aerospace

Article
Use of Cost-Adjusted Importance Measures for
Aircraft System Maintenance Optimization
Michail Bozoudis 1,† , Ilias Lappas 2, *,† ID
and Angelos Kottas 1,†
1 Hellenic Ministry of National Defense, 15451 Athens, Greece; m.bozoudis@gdaee.mil.gr (M.B.);
angelos.kottas@haf.gr (A.K.)
2 Department of Mechanical and Aeronautical Engineering, School of Engineering, University of South Wales,
Treforest Campus, Treforest CF37 1DL, UK
* Correspondence: ilias.lappas@southwales.ac.uk; Tel.: +44-01443-482565
† These authors contributed equally to this work.




Received: 28 May 2018; Accepted: 21 June 2018; Published: 24 June 2018


Abstract: The development of an aircraft maintenance planning optimization tool and its application
to an aircraft component is presented. Various reliability concepts and approaches have been analyzed,
together with objective criteria which can be used to optimize the maintenance planning of an aircraft
system, subsystem or component. Wolfram® Mathematica v10.3 9 (Witney, UK) has been used
to develop the novel optimization tool, the application of which is expected to yield significant
benefits in selecting the most appropriate maintenance intervention based on objective criteria,
in estimating the probability of nonscheduled maintenance and in estimating the required number of
spare components for both scheduled and nonscheduled maintenance. As such, the results of the
application of the tool can be used to assist the risk planning process for future system malfunctions,
providing safe projections to facilitate the supply chain of the end user of the system, resulting in
higher aircraft fleet operational availability.

Keywords: aircraft system; reliability; life cycle cost; maintenance planning optimization; reliability
centered maintenance; importance measures

1. Introduction
Maintaining a fleet of aircraft poses significant challenges for any organization in the aircraft
operations business, as multiple and, many times, conflicting requirements are set regarding to the
maintenance and operation costs and the desired service levels. Existing approaches to aircraft
maintenance planning and scheduling are limited in their capacity to deal with contingencies arising
out of tasks carried out during the implementation of maintenance projects [1].
At the aircraft system level, recent methods have been proposed [2–4] that aim to optimize the
outcome of an aircraft maintenance plan with respect to various aircraft operational requirements.
These studies are based on deterministic mathematical models describing flight and maintenance
procedures, without taking into account failures and corrective maintenance requirements. An attempt
to incorporate effects of stochastic events in optimizing the maintenance scheduling of an aircraft’s
main system (engine) to achieve robust flight maintenance planning solutions, has employed Monte
Carlo simulations [5]. Further studies have extended the scope to include modelling unscheduled event
consequences [6] and up to using a fuzzy analytic hierarchy process (AHP) to improve staff allocation,
as well as the support of decision-making process within the aircraft maintenance industry [7].
At the aircraft subsystem and multicomponent subsystem level (a Low Pressure Turbine-LPT of an
aircraft jet engine for example), there is an increasing interest on maintenance optimization that would

Aerospace 2018, 5, 68; doi:10.3390/aerospace5030068 www.mdpi.com/journal/aerospace

Aerospace 2018, 5, 68 2 of 20

address preventive maintenance scheduling under various constraints [8], minimize the maintenance
cost and the unexpected maintenance stop occasions [9], or would minimize the operational costs [10].
Maintaining an aircraft system is necessary for achieving sustained performance levels during
its operational life. The maintenance should be implemented with the minimum possible cost,
while adhering to the highest possible quality standards to guarantee the uninterrupted operation of
the systems, minimizing their downtime and the cost-efficient operation of the systems, by careful
allocation of the available resources.
The life cycle cost (LCC) of a system can be classified as [11]:

• Research, development, test and evaluation (RDT&E) cost.

• Investment cost, which is related to production, procurement, manufacture, and infrastructure
maintenance activities among others.
• Operating and support (O&S) cost, which also includes the maintenance cost.
• Disposal cost.

The outcome of the cost assessment and program evaluation of the Department of Defense of the
United States of America [11] has shown that the operating and support cost has the biggest share at
the life cycle cost (nearly 60%). This indicates that the maintenance of a system not only influences its
operational capabilities, but it also determines significantly its life cycle cost.
This is why nowadays an immense pressure is applied to the aerospace industry for developing
systems with very high maintainability levels throughout their operational life. The maintainability
indicates first and foremost how easy it is for a system to be maintained, and secondly how costly it is.
The desired maintainability levels shall be part of the specifications of the system from
the development and production stages, aiming at reducing the cost and the complexity of the
maintenance procedures which are going to be implemented during the system’s operational life.
The achieved maintainability levels can be positively influenced by the following characteristics and
design philosophies:

• Modular design architecture, which facilitates the removal and installation process for the
subsystems and components of the system, which can then be forwarded to the respective
repair shops, thus eliminating the need for ‘on-board’ repair work.
• Interoperability of subsystems and components with the use of standard interface protocols,
which facilitates the prompt repair or upgrade of the system simply by installing a new and/or
upgraded subsystem or component.
• Prognostics, which enables the monitoring, tracking and recording of the operational data,
a feature which helps the user to identify operational limit exceedances and potential failures,
while suggesting preventive actions.
• ‘Fail-safe’ design, which isolates the subsystems and components in case of a system failure,
protecting them from further failures and malfunctions.
• Accessibility, especially for subsystems and components that need to be inspected in
frequent intervals.
• Commonality with other systems.
• Standardization of subsystems, components and support tools and equipment.
• Opportunistic maintenance and maintenance-free operating periods.
• Commercial off-the-shelf (COTS) support.

Various exogenous factors can also play an important role to the maintainability of a system,
such as:

• Low probability of diminishing manufacturing sources (DMS), material shortage, intellectual

property rights and monopolies.
• Use of technical orders and digital training of the maintenance personnel.
Aerospace 2018, 5, 68 3 of 20

• Centralized and automated analysis and reporting of the operational and support data,
using appropriate key performance indicators (KPIs).
• Network-centric management of the supply chain.
• Appropriate packaging, handling, storage and transportation.
• Spares optimization as well as personnel allocation optimization.
• Follow-on support programs.

