 
dollars per month, due to system downtime. The parameter p is the conditional 
probability of actual system downtime given that the valve is unavailable. A value of p 
equals to 0.4 indicates a 40%-chance of economic impact due to valve unavailability. 
Further discussion of this aspect will be discussed in next chapter. 
The last two columns of Table 5 show the duration of test and repair tasks. Based on 
experience, a periodic valve inspection is assumed to take 1.5 days (0.05 moths), while an 
unplanned corrective maintenance is taken as 1 week (0.25 months) in length. 
In general, the values presented in Table 5 are reference numbers for this academic work. 
They were selected to illustrate the use of the model and are not intended to be used as 
exact values in real projects.  Real life analysis should be based on specific data and will 
vary according to the particular characteristics of the unit and system under study. 
Using the inputs presented in Table 5, and optimizing equation 34 for different values of 
overhaul frequency (N), different results were obtained for the optimal inspection interval 
(T
opt
) for this case study. 
The following chart shows the results: 
 
 
41 
 
 
 
 
 
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
12345678910
N
Topt
 (
m
ont
hs
)
 
Figure 14. T
opt
 versus overhaul frequency for a safety relief valve 
 
 
Figure 14 shows the optimal inspection interval for different values of the overhaul 
frequency.  As discussed before, the value of T
opt
 decreases and tends to stabilize for 
large values of N. 
If the inspection/maintenance policy is to pre-set the overhaul frequency based on 
management strategy, this chart can be used establish the corresponding optimal test 
interval for a given value of N. This may be the case of some industrial facilities were the 
frequency of overhaul (preventive maintenance) has to meet certain standards or 
regulations. 
Now, in order to identify the optimal overhaul frequency, it is important to explore the 
value of the cost rate function, crf, for each N.  Accordingly, Figure 15 plots the crf 
evaluated at T
opt
 for different values of overhaul frequency. 
 
42 
 
 
 
43 
 
 
This optimal combination of time between inspection (T) and overhaul frequency (N) can 
also be seen when plotting the crf as a function of these two variables. Figure 16 
illustrates, by using a surface chart, how the optimal combination (minimal point in the 
crf surface) is reached at T = 51 months and N = 2. 
Figure 15 indicates that N = 2 constitutes the optimal overhaul frequency, since it yields 
the minimum value of cost rate function for this case study.  Thus, the optimal inspection 
policy for this safety relief valve is to inspect the valve every 51 months (from Figure 14) 
with preventive overhaul every 2 test cycles. 
 
 
 
Surface plotting constitutes a useful way to represent the general idea of optimizing the 
total cost per unit time as a basis to establish the best inspection and preventive 
maintenance policy. 
600.00
1234
700.00
800.00
900.00
1000.00
1100.00
1200.00
5678910
N
c
r
f(T
o
p
t)  [$
/
m
o
t
h
]
 
Figure 15. crf(T
opt
) versus overhaul frequency for a safety relief valve 
20
23
26
29
32
35
38
41
44
47
50
53
56
59
1 2
3 4
5
500.00
650.00
800.00
950.00
1100.00
1250.00
1400.00
1550.00
1700.00
1850.00
2000.00
crf [$/month]
T [months]
N
 
 
44
 
Figure 16. Cost rate function versus T and N for a safety relief valve 
 
 
 
 
4.4 Availability versus cost-based optimization 
The inspection interval of periodically tested components has been usually selected 
without considering cost aspects.  Traditionally, the inspection policy has been based on 
maximizing the unit availability with almost no attention to inspection, repair and failure 
costs. 
Using the relief valve example, it is possible to identify the value of the test interval (T) 
that yields the maximal average availability over the renewal cycle.   
This calculation can be done for any value of overhaul frequency.  For instance, taking N 
equals to 10, the average availability for the renewal cycle is defined as follows: 
 
?(T) = 
?
?
=
=
=
10
1
10
1
][
][
i
i
i
i
LE
UE
lengthcycleExpected
uptimeExpected
                               (49) 
 
where E[L]
i
 
 
and E[U]
i
 
 
are calculated according to equations 30 and 31 respectively. 
 