1.1. Basic Terminology

The maintenance practice has evolved throughout the ages and various philosophies and
approaches have been introduced mainly due to the increasing complexity and technological
enhancement of the aircraft systems. The most fundamental philosophies/approaches are
the following:
• Run-to-Failure or Breakdown Maintenance. It is implemented on a nonscheduled basis,
following the failure of a system/subsystem/component. Its objective is the identification,
isolation and rectification of a failure to return the system/subsystem/component within its
established operating limits [12].
• Preventive Maintenance. Its objective is to reduce the probability of a nonscheduled maintenance,
which typically incurs high costs and considerably lengthy times to return the system to service.
Preventive maintenance is implemented through a variety of tools, such as non-destructive
inspections (NDI) and planned component replacement (PCR) [12].
• Opportunistic Maintenance. It is a combination of breakdown and preventive maintenance
philosophies. Its objective is to reduce the maintenance cost, by taking advantage of any failure
and its subsequent downtime, so as to intervene and implement preventive maintenance of
subsystems/components which have not failed yet, aiming to reduce the probability of future
failure [13].
• Upgrade or Modification. It aims to upgrade the system to enhance its performance and
maintainability. It might be required as a solution to a design or manufacturing problem.
• Predictive or Condition Based Maintenance (CBM). It is based on continuous condition and
operational data monitoring of a system with an objective to predict its future failure [14].
• Reliability Centered Maintenance (RCM). It is a structured process that aims to optimize the
management of the failures of a system [12]. Its objective is to sustain the operation of the system
within the desired performance levels, to manage the consequences of the failures and to define
the optimum and applicable maintenance policy, by taking into account existing constraints with
regard to resources, environmental, health and safety legislation [15].
• Risk Based Maintenance (RBM). It focuses on the management of the risk of active and potential
damage mechanisms and its effects to the health, safety and the environment. Severity and
probability are assessed for each identified risk, with an objective to define the maintenance
schedule which minimizes the overall risk [16].
• Design-Out Maintenance. Its aim is to detect possible defects during the design phase of a
system, thus avoiding future system failures. It focuses on possible critical failures which should
incur costs which are higher than certain affordable levels.

1.2. Reliability
Reliability is an expression of the ability of a system/subsystem/component to operate according
to its specification and within the established operating limits, without being subjected to nonscheduled
inspection. In other words, it is the probability to operate without failures within a defined period [17].
In analytical form, the reliability or survival function R(t) expresses the probability that a system is
operating beyond time t:
R(t) = Prob( T > t) = 1 − F (t), (1)
Aerospace 2018, 5, 68 4 of 20

where F (t) is the cumulative distribution function (CDF) that describes the intervals between
successive failures?
The mean time to/between failure (MTTF or MTBF) serves as an indicator for the reliability of a
system/subsystem or component. High MTTF values indicate high reliability levels. If a system begins
to operate at time t0 and fails n times at temporal points t1 , t2 , . . . , tn , where Times To Failure (TTF) are
defined as the intervals TTF1 = t1 − t0 , TTF2 = t2 − t1 , . . . , TTFn = tn − tn−1 , then MTTF is given by:
n
1
MTTF =
n ∑ TTFi . (2)
i =1

MTTF derives from the reliability function, as follows:

+∞
Z
MTTF = R(t)dt. (3)
0

Reliability is an intrinsic qualitative characteristic of a system or component, forged during

the design, development, test, and production phases. However, reliability can be affected from
extrinsic factors, such as environment, operations, maintenance and support, etc. It is therefore
important that these factors correspond to the technical and operational specifications of the system
and its components.

1.3. An Overview of the ‘Component Importance Measures’

Importance measures are objective criteria that classify the significance of the components within
the structure of a system, depending on the purpose of the analysis. These measures may serve as
resource allocation factors for scheduled maintenance and can help to estimate the spare parts required
for both scheduled and unscheduled component replacements. Not all existing importance measures
are utilized within the context of the analysis of this study. However, because the selection of the
importance measure is subjective and might serve different purpose of analysis, it is believed that an
overview of existing importance measures is beneficial for the reader and may trigger ideas on how
to expand the applicability of this study. The overview is based on existing literature on ideas and
developments in importance measures for reliability and risk analysis [18–24].
In order to describe the importance measures that follow, the notations of Table 1 are used.

Table 1. Notations for the basic functions.

Notation Description
R(t) System reliability function
Q(t) System unreliability function
ri ( t ) Component i reliability function
qi ( t ) Component i unreliability function
Ri,1 (t) System reliability function, whereas ri (t) = 1 (perfect component i)
Ri,0 (t) System reliability function, whereas ri (t) = 0 (failed component i)
Qi,1 (t) System unreliability function, whereas ri (t) = 1 (perfect component i)
Qi,0 (t) System unreliability function, whereas ri (t) = 0 (failed component i)

1.3.1. Importance Measures Based Solely on System Structure

These importance measures depend solely on the system structure and are independent of time.
Structural Importance (Str) or Birnbaum’s Structural Importance
Structural importance (also known as Birnbaum’s structural importance) for component i is the
fraction of the system states in which component i is working, where a failure of component i will
result in a failure of the system.
Aerospace 2018, 5, 68 5 of 20

Barlow-Proschan Importance (BP)

The Barlow-Proschan importance for component i is the probability that the failure of component
i coincides with the failure of the system.

1.3.2. Importance Measures Based on System Structure and Component’s Reliability

These importance measures depend on the system structure and time, because they take into
consideration both the system structure and the components reliability functions.
Birnbaum Importance (B) or Reliability Importance
Birnbaum importance (also known as reliability importance) at time t for component i is the
improvement in the system reliability that would be gained by replacing a failed component i with a
perfect component i:
Bi (t) = Ri,1 (t) − Ri,0 (t). (4)

The Birnbaum importance measure does not take into consideration the reliability for component
i at time t.
Improvement Importance (Imp) or Improvement Potential
Improvement importance (also known as improvement potential) at time t for component i is the
increase of the system reliability if component i is replaced with a perfect component:

Impi (t) = Ri,1 (t) − R(t). (5)

For the purpose of this study, whenever a planned component replacement (PCR) is required,
we use the improvement importance measure after a ‘cost adjustment’. As such, the decision which is
being made upon scheduled maintenance is to replace the component for which the highest value of
‘benefit/cost’ is achieved. More specifically, if the value of the improvement importance measure for
component i is Impi (t), and the cost of scheduled replacement for component i is CSi , we define the
cost-adjusted improvement importance measure as: CAImpi (t) = Impi (t)/CSi .
Risk Achievement Importance or Risk Achievement Worth (RAW)
Risk achievement importance (also known as risk achievement worth) at time t for component i
expresses the relative increase of the system unreliability, if component i failed:

Qi,0 (t)
RAWi (t) = . (6)
Q(t)

Risk Reduction Importance or Risk Reduction Worth (RRW)

Risk reduction importance (also known as risk reduction worth) at time t for component i
expresses the relative decrease of the system unreliability, if component i were perfect:

Qi,1 (t)
RRWi (t) = . (7)
Q(t)

Failure-Based Criticality Importance (FBC)

Failure-based criticality importance (also known as criticality failure importance or failure
criticality or criticality importance factor) at time t for component i is the probability that component i
has caused system failure, when the system has failed at time t:

Bi (t)qi (t)
FBCi (t) = . (8)
Q(t)
Aerospace 2018, 5, 68 6 of 20

Proposition 1. The improvement importance and failure-based criticality importance measures obtain analogous
weights. Specifically, for any given component, the improvement importance measure equals the system
unreliability times the failure-based criticality importance measure.

Proof of Proposition 1.

Bi (t)qi (t) = [ Ri,1 (t) − Ri,0 (t)]qi (t) = Ri,1 (t)qi (t) − Ri,0 (t)qi (t)
(eq.(4))
= Ri,1 (t)[1 − ri (t)] − Ri,0 (t)qi (t)
= Ri,1 (t) − [ Ri,1 (t)ri (t) + Ri,0 (t)qi (t)]
= Ri,1 (t) − R(t) = Impi (t).
(eq.(5))

Hence, Equation (8) becomes:

Impi (t) = Q(t) FBCi (t). (9)

Success-Based Criticality Importance (SBC)

Success-based criticality importance (also known as criticality success importance or criticality
importance factor) at time t for component i is the probability that component i contributes to system
success, given that the system is operating:

Bi (t)ri (t)
SBCi (t) = . (10)
R(t)

Fussell-Vesely Importance (FV)

The Fussell-Vesely importance at time t for component i is given by:

pi ( t )
FVi (t) = , (11)
Q(t)

where pi (t) is the probability that at least one minimal cut set containing component i has failed at
time t . A minimal cut set is a minimal set of components which, if failed, causes the system to fail.
Partial Derivative Importance (PD)
The partial derivative importance at time t for component i expresses the sensitivity of the system
reliability on a marginal change of the component i reliability [25]:

∂R(t)
PDi (t) = . (12)
∂ri (t)

An advantage of this importance measure is that it may consider marginal changes in a

component’s reliability, whereas the previous importance measures consider either a failed or a
perfect component.

2. Materials and Methods

The developed optimization software tool is demonstrated as applied to the maintenance planning
of a fuel pump of an aircraft jet engine. The fuel pump consists of two subsystems (subsystem 1,
subsystem 2) which operate in parallel, hence for the pump to operate at least one subsystem
must operate.
Aerospace 2018, 5, 68 7 of 20

Subsystem 1 consists of an electrical valve (a) and a mechanical fuel regulator (b). For subsystem
1 to operate, both (a) and (b) must operate. Subsystem 2 consists of an electronic fuel regulator (c),
an electrical compressor (d) and a hydraulic valve (e). For Subsystem 2 to operate, all (c), (d), and (e)
must operate. In case of a failure of the electronic fuel regulator (c), Subsystem 2 can use the mechanical
fuel regulator (b), which belongs to Subsystem 1.

2.1. Basic Assumptions

The fuel pump is represented by the schematic of Figure 1.
Aerospace 2018, 5, x FOR PEER REVIEW 7 of 20

Figure 1. Schematic of the fuel pump.

Figure 1. Schematic of the fuel pump.

All
All its
its components
components operate
operate independently
independently and
and aa failure
failure of
of one
one or
or more
more components
components is is not
not going
going
to
to affect
affect the operation of
the operation the rest.
of the rest. Furthermore,
Furthermore, the
the pump
pump can can operate
operate even
even when
when one
one or
or more
more ofof its
its
components
components have failed. The first assumption here is that a component failure cannot be detected if
have failed. The first assumption here is that a component failure cannot be detected if
the pump keeps operating, as such, corrective maintenance will be implemented only in case
the pump keeps operating, as such, corrective maintenance will be implemented only in case that the that the
pump stops
pump stops operating.
operating.

2.2.
2.2. System
System Structure
Structure Function
Function
The
The fuel
fuel pump
pump structure function is:
structure function is:
( a, b, c, d, e) = a· b + b· d· e - a· b· d· e + c· d· e - b· c· d· e. (13)
ϕ ( xa , xb , xc , xd , xe ) = xa · xb + xb · xd · xe − xa · xb · xd · xe + xc · xd · xe − xb · xc · xd · xe . (13)

2.3. Minimal Cut Sets

The fuel pump minimal cut sets are: {{a, d}{a,
a, d}, e},e{b,
, {a, {b,{b,c}d},
}, c}, , {{b,
b, e}
d}, {b, e}

2.4. Components
2.4. Components Reliability
Reliability Functions
Functions
We accept
We accept that
that every
every component , withi ∈ ∈{a,
componenti, with {a,b,b,c,c, d, e},isisgoing
d, e}, going through
through its
its useful
useful life,
life, hence
hence
its failure
its failure rate
rate is
is constant
constant and
and the
the respective
respective failure
failure intervals
intervals follow
follow thethe exponential
exponential distribution
distribution with
with
parameter
parameter i = 1/
λ = 1/MTTF . In other words, each component
i. In other words, each component i fails randomly and its failures follow
fails randomly and its failures follow the
Poisson
the distribution
Poisson withwith
distribution parameter λi . The. reliability
parameter function
The reliability ri (t) of component
function i is given is
( ) of component by:given
by:
r i ( t ) = e − λi t (14)
( )= (14)
The component reliability information is shown in Table 2.
The component reliability information is shown in Table 2.

Table 2. Component reliability and cost information.

Component Component Cost of Scheduled Cost of Unscheduled

Component
Failure Rate Reliability Replacement of a Single Replacement of a Single
MTTF (Hours)
(Constant) Function Component (Euros) Component (Euros)
.
a = 500 a = 0.002 a( ) = a = 2000 a = 4000
Aerospace 2018, 5, 68 8 of 20

Table 2. Component reliability and cost information.

Component Component Cost of Scheduled Cost of Unscheduled

Component
Failure Rate Reliability Replacement of a Single Replacement of a Single
MTTF (Hours)
(Constant) Function Component (Euros) Component (Euros)
MTTFa = 500 λa = 0.002 ra (t) = e−0.002t CSa = 2000 CUa = 4000
MTTFb = 1000 λb = 0.001 rb (t) = e−0.001t CSb = 3000 CUb = 6000
MTTFc = 667 λc = 0.0015 rc (t) = e−0.0015t CSc = 4000 CUc = 8000
MTTFd = 400 λd = 0.0025 rd (t) = e−0.0025t CSd = 6000 CUd = 12, 000
MTTFe = 2000 λe = 0.0005 re (t) = e−0.0005t CSe = 7000 CUe = 14, 000

Aerospace 2018, 5, x FOR PEER REVIEW 8 of 20

2.5. System Reliability Function
2.5. System Reliability Function
The reliability function R(t) of the fuel pump for t ≤ 1.000 is shown at the Figure 2 below.
The reliability function ( ) of the fuel pump for ≤ 1.000 is shown at the Figure 2 below.