By definition, the expected uptime and the expected renewal cycle length are functions of 
the inspection interval. Thus, the average availability for the renewal cycle can be plotted 
as a function of the time between inspections (T). 
Figure 17 shows the behavior of ?(T) versus the inspection interval for N = 10.  This 
chart also indicates the value of T that maximizes the average availability for the renewal 
cycle assuming overhaul after 10 surveillance tests. 
 
 
 
45
 
 
 
1
0.95
AT()
1001 T
0 10 20 30 40 50 60 70 80 90 100
0.95
0.96
0.97
0.98
0.99
1
T = 29 months
a(T)
 
 
 
 
 
 
 
 
 
 
 
Figure 17. Average availability versus T for a relief valve with N = 10  
 
Here, it is important to point out that the value of T that yields the maximum availability 
(29 months) is smaller than the one that minimizes the cost per unit time for N equals 10 
(31 months, from Figure 14). Moreover, since the cost rate function (crf) is decreasing to 
the left of T
opt
, it is important to notice that the value of the crf evaluated at T for maximal 
availability is greater than the one evaluated at T for minimal cost (Figure 18). 
 
 1100
980
crf T()
4020 T
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
980
992
1004
1016
1028
1040
1052
1064
1076
1088
1100
[mont hs]
[$
/
m
o
n
t
h
]
[$
/
m
o
n
t
h
]
 
 
 
 
 
 
 
Figure 18. Cost rate function for N = 10 for a safety relief valve 
 
46
 
 
According to this, to establish the inspection interval based on cost optimization instead 
of maximizing availability (traditional approach) yields to important savings, especially if 
we consider the large number of periodically tested units that may exist in a system or 
facility.  For instance, if we consider the optimal combination obtained for the case study 
(N = 2, T
opt
 = 51 months), and a plant with 100 valves, we would get: 
 
Table 6. Availability versus cost-based optimization results for a relief valve 
 
T
opt
 crf(T
opt
) T
max. availability 
crf(T
max. availability
) Savings per valve 
per year 
51 months 869 $/month 42 months 895 $/month 312 $ 
 
Here, the amount saved per month and unit would represent a total saving of 31200$/year 
in a plant with 100 periodically tested valves. 
 
4.5 Sensitivity evaluation 
To evaluate the model sensitivity to changes in the different inputs, marginal variations 
(one input at a time) were considered for the case study.  Results of this evaluation are 
presented in Table 7. 
We can note from Table 7 that the case study results do not change with moderate 
variations (? 40%) of the repair duration, and direct cost of inspection and repairs.  On 
the other hand, the optimal combination of N and T is even sensitive to small changes of 
Ti, Cp or Co.  The optimal value of N decreases with increments in Ti or Cp, and 
increases when Co is augmented.  
As anticipated, the optimal time between inspections increases with increments in the test 
duration. 
 
47
 
 
 
Table 7. Results of sensitivity evaluation for the case study 
 
Input N
opt 
T
opt
 (months) 
0.03 3 39
0.04 2 48 
0.05 2 51
0.06 1 71 
Ti 
(months) 
0.07 1 74
0.15 2 51 
0.20 2 51
0.25 2 51 
0.30 2 51
Tr 
(months) 
0.35 2 51 
300 2 51
400 2 51 
500 2 51
600 2 51 
Ci
0
*
 
(dollars) 
700 2 51
3000 2 51 
4000 2 51
5000 2 51 
6000 2 51
Cr
0
* 
(dollars) 
7000 2 51 
6x10
5
 2 54
7x10
5
2 52 
8x10
5 
2 51
9x10
5
 1 68 
Cp 
($/month) 
10x10
5 
1 66
10000 1 61 
15000 1 66
20000 2 51 
25000 2 53
Co 
(dollars) 
30000 3 46 
           *with linear increment after every test cycle according to eq. 35 and 36 (m = 50, b = 500) 
 
T
opt
 is also very sensitive to changes in the cost of unavailability (Cp). According to the 
Table, for a given optimal overhaul frequency, the optimal test interval decreases when 
Cp is augmented. Conversely, T
opt 
becomes larger with increments in the cost of overhaul 
(Co) for the same N
opt. 
 