Aerospace 2018, 5, x FOR PEER REVIEW 8 of 20

2.5. System Reliability Function

The reliability function ( ) of the fuel pump for ≤ 1.000 is shown at the Figure 2 below.

Figure 2. Reliability function ( ) of the fuel pump for ≤ 1.000.

Figure 2. Reliability function R(t) of the fuel pump for t ≤ 1.000.
At = 0 all components are assumed to be new (‘perfect’). Following Equations (13) and (14)
At t =
the0fuel
all pump
components
reliabilityare assumed
function is giventoby:be new (‘perfect’). Following Equations (13) and (14) the
fuel pump reliability ( ) of the fuel pump for ≤ 1.000.
( ) = a ( )· function is given by:
Figure 2. Reliability function
b ( ) + b ( )· d ( )· e ( ) - a ( )· b ( )· d ( )· e ( ) + c ( )· d ( )· e ( ) - b ( )· c ( )· d ( )· e ( ) (15)
At = 0 all components are assumed to be new (‘perfect’). Following Equations (13) and (14)
R(t) = ra2.6. b (the + rpump
(t)·rCalculation
t) fuel (tthe
bof t) −Importance
rdComponents’
(t)·re (function
)·reliability rais(tgiven )·rd (t)·re (t) + rc (t)·rd (t)·re (t) − rb (t)·rc (t)·rd (t)·re (t) (15)
)·rb (by:
tMeasures

As an( example,
) = a ( )· b ( ) + b ( )· d ( )· e ( ) - a ( )· b ( )· d ( )· e ( ) + c ( )· d ( )· e ( ) - b ( )· c ( )· d ( )· e ( )
Figure 3 shows the components’ importance measures percentages at(15)= 40.
2.6. Calculation of the Components’ Importance Measures
2.6. Calculation of the Components’ Importance Measures
As an example, Figure 3 shows the components’ importance measures percentages at t = 40.
As an example, Figure 3 shows the components’ importance measures percentages at = 40.

Figure 3. Cont.
Aerospace 2018, 5, 68 9 of 20

Aerospace 2018, 5, x FOR PEER REVIEW 9 of 20

Figure 3. Components’ importance measures percentages at = 40.

Figure 3. Components’ importance measures percentages at t = 40.
Notably, the percentages of the failure-based criticality and the improvement importance
measures are equal (see Proposition 1).
Notably, the percentages of the failure-based criticality and the improvement importance measures
are equal (see Proposition
2.7. Estimation of1).
the Optimum Maintenance Plan
Importance measures can be used as objective criteria to make the optimum decision regarding
2.7. Estimation of
thethe Optimum
maintenance Maintenance
planning PlanHowever, since the importance measures do not consider
of the fuel pump.
the associated cost of components, we introduce a cost adjustment. Thus, we expand the applicability
Importance of measures
the importancecan
measures as a ‘benefit
be used over cost’criteria
as objective criterion for
to decision
make themaking, while trying
optimum to
decision regarding
determine the optimum maintenance scenario.
the maintenance planning of the fuel pump. However, since the importance measures do not consider
2.7.1. of
the associated cost Inputs
components, we introduce a cost adjustment. Thus, we expand the applicability
of the importanceThe measures as a ‘benefit
proposed software over
tool uses the cost’
following criterion for decision making, while trying to
inputs:
•
determine the optimum maintenance
The structure of the system.scenario.
•The life cycle of the system, more specifically the timeframe of the maintenance plan of the
system.
2.7.1. Inputs • The reliability distribution of each component.
• The importance measure which is going to be used as an objective criterion to determine the
The proposedcomponent
softwarewhich toolwill
uses the following
be replaced inputs:
on a preventive basis during the implementation of the
scheduled maintenance of the system. The specified importance measure will be cost-adjusted
• The structure according
of the system.
to the cost inputs that follow.
• The cost of the scheduled preventive replacement of each component with a brand new one. In
• The life cycle of the system, more specifically the timeframe of the maintenance plan of the system.
other words, the cost of scheduled preventive maintenance for each component after which the
• The reliabilitycumulative
distribution
time of of each of
operation component.
the component is zero.
• The cost of the nonscheduled replacement of each component with a brand new one. In other
• The importance measure which is going to be used as an objective criterion to determine the
words, the cost of nonscheduled maintenance for each component after which the cumulative
component which will beofreplaced
time of operation the component onis zero.
a preventive basis during the implementation of the
scheduled•maintenance
The cost of scheduled preventive replacement of all the components, at once, with brand new
of the system. The specified importance measure will be cost-adjusted
ones. In other words, the cost of scheduled preventive maintenance for all components
according to the cost inputs
simultaneously, afterthat
whichfollow.
the cumulative time of operation of the components is zero.
• The confidence level for the fulfilment of the nonscheduled maintenance requirements, in case
• The cost of the scheduled preventive replacement of each component with a brand new one.
of system failures.
In other words, the cost of scheduled preventive maintenance for each component after which the
cumulative2.7.2.
timeProcessing
of operation of the component is zero.
The software tool will use an algorithm to assess all the potential scheduled maintenance
• The cost ofscenarios;
the nonscheduled replacement of each component with a brand new one. In other
for each scenario it will calculate the value of the criterion ‘lowest total maintenance
words, thecost/average
cost of nonscheduled
reliability outcome’.maintenance
The lowest valuefor each
of the component
criterion after
will determine the which
optimumthe cumulative
scheduled of
time of operation maintenance scenario. Specifically,
the component is zero.the structure of the algorithm is:
• The cost of scheduled preventive replacement of all the components, at once, with brand new ones.
In other words, the cost of scheduled preventive maintenance for all components simultaneously,
after which the cumulative time of operation of the components is zero.
• The confidence level for the fulfilment of the nonscheduled maintenance requirements, in case of
system failures.