48
 
 
As mentioned above, the optimal test policy does not change with moderate variations of 
the inspection cost (Ci
0
) and repair cost (Cr
0
) for the case of linear increments in these 
values.  The effect of the different incrementing cost models (see Section 3.3) in this case 
study is presented in Figure 19: 
 
25.00
30.00
35.00
40.00
45.00
50.00
55.00
60.00
65.00
70.00
75.00
12345678910
N
To
pt [m
o
n
ths
]
Linear
Non-linear 1
Non-linear 2
 
Figure 19. T
opt
 versus N for different incrementing cost models for a relief valve 
 
 
Figure 19 shows that the results are almost identical for the three models. Thus, for this 
particular case study, the optimal test policy is insensitive not only to moderate changes 
in the values of the direct costs Ci
0
 and Cr
0
, but also to the model considered for the 
increment of these costs after every test cycle.  
According to this observation, it is important to note that even uncertain approximations 
of these inputs will yield acceptable results for this example. 
 
49
 
 
In general, the values of the test duration and cost of overhaul are well known in most 
cases (this helps to control the sensitivity). However, the impact of unavailability (p?Cp) 
is often uncertain.  
The uncertainty of this last value may be controlled by taking into account different risk 
scenarios for this input. That is, by considering an event analysis with all the potential 
scenarios, with their corresponding consequence and likelihood. This analysis can be 
done with an event tree and probabilistic risk assessment [12].  By using this approach, 
instead of a single input for the expected impact of unavailability (p?Cp), there would be 
an expectation composed by many scenarios, e.g.  p
1
?Cp
1
 + p
2
?Cp
2
 + p
3
?Cp
3
 +.... p
n
?Cp
n
. 
A special situation regarding the consequence of unavailability is the case of no impact 
(p?Cp = 0). Under this assumption (see Figure 20) the cost rate function does not reach a 
finite minimum.   
 
100
0
crf1 T()
crf5 T()
crf10 T()
100000T
0 2000 4000 6000 8000 1 10
4
0
10
20
30
40
50
60
70
80
90
100
[months]
[$/month]
 
 
   Figure 20. crf vs. T for different N for a relief valve with no unavailability impact 
 
50
 
 
4.6 Uncertainty in Weibull parameters 
Although not shown in Table 7, the parameters of the time-between-failure distribution 
(? , ? ) constitute the basic inputs for the inspection policy analysis.  In most cases, 
especially when no failure history is available, these values are difficult to establish. The 
following paragraphs discuss the variability of these parameters through the consideration 
of uncertainty. 
To account for the variability of the time-between-failure distribution, the parameters ?  
and ?  of the Weibull pdf were treated as random variables. To conduct this analysis, a 
conventional Monte Carlo Simulation (MCS) was carried out. 
Monte Carlo Simulation is a technique based on the use of generated random numbers 
[20]. For this analysis, uniformly distributed random numbers between 0 and 1 were 
sampled by using Mathcad? Professional in order to generate a set of random values for 
the parameters ?  and ? . 
Within the process, the zero-to-one sampled numbers were converted into ?  or ?  values 
by using inverse cumulative distribution functions. For this particular study, the 
following probability distributions were considered (see Mathcad sheet in Appendix): 
 
? :  normally distributed, with mean equals to 3500 months and standard deviation equals 
to 350 months. 
? :  normally distributed, with mean equals to 1.5 and standard deviation equals to 0.1. 
 
Then, using 1000 repetitions of the MCS-sampling process, the following results were 
obtained for the case study: 
 
 
51
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
0.00
20.00
40.00
60.00
80.00
100.00
120.00
12345678910
N
To
pt
 (
m
on
t
h
s
)
Topt
f
Point estimate
To
pt
 (
m
on
t
h
s
)
 
Figure 21. Point estimates and 90% confidence bounds for T
opt
 vs. N for a relief valve 
 
 
As presented in Figure 21, with uncertainty in the Weibull parameters, instead of a single 
number for the optimal test interval, we get a range of values (random variable) for each 
N.  The dots of the figure indicate point estimates (mean of the distributions), while the 
bars show the 90% confidence intervals (5 and 95 percentiles). 
 