2.7.2. Processing
The software tool will use an algorithm to assess all the potential scheduled maintenance
scenarios; for each scenario it will calculate the value of the criterion ‘lowest total maintenance
cost/average reliability outcome’. The lowest value of the criterion will determine the optimum
scheduled maintenance scenario. Specifically, the structure of the algorithm is:
Aerospace 2018, 5, 68 10 of 20

Estimate system structure function

For N = 0 to 98:
Set R_lower limit = 0.99 − 0.01N
While t ≤ T:
Estimate tj for next scheduled maintenance, solving R(t) = R_lower limit
Estimate components cost-adjusted improvement importance measures at tj
Replace component with highest cost-adjusted improvement importance measure
Estimate R(t) for t > tj , given the component replacement at tj
Estimate cost of scheduled component replacements
Estimate R(t), λ(t), λ(t), MTTF, and system failures at desired confidence level
Estimate components improvement importance measure integrals, from t = 0 to T
Allocate the system failures to each component according to the above integrals
Estimate the cost of unscheduled component replacements
Estimate the total cost of replacements (scheduled + unscheduled)
Estimate the value of the optimization criterion: total cost of replacements/R(t)
Display the maintenance scenario with the lowest value of the optimization criterion

2.7.3. Outputs
The outputs of the tool are the following:

• The diagram of the procedure for which the lowest value of the criterion ‘cost/benefit’ is achieved.
• The reliability function diagram of the system for the optimum scheduled maintenance scenario.
• The average reliability of the system.
• The lowest value of the reliability of the system, at which the system has to be grounded for
scheduled maintenance.
• The MTTF of the system.
• The replacement schedule for the system’s components.
• The required number of spare parts for each component, for both scheduled and nonscheduled
maintenance (at the determined confidence level).
• The cost analysis for both scheduled and nonscheduled replacement of the components.

3. Exemplified Example/Results
The following is an exemplified example of an optimization process that uses the proposed
software tool.

3.1. Task
The optimum maintenance plan for the fuel pump needs to be established for its first 3000 h of
operation with the following constraints: Only one component will be replaced by a new one (or its
cumulative time of operation will be considered as zero following an inspection/rectification) during
the implementation of the scheduled maintenance of the system. The criterion that will determine the
component to be replaced is the ‘cost-adjusted improvement importance measure’, which corresponds
to the cost of the scheduled replacement of each individual component, as illustrated in Table 2.

3.2. Components Replacement Cost

The replacement cost of the components is estimated for three different cases:

• Scheduled replacement of a unique component (preventive maintenance)

• Nonscheduled replacement of a unique component (corrective maintenance)
• Scheduled replacement of all the components simultaneously (preventive maintenance)
Aerospace 2018, 5, 68 11 of 20

The estimation of the scheduled replacement cost for each component takes into account the
purchase price
Aerospace 2018,of
5, xthe
FORcomponent
PEER REVIEW and of all the consumables required for its replacement, 11as
of well
20 as
the total cost of the required maintenance work, such as depreciation of special tools and equipment,
energy cost,Theman-hours
estimation cost,of thethescheduled replacementand
system down-time costits
foreffect
each on
component takes into
the operational account the
availability, as well
as anypurchase price of thecost
other associated component and of all theprocedures
(safe maintenance consumablescost, required for its replacement,
accessibility as well as
cost, operational checks
the total cost of the required maintenance
cost, transportation cost for involved staff and materiel). work, such as depreciation of special tools and equipment,
energy cost, man-hours cost, the system down-time and its effect on the operational availability, as
The cost of the nonscheduled replacement of each component in case of a pump failure should
well as any other associated cost (safe maintenance procedures cost, accessibility cost, operational
be considered higher than the respective scheduled replacement cost. Further to the cost categories
checks cost, transportation cost for involved staff and materiel).
which have Thebeen
cost ofmentioned previously,
the nonscheduled the riskofofeach
replacement unintended
componentdamagein case ofand/or
a pumpfailures of other jet
failure should
enginebe considered higher than the respective scheduled replacement cost. Further to the cost categories that
subsystems due to the failure of the pump, should also be considered. It is also possible
the fuel pump
which havefails
beeninmentioned
a locationpreviously,
at a distance the from
risk ofthe maintenance
unintended damage base station,
and/or a situation
failures of otherthat
jet will
potentially
engine incur higher
subsystems duecosts and
to the disruption
failure to the should
of the pump, aircraftalso
fleet
beoperations,
considered. It due to the
is also nonscheduled
possible that
the fuelofpump
grounding the jetfails in a location
engine at a distance from
and, consequently, the maintenance base station, a situation that will
the aircraft.
The information regarding the replacement costaircraft
potentially incur higher costs and disruption to the fleet operations,
(scheduled due to theof
and nonscheduled) nonscheduled
each component
grounding of the jet engine and,
of the fuel pump is presented at the Table 2. consequently, the aircraft.
The information regarding the replacement cost (scheduled and nonscheduled) of each
The estimation of the replacement cost, at once, of more than one components, should consider
component of the fuel pump is presented at the Table 2.
the fact that the total replacement cost should be less than the sum of the scheduled replacement
The estimation of the replacement cost, at once, of more than one components, should consider
cost for eachthat
the fact component (Table 2). cost
the total replacement Thisshould
is due beto the
less economies
than the sum ofofthe scale, whichreplacement
scheduled materialize costdue to
maintenance work which
for each component is common
(Table 2). This for
is due some or all
to the the components.
economies For example,
of scale, which materializewhen due to all the
components
maintenanceare replaced
work which at once, safe maintenance
is common for some or all procedures and theFor
the components. operational
example, when checkall of the
the fuel
pump take placeare
components only once. at
replaced Inonce,
addition, the total downtime
safe maintenance proceduresofand thethe
pump is less check
operational and the cumulative
of the fuel
pump take place only once. In addition, the total downtime of
effect to the operational availability of the jet engines/aircraft fleet is less severe. the pump is less and the cumulative
effect to the operational availability of the jet engines/aircraft fleet is less severe.
3.3. Optimization Criterion
3.3. Optimization Criterion
As optimum maintenance plan is considered the one with the lowest possible total maintenance
As optimum maintenance plan is considered the one with the lowest possible total maintenance
cost (for both scheduled and nonscheduled maintenance) for the average reliability, which is achieved
cost (for both scheduled and nonscheduled maintenance) for the average reliability, which is
during the first 3000 h of operation of the fuel pump. In other words, the optimization criterion is the
achieved during the first 3000 h of operation of the fuel pump. In other words, the optimization
lowest possible
criterion value
is the of ‘cost
lowest over value
possible benefit’. The optimization
of ‘cost of the
over benefit’. The preventiveofmaintenance
optimization plan is
the preventive
shown in Figure 4.
maintenance plan is shown in Figure 4.