 
0.00
200.00
400.00
600.00
800.00
1000.00
1200.00
1400.00
1600.00
1800.00
12345678910
N
c
r
f
(
Topt
)
  [
$
/
m
ot
h]
 
 
Figure 22. Point estimates and 90% confidence bounds for crf(T
opt
) vs. N for a relief valve 
 
52
 
 
Similarly, Figure 22 shows the point estimates and 90% confidence intervals for the cost 
rate function evaluated at T
opt
 for different N. 
It can be seen in both Figure 21 and 22 that the point estimates are consistent with the 
results plotted in Figures 14 and 15.  It also can be noticed (especially in Figure 21) that 
the 90% confidence intervals become narrower as N increases; so that, we can note that 
the effect of uncertainty in the Weibull parameters is more significant for small values of 
the overhaul frequency. 
Although the optimal N is obvious in Figure 15, we can observe in Figure 22 that the 
optimal frequency becomes also a random variable when considering uncertainty in the 
Weibull parameters.   
From the Monte Carlo Simulation (see Figure 23), N
opt
 is a discrete random variable with 
mode equals 2 and standard deviation equals 0.8 (5-percentile = 1, 95-percentile = 3). 
 
0
0.1
0.2
0.3
0.4
0.5
12345
Nopt
Fr
e
que
nc
y
 
Figure 23. Frequency chart for the optimal overhaul frequency for a relief valve 
 
 
 
53
 
 
Summing up, for this practical example we can say with 90% confidence that the optimal 
inspection interval for the relief valve is between 36 and 70 months (3 and 5.8 years), 
with an optimal overhaul frequency that falls between 1 and 3. 
The use of uncertainty in the inputs of the model provides important information for 
decision making, especially when there is no failure history available.  This kind of 
analysis provides results in terms of range of values (confidence intervals) instead of 
single point estimates.  Unlike the deterministic analysis, the consideration of certainty 
bounds usually makes inspection planners confident about the results. 
 
54
 
 
Chapter 5: Extensions 
Previous chapters described the analytical model developed for this work and discussed 
important results based on general inputs and particular examples related to a case study.  
The following paragraphs explain possible extension of this work that can be considered 
in future studies.  Among the aspects that can be expanded or added, the following 
elements can be taken into account: 
 
5.1 Generalized Renewal Process after test cycles 
The model presented in this report assumes that the component returns to an ?as bad as 
old? condition after every test cycle (NHPP model).  Nevertheless, in real situations it is 
possible to find cases where the inspection/maintenance tasks restore the unit to 
conditions different than ?as bad as old?.  Even though it has been shown that the NHPP 
model provides results very similar to those obtained with Generalized Renewal Process 
(GRP) with 0.1 < q < 1 [17], the GRP approach is still useful for situations with q > 1 
(worse-than-old restoration) and  q < 0 (better-than-new restoration). 
To consider the GRP model after the inspection cycle, instead of the NHPP expression 
(equation 24), the following equation should be considered [15]: 
 
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?=
?
?
?
=
?
=
1
1
1
1
exp1)(
i
j
ji
i
j
ji
tqt
t
q
tF                            (50) 
 
 
55
 
 
Like equation 24, equation 50 assumes that the interarrival times between successive 
failures follow a Weibull distribution.  However, the GRP expression incorporates an 
additional parameter (q) to account for the quality of the maintenance tasks. 
 
5.2 Imperfect surveillance inspections 
The analytical model described in this document is based on perfect surveillance tests. 
According to this assumption, unraveled failures can always be detected by the inspection 
tasks (probability of detection equals to 1).  A possible extension for this analysis may 
incorporate the probability of failures remaining undiscovered after the test, to account 
for the possibility of imperfect inspections. Considerations for optimal inspection and 
maintenance policies of units with unrevealed failures with less than perfect tests are 
given by Bad?a et al. [8]. 
 
5.3 Further uncertainty and risk analysis 
In chapter 4 the model sensitivity was discussed.  As presented in Table 7 and explained 
in section 4.5, the results for the optimal inspection policy are very susceptible to changes 
in the values of test duration (Ti), Cost of overhaul (Co) and the potential cost of 
unavailability (p?Cp).   
As done for the Weibull parameters of the time-between-failure distribution, random 
variables maybe used to account for the variability of Ti and Co.  As presented in the 
Appendix (Mathcad work sheet) this extended analysis can be modeled by using random 
numbers and Monte Carlo Simulation. 
 
56
 
 
The uncertainty associated with the impact of unavailability maybe handled with a 
rigorous risk analysis that considers different scenarios.  Moreover, the analytical model 
and risk analysis may be extended to incorporate not only the impact of unavailability in 
terms of cost per unit time, but also potential accidents with instant safety consequences. 
 
5.4 Systems of periodically tested components 
This work is focused on concepts and equations for availability and cost analysis at 
component level.  However, future studies may be extended to system analysis. 
The analysis of systems consists of studying multiple components simultaneously, 
considering not only the properties of the single items but also the way they are related.  
Besides single periodically tested units, situations are also encountered for systems with 
components whose failures are only detected upon inspection. In these cases, the 
evaluation of availability becomes more complex, for different testing strategies may be 
considered.   
To analyze the availability of this type of systems it is important to give especial attention 
to the test interval, T, and also to the inspection strategy.  Depending on the inspection 
crew and maintenance plan, units can be tested simultaneously or in a staggered manner, 
which will affect the estimation of the system availability. 
Lewis [18] developed simple expressions assuming negligible test and repair times with 
respect to the inspection interval, just to evaluate the effect of staggered testing in system 
availability.  Table 8 presents approximated equations for typical system configurations 
of periodically tested components with identical constant failure rate (?): 
 
57
 
 
Table 8. Average availability for systems with periodically tested units 
 
Test strategy  2-unit series system 2-unit parallel system 
Simultaneous 1 - ? T + 2/3(? T)
2
 1 - 1/3(? T)
2
 
Staggered 1 - ? T + 13/24(? T)
2
 1 - 5/24(? T)
2
 
 
 
From Table 8, it is important to point out that the availability of redundant systems is 
grater with staggered tests, while the availability of series configuration is better with 
simultaneous inspections.  
 
Redundant system with common cause failures 
Redundant systems are usually subject to dependencies and common cause failures that 
may affect more than one unit at the same time.  For these cases, the inspection strategy 
plays an important role in system availability, especially for those systems where the 
component failures remain unrevealed until next inspection. 
Resent works have explored the effect of test strategies in the availability of redundant 
system. Particularly, Vaurio [21] presented a set of equations to evaluate the common 
cause failure probabilities in standby systems, and an economic model to optimize the 
inspection interval according to different inspection policies. 
The cost-based optimization for the inspection policy of systems with periodically tested 
components is still an area of research. Given the additional complication that represents 
the system itself, research works usually make simplifications at component level when 
studying complex systems. In general, the optimal inspection policy of systems with 
periodically tested units allows for advance considerations. Analysis on this topic should 
take into account not only the optimal test interval and maintenance frequency, but also 
the optimal degree of test staggering. 
 
58
 
 
Chapter 6: Conclusion  
 
Inspection and maintenance strategies have been widely studied for monitored 
components; however, less attention has been paid to units whose failures are detected 
upon surveillance tests. 
In this work, a cost rate function model was developed to identify the optimal inspection 
policy of periodically tested repairable components under aging process. 
The model presented in this report is based on minimizing the total cost per unit time 
during the component renewal cycle. It considers the unit availability assuming that the 
item is ?as old? after tests and repairs and ?as new? after overhauls. The model takes into 
account costs associated not only with surveillance tests and maintenance, but also with 
the potential losses related to component unavailability.  The model also considers the 
duration of inspections and repairs, and allows for uncertainty in the parameters of the 
probability density function of the time between failures. 
The effect of overhaul policy on the optimal test interval, T
opt
, was studied considering 
different values of the overhaul frequency, N. Results obtained from diverse sets of costs 
and time-to-failure parameters suggest that T
opt
 decreases and tends to stabilize when N is 
increased.  
The effect of N on the total cost per unit time was studied by evaluating the cost rate 
function, crf, in T
opt
 for different values of N. The analysis reveals that an optimal N can 
be identified, so that, the use of the model provides not only an optimal time between 
surveillance tests, but also an optimal overhaul frequency for periodically tested 
components. 
 