Figure 4. Optimization of the preventive maintenance plan for the fuel pump for the first 3000 h of its
Figure 4. Optimization of the preventive maintenance plan for the fuel pump for the first 3000 h of
operation.
its operation.
3.4. Optimum Number of Spare Components for Scheduled Maintenance
3.4. Optimum Number of Spare Components for Scheduled Maintenance
Every time that the reliability of the fuel pump approaches the lowest acceptable limit, the pump
Every time for
is grounded thatscheduled
the reliability of the fuel
maintenance. pump
In that case,approaches
a preventivethe lowest acceptable
replacement limit,forthe
will take place thepump
is grounded for scheduled maintenance. In that case, a preventive replacement will take place
component for which it is estimated to achieve the highest possible improvement of the reliability of for the
component for which it is estimated to achieve the highest possible improvement of the reliability of
Aerospace 2018, 5, 68 12 of 20

the pump for the associated cost of the improvement. The importance measure of the improvement is
actually ‘cost adjusted’ and as such the decision which has been made is to replace the component for
which the lowest value of ‘cost/benefit’ is achieved.
More specifically, if at the time tk of the scheduled downtime, the value of the improvement
importance measure for the component i is Impi (tk ), with i ∈ {a, b, c, d, e}, and the replacement
cost of each component is CSi , then, as shown in Table 2, the component replaced is not the one with
the highest value of Impi (tk ), but the one with the highest value of Impi (tk )/CSi . This is how an
estimation can be made for the complete list of the components which are going to be replaced during
the first 3000 h of operation of the pump. Having assessed that, an estimation can be made as well for
the number of spare components which will be needed for the scheduled maintenance of the pump.

3.5. Optimum Number of Spare Components for Non-Scheduled Maintenance

The mean time between two successive occurrences of nonscheduled maintenance for the first
3000 h of operation of the fuel pump can be estimated by Equation (3) as MTTF = 2040.3 h. The number
of failures is: µ = 8.26.
The number of failures, for a certain confidence level, and the respective number of spare
components needed for the nonscheduled maintenance for the first 3000 h of operation can be estimated
by using Poisson distribution with parameter µ = 8.26. As per the results presented at the Table 3 (for a
95% confidence level), a number of 13 nonscheduled component replacements will be needed for a
single fuel pump.

Table 3. Spare components required to fulfil the nonscheduled maintenance requirements (95%
confidence level).

Part Type Items Cost

A 5.02131 20,085.20
B 4.36118 26,167.10
C 0.462472 3699.78
D 2.6525 31,830.00
E 0.502543 7035.60
Fails
(95.7391%) 13 88,817.70

Having estimated this, the quantity of each specific type of spare part for the corrective
maintenance can now be estimated. This time, the failure-based criticality importance measure
is considered to better express the contribution of each type of spare component to the occurrence of a
fuel pump failure. This importance measure represents the probability of a fuel pump failure due to a
component failure. Equivalently, according to the Proposition 1, the (more conveniently estimated)
improvement importance measure can be used as well.
For the occasion of unscheduled maintenance requirements, cost adjustment on the improvement
importance measure is not required. Indeed, cost does not affect the probability of fuel pump failure
due to a specific component failure. However, since the improvement importance measure is now
applied through the whole 3000 operational hours timeframe, and not to a specific instance in time (such
as in the case of scheduled maintenance temporal points), the integral of the improvement importance
measure for every component is calculated, from 0 to 3000 h of the pump’s operation. The integration
of the improvement importance measures serves as the allocation base for the previously estimated
total 13 spare components (unscheduled maintenance), in order to determine the quantity for each
component type.
After the allocation of the 13 system failures to each spare part, it is strongly recommended
that no rounding should be performed for a single fuel pump’s spares (as seen in Table 3). Such an
analysis normally aims at making provisions for a relatively large population of fuel pumps, hence it
Aerospace 2018, 5, 68 13 of 20

Aerospace 2018, 5, x FOR PEER REVIEW 13 of 20

is suggested that any rounding should be performed after the calculation of the total number of each
suggested that any rounding should be performed after the calculation of the total number of each
spare part type.
spare part type.
TableTable
3 shows that a that
3 shows confidence level oflevel
a confidence 95% to of fulfill
95% to thefulfill
nonscheduled maintenance
the nonscheduled requirements
maintenance
of therequirements
pump for the first 3000 h of operation, is going to require 13 spare components.
of the pump for the first 3000 h of operation, is going to require 13 spare components. Using the
integral of the improvement importance measure as an allocation basis for the
Using the integral of the improvement importance measure as an allocation basis for the 13 spare13 spare components,
it is components,
concluded that five nonscheduled
it is concluded replacements
that five nonscheduled of component
replacements (a) are(a)going
of component to take
are going place),
to take
place), 4.4 nonscheduled
4.4 nonscheduled replacements replacements of the component
of the component (b), 0.5 nonscheduled
(b), 0.5 nonscheduled replacements
replacements of the
of the component
component
(c), 2.7 (c), 2.7
nonscheduled nonscheduledof replacements
replacements the component of (d)
the and
component (d) and 0.5replacements
0.5 nonscheduled nonscheduledof the
component (e). The total cost of all the above nonscheduled replacements is 88,817.70 e, and thisisis the
replacements of the component (e). The total cost of all the above nonscheduled replacements
88,817.70 €, and this is the total cost of the nonscheduled maintenance of the pump for the first 3000
total cost of the nonscheduled maintenance of the pump for the first 3000 h of operation.
h of operation.
3.6. Results
3.6. Results
The optimum maintenance
The optimum plan
maintenance planisisshown
shown in
in Figure
Figure 55and
andininTables
Tables 3–5.
3–5.

Figure 5. The reliability function under the optimum maintenance plan for the fuel pump for its first
Figure 5. The reliability function under the optimum maintenance plan for the fuel pump for its first
3000 h of operation.
3000 h of operation.
The reliability function of the first 3000 h of operation of the fuel pump is presented in Figure 5.
The reliability
The local maxima function of the
of the curve first 3000
represent h of operation
the improvement of the
of the fuel pump
reliability of the is presented
fuel in Figure 5.
pump following
The local maxima of the curve represent the improvement of the reliability of the
the scheduled down time during which a preventive replacement of a component has taken place. fuel pump following
It
the scheduled
is reminded that the decision to replace a component has been made based to the lowest value of the It is
down time during which a preventive replacement of a component has taken place.
cost/benefit
reminded optimization
that the decision criterion,
to replace using the cost-adjusted
a component improvement
has been made based importance measure.
to the lowest value of the
Theoptimization
cost/benefit horizontal redcriterion,
line showsusing
the lowest acceptable reliability
the cost-adjusted limit for importance
improvement the fuel pump, which is
measure.
53%.horizontal
The Whenever red the line
limitshows
is reached, the pump
the lowest is grounded
acceptable reliability forlimit
preventive maintenance
for the fuel or
pump, which is
replacement of a specific component. The horizontal dotted black line shows the mean reliability of
53%. Whenever the limit is reached, the pump is grounded for preventive maintenance or replacement
68%, which is achieved during the 3000 h of operation.
of a specific component. The horizontal dotted black line shows the mean reliability of 68%, which is
achieved during the 3000 h of operation.
Aerospace 2018, 5, 68 14 of 20

Table 4. Components replacement plan for the scheduled maintenance of the fuel pump for its first
3000 h of operation.