59
 
 
A practical numerical example carried out for a typical tested unit (a safety relief valve) 
with mean time between failures of 105 days shows that the optimal policy would be to 
inspect the valve every 51 months, with preventive overhaul every two test cycles. 
Comparison between the optimal test interval obtained from cost optimization and the 
one got for maximal availability (traditional approach) shows that the cost-based 
optimization may provide significant savings especially in facilities with several 
periodically tested units. 
Sensitivity evaluation in the case study indicates that the results are almost insensitive to 
moderate variations of the repair duration, as well as variations of the direct cost of 
inspection and repairs (this behavior may not be the case in other analyses).  On the other 
hand, the optimal inspection policy is very susceptible to changes in test duration (Ti), 
cost of overhaul (Co), cost of unavailability (Cp), and the Weibull parameters of the time 
between failures (?  and ? ).  The optimal value of N decreases with increments in Ti or 
Cp, and increases when Co is augmented. This confirms the significance of these inputs 
and the importance of considering the test duration (often neglected in literature) in the 
analysis of periodically tested units. 
Monte Carlo Simulation was used to study the effect of uncertainty in the Weibull 
parameters of the relief valve failure distribution. Results obtained from this analysis are 
consistent with those obtained from deterministic parameters.  
The use of uncertainty in the inputs of the model provides important information for 
inspection decision making, especially when there is no failure history available. This 
kind of analysis offers results in terms of confidence intervals instead of single point 
estimates.   
 
60
 
 
Possible extensions of this work may include the consideration of Generalized Renewal 
Process (GRP) for modeling the component maintenance (both repairs and preventive 
overhaul) as well as the inclusion of non-perfect inspections. 
The use of uncertainty in other model inputs, such as time to inspect and cost of overhaul, 
and the analysis at system level (arrangements of many tested components) are also 
important aspects to be considered in future research works. 
 
61
 
 
 
 
 
 
 
 
 
 
Appendix 
 
Mathcad work sheets 
 
 
62
 
 
RISK AND ECONOMIC ESTIMATIN OF INSPECTION INTERVAL FOR PERIODICALLY 
TESTED REPAIRABLE COMPONENTS
i 1 1000..
r
i
rnd 1() Monte Carlo
Simulation
? i( ) qnorm r
i
1.5, 0.1, ? i( ) qnorm r
i
3571, 357,
fti,()
? i()
? i()
t
? i()
? i() 1
.
e
t
? i()
? i()
.
?
 is Weibull scale 
parameter
RTi,()1
0
T
tfti,()d FTi,()1RTi,()
First cycle
Ti 0.05 Tr 0.25 n1
Cio 500 mi 50 Cro 5000 mr 500 Cp 800000 p 0.4
Ci1 Cio mi n
.
Cr1 Cro mr n
.
Co 20000
L1 T i,( ) Ti T Tr F T i,()
.
U1 T i,()TRTi,()
.
0
T
ttft i,()
.
d
D1 T i,()L1Ti,()U1Ti,()
Cost1 T i,( ) Ci1 Cr1 F T i,()
.
pCp
.
D1 T i,()
.
Co
Cost rate function
y1 T i,()
Cost1 T i,()
L1 T i,()
 
 
 
 
 
63
 
 
Second cycle
n2
Weibull conditional PDF
(Power Law Model)
f2 t i, T,()
? i()
? i()
t
? i()
? i() 1
.
e
T
? i()
? i()
t
? i()
? i()
.
R2 T i,()1
T
2T
.
tf2 t i, T,()d F2 T i,()1R2Ti,()
Ci2 Cio mi n
.
Cr2 Cro mr n
.
L2 T i,( ) Ti T Tr F2 T i,()
.
U2 T i,()TR2Ti,()
.
T
2T
.
ttT()f2ti, T,()
.
d
D2 T i,()L2Ti,()U2Ti,()
Cost2 T i,( ) Ci2 Cr2 F2 T i,()
.
pCp
.
D2 T i,()
.
Cost rate function
Cost T i,( ) Cost1 T i,( ) Cost2 T i,()
LTi,()L1Ti,()L2Ti,()
y2 T i,()
Cost T i,()
LTi,()
 
64
 
 
Third cycle
n3
Weibull conditional PDF
(Power Law Model)
f3 t i, T,()
? i()
? i()
t
? i()
? i() 1
.
e
2T
.
? i()
? i()
t
? i()
? i()
.
R3 T i,()1
2T
.
3T
.
tf3 t i, T,()d F3 T i,()1R3Ti,()
Ci3 Cio mi n
.
Cr3 Cro mr n
.
L3 T i,( ) Ti T Tr F3 T i,()
.
U3 T i,()TR3Ti,()
.
2T
.
3T
.
tt2T
.
()f3ti, T,()
.
d
D3 T i,()L3Ti,()U3Ti,()
Cost3 T i,( ) Ci3 Cr3 F3 T i,()
.
pCp
.
D3 T i,()
.
Cost rate function
Cost T i,( ) Cost1 T i,( ) Cost2 T i,()Cost3 T i,()
LTi,()L1Ti,()L2Ti,()L3 T i,()
y3 T i,()
Cost T i,()
LTi,()
 
65
 
 
Fourth cycle
n4
Weibull conditional PDF
(Power Law Model)
f4 t i, T,()
? i()
? i()
t
? i()
? i() 1
.
e
3T
.
? i()
? i()
t
? i()
? i()
.
R4 T i,()1
3T
.
4T
.
tf4 t i, T,()d F4 T i,()1R4Ti,()
Ci4 Cio mi n
.
Cr4 Cro mr n
.
L4 T i,( ) Ti T Tr F4 T i,()
.
U4 T i,()TR4Ti,()
.
3T
.
4T
.
tt3T
.
()f4ti, T,()
.
d
D4 T i,()L4Ti,()U4Ti,()
Cost4 T i,( ) Ci4 Cr4 F4 T i,()
.
pCp
.
D4 T i,()
.
Cost rate function
Cost T i,( ) Cost1 T i,( ) Cost2 T i,()Cost3 T i,()Cost4 T i,()
LTi,()L1Ti,()L2Ti,()L3 T i,()L4 T i,()
y4 T i,()
Cost T i,()
LTi,()
 
66
 
 
Fifth cycle
n5
Weibull conditional PDF
(Power Law Model)
f5 t i, T,()
? i()
? i()
t
? i()
? i() 1
.
e
4T
.
? i()
? i()
t
? i()
? i()
.
R5 T i,()1
4T
.
5T
.
tf5 t i, T,()d F5 T i,()1R5Ti,()
Ci5 Cio mi n
.
Cr5 Cro mr n
.
L5 T i,( ) Ti T Tr F5 T i,()
.
U5 T i,()TR5Ti,()
.
4T
.
5T
.
tt4T
.
()f5ti, T,()
.
d
D5 T i,()L5Ti,()U5Ti,()
Cost5 T i,( ) Ci5 Cr5 F5 T i,()
.
pCp
.
D5 T i,()
.
Cost rate function
Cost T i,( ) Cost1 T i,( ) Cost2 T i,()Cost3 T i,()Cost4 T i,()Cost5 T i,()
LTi,()L1Ti,()L2Ti,()L3 T i,()L4 T i,()L5 T i,()
y5 T i,()
Cost T i,()
LTi,()
 
 
 
NOTES: 
 
The procedure is similar for subsequent test cycles. 
 
Optimizations were done by using Mathcad commands MAXIMIZE and MINIMIZE. 
 
67
 
 
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