Part Type Replace at

d 353.653
b 500.406
a 607.353
a 829.207
b 951.600
a 1104.010
b 1273.250
a 1381.720
a 1560.730
b 1678.180
a 1813.860
a 1981.150
b 2092.210
a 2230.450
a 2396.240
b 2506.640
a 2644.860
a 2810.480
b 2920.860

Table 5. Total replacement cost breakdown for the scheduled maintenance of the fuel pump for its first
3000 h of operation.

Part Type Items Cost

a 11 22,000
b 7 21,000
c 0 0
d 1 6000
e 0 0
Total sched. 19 49,000

The results shown in Tables 4 and 5 indicate that fulfilling the scheduled maintenance
requirements of the first 3000 h of operation, is going to require 11 stand-alone replacements of the
component (a), seven stand-alone replacements of the component (b) and one stand-alone replacement
of the component (d). The total cost of all the above replacements is 49,000 e and this is the total
cost of the scheduled maintenance of the pump for the first 3000 h of operation. Notably, no PCRs
for components (c) and (e) are required, because any potential replacement of these components
would have brought relatively poor improvement in the fuel pump’s reliability, given the associated
replacement cost. On the other hand, Table 5 shows that the components (a) and (b) PCRs usually
have the highest impact on the fuel pump’s reliability improvement, compared to the associated
replacement cost.
The total maintenance cost (scheduled and nonscheduled) can now be calculated for the first 3000
operation hours of the pump:
49, 000 + 88, 817.7 = 137, 817.7 e and the mean reliability for the same timeframe is 68%. As such,
the optimization criterion of (cost/benefit) takes its optimum value of 137, 817.7/0.68 = 202, 673.
Figure 6 shows the failure rate λ(t) and the mean failure rate λ(t) of the fuel pump.
Aerospace 2018, 5, 68 15 of 20
Aerospace 2018, 5, x FOR PEER REVIEW 15 of 20

Aerospace 2018, 5, x FOR PEER REVIEW 15 of 20

Figure 6. Failure rate λ( ) (blue curve) and the mean failure rate λ( ) (horizontal dotted line) of the
Figure 6. Failure rate λ(t) (blue curve) and the mean failure rate λ(t) (horizontal dotted line) of the
fuel pump for its first 3000 h of operation.
fuel pump for its first 3000 h of operation.
Figure 6. Failure rate λ( ) (blue curve) and the mean failure rate λ( ) (horizontal dotted line) of the
It is pump
fuel observed
for itsthat
firstafter the
3000 h of first 1500 h of operation, which seems as a ‘warm-up period’, the
operation.
It is observed
failure thatpump
rate of the afterconverges
the first 1500
to a hconstant
of operation,
value λwhich seems
≈ 0.003 (CFRasperiod).
a ‘warm-up period’, the failure
rate of theItpump converges
is observed to athe
that after constant value
first 1500 ≈ 0.003 (CFR
h ofλoperation, whichperiod).
seems as a ‘warm-up period’, the
3.7. An rate
failure Alternative Scenario:
of the pump All Components
converges value λSimultaneously
Are Replaced
to a constant ≈ 0.003 (CFR period).
3.7. An Alternative Scenario: All Components Are Replaced Simultaneously
At this scenario, all the components are replaced simultaneously at any time the pump is
3.7. An Alternative Scenario: All Components Are Replaced Simultaneously
At this scenario,
subjected all themaintenance.
to scheduled componentsStill,
arethereplaced
timingsimultaneously
of the scheduled at any time the
maintenance pump is subjected
is determined by
At this
lowest scenario,
reliability all
limit the components
which has been are
set forreplaced
the pump. simultaneously
After each timeat any
to scheduled maintenance. Still, the timing of the scheduled maintenance is determined by the to
the the time
pump is the pump
subjected is
lowest
subjected
scheduled to scheduled
maintenance, maintenance.
the pump is Still, the
considered timing
as of
brand the scheduled
new and, as maintenance
such, the time is determined
interval
reliability limit which has been set for the pump. After each time the pump is subjected to scheduled betweenby
the lowest
two
maintenance, reliability
subsequent
the limit
scheduled
pump which has been
maintenance
is considered set for the
asoccurrences
brand newpump.
and,After
is considered each time the pump
as constant.
as such, the time is subjected
interval betweento two
scheduled maintenance,
The optimization ofthe
thepump is considered
maintenance as this
plan for brand new and,scenario
alternative as such, the time interval
is shown between
at the Figure 7.
subsequent scheduled maintenance occurrences is considered as constant.
two subsequent scheduled maintenance occurrences is considered as constant.
The optimization of the maintenance plan for this alternative scenario is shown at the Figure 7.
The optimization of the maintenance plan for this alternative scenario is shown at the Figure 7.

Figure 7. Optimization of the preventive maintenance plan for the fuel pump for the first 3000 h of its
operation (alternative scenario).
Figure 7. Optimization of the preventive maintenance plan for the fuel pump for the first 3000 h of its
Figure 7. Optimization
Following this, theofdeveloped
operation (alternative
the preventive maintenance plan for the fuel pump for the first 3000 h of its
scenario). software tool investigates thoroughly the optimum maintenance
operation (alternative
plan and returns scenario).
to Figure 8 as well as Tables 6–8.
Following this, the developed software tool investigates thoroughly the optimum maintenance
Following this, the
plan and returns developed
to Figure software
8 as well tool
as Tables 6–8.investigates thoroughly the optimum maintenance
plan and returns to Figure 8 as well as Tables 6–8.
Aerospace 2018, 5, 68 16 of 20
Aerospace 2018, 5, x FOR PEER REVIEW 16 of 20

Figure 8. Fuel pump reliability for the first 3000 h of its operation (alternative scenario).
Figure 8. Fuel pump reliability for the first 3000 h of its operation (alternative scenario).
Table 6. Components replacement plan (alternative scenario).
Table 6. Components replacement plan (alternative scenario).
Part Types Replace at
a, b, c, d, e
Part Types 429.21 Replace at
a, b, c, d, e 858.42
a, b, c, a,
d,b,e c, d, e 1287.64 429.21
a, b, c, d, e 858.42
a, b, c, d, e 1716.85
a, b, c, d, e 1287.64
a, b, c, a,
d,b,e c, d, e 2146.06 1716.85
a, b,
a, b, c, d, e c, d, e 2575.27 2146.06
a, b,
Table 7. Total c, d, e
replacement 2575.27 scenario).
cost breakdown (alternative

Part Typecost Items

Table 7. Total replacement breakdownCost
(alternative scenario).
a 6
Part Type b 6
Items Cost
a c 66
b d 66
c e 66
d Total sched. 306 69,049.20
e 6
Total sched. 30 69,049.20

Table 8. Total replacement cost breakdown (alternative scenario, confidence level 95%).

Part Type Items Cost

a 3.200540 12,802.10
b 1.983480 11,900.90
c 0.618516 4948.13
d 3.661420 43,937.10
e 0.536043 7504.60
Fails. 10 81,092.80
Aerospace 2018, 5, 68 17 of 20

According to the assumption of Section 3.2, the cost of the scheduled replacement of more than
one components simultaneously, is going to be less than the sum of the costs of each stand-alone
scheduled replacement as shown in Table 2. More specifically, it is deducted that since the cost of the
preventive simultaneous replacement of all the components is less than the 52.31% of the sum of the
costs of each stand-alone preventive replacement, then the scenario of the preventive replacement of all
the components simultaneously is more efficient. In case the scheduled simultaneous replacement of
all the components maintains the cost at 52.31%, as compared with the original scenario, then the two
scenarios are almost equivalent with regards to the optimization criterion being used. Of note, the case
under examination serves as an example for the application of the software tool; no actual data were
used, including the system structure, components reliability functions, and component replacement
costs. The equivalence threshold is sensitive to structure, reliability, and cost inputs—hence the
above-estimated 52.31% threshold should not be generalized.
Table 6 results indicate that the optimum maintenance is achieved when the pump is grounded
every 429.21 h of operation and all its components are replaced at once, with reliability approaching
the lowest acceptable limit of 43%.
Table 7 shows that to fulfil the preventive maintenance requirements of the pump during its first
3000 h of operation, six components of each type need to be scheduled to be replaced. In that case,
the total replacement cost of the 30 involved components is 69, 049.2 e. For this alternative scenario,
this is the total cost of the preventive maintenance of the pump for the first 3000 h of operation.
Table 8 shows that, a confidence level of 95% to fulfil the nonscheduled maintenance requirements
for the first 3000 h of operation of the fuel pump, is going to require a nonscheduled replacement of 10
components. In more detail, the alternative scenario is going to require the nonscheduled replacement
of 3.2 (a) components, 1.98 (b) components, 0.62 (c) components, 3.66 (d) components and 0.54 (e)
components. The total cost of all the above nonscheduled replacements is 81,092.8 e, and this is the
total cost of the nonscheduled maintenance of the pump for the first 3000 h of operation.
Figure 9 shows the failure rate λ(t) and the mean failure rate λ(t). The failure rate takes its highest
value (0.00284) just before the scheduled grounding of the pump. Another observation is that between
two successive groundings, the pump is always at IFR period.
Aerospace 2018, 5, x FOR PEER REVIEW 18 of 20

Figure 9. Failure rate λ( ) (blue curve) and the mean failure rate λ( ) (horizontal dotted line) of the
Figure 9. Failure rate λ(t) (blue curve) and the mean failure rate λ(t) (horizontal dotted line) of the
pump for its first 3000 h of its operation (alternative scenario).
pump for its first 3000 h of its operation (alternative scenario).

Table 9. Summary of the two maintenance concepts.

Initial Scenario Alternative Scenario

Average reliability 0.68 0.74
Reliability lower limit 0.53 0.43
Cost of scheduled maintenance 49,000 69,049
Cost of unscheduled maintenance 88,817 81,093
Total maintenance cost 137,818 150,142
Aerospace 2018, 5, 68 18 of 20

The total maintenance cost (scheduled and nonscheduled) can now be calculated for the first 3000
operation hours of the pump:
69, 049.2 + 81, 092.8 = 150, 142 e and the mean reliability for the same timeframe is 74.1%.
As such, the optimization criterion of cost/benefit takes its optimum value of 150, 142/0.741 = 202, 636.
The value is almost equivalent to the value of the optimization criterion of the first scenario.
The outputs of the two maintenance concepts are summarized in Table 9.

Table 9. Summary of the two maintenance concepts.

Initial Scenario Alternative Scenario

Average reliability 0.68 0.74
Reliability lower limit 0.53 0.43
Cost of scheduled maintenance 49,000 69,049
Cost of unscheduled maintenance 88,817 81,093
Total maintenance cost 137,818 150,142
Total maintenance cost/average reliability 202,643 202,636
Average failure rate per 1000 h of operation 2.75 1.97
MTTF 2040 2223
Component Types Sched. Unsched. Total Sched. Unsched. Total
A 11 5.0 16.0 6 3.2 9.2
B 7 4.4 11.4 6 2.0 8.0
c 0.5 0.5 6 0.6 6.6
d 1 2.7 3.7 6 3.7 9.7
e 0.5 0.5 6 0.5 6.5
Total 19 13 32 30 10 40

4. Discussion
Combination of dependencies and simulation optimization have been considered as promising
areas for future work for optimal maintenance of multicomponent systems [26]. On the other
hand, most optimal maintenance models in the literature use as optimization criterion the system
maintenance cost rate, but they ignore the reliability performance [27]. Minimizing the system
maintenance cost rate does not necessarily imply that the system reliability performance is optimized
with regards to the cost, especially for multicomponent systems, and sometimes minimal maintenance
cost is associated with very low system reliability measures. This is one of the effects of having various
components in the system which may have different maintenance costs and different importance
for the system [28]. As such, the optimal maintenance should always take into account both the
maintenance cost and reliability and this is the rationale of introducing the cost adjusted importance
measure to determine the optimal maintenance plan.
In the specific example of the fuel pump, it is noticeable that the first scenario (replacement
of one component at a time) essentially brings the fuel pump at a state of a constant failure rate,
after a ‘warmup’ period. On the contrary, the alternative scenario (simultaneous replacement of all
components) does not have the same effect on the failure rate. Further investigation is needed, in order
to assess the mechanism behind the convergence of the failure rate of the first scenario.
Other preventive maintenance scenarios may be examined as well, for example the simultaneous
replacement of two or more components, at each time the pump is grounded for scheduled
maintenance. In such cases, the calculation of combined importance measures is required.
Furthermore, the required confidence level for the number of components spare parts, which are
required for the nonscheduled maintenance, affects the respective maintenance cost. This is a key
cost driver for the maintenance optimization process; it is also a risk source for potential availability
shortfalls, therefore careful attention should be paid to determine an appropriate confidence level.
Aerospace 2018, 5, 68 19 of 20

Author Contributions: M.B. contributed to the data curation and all authors contributed to the conceptualization,
project administration, formal analysis, writing, editing and reviewing the manuscript.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflicts of interest.

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