beyond its cost

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Reliability Engineering and System Safety 94 (2009) 644–657

Contents lists available at ScienceDirect

Reliability Engineering and System Safety

0951-83

doi:10.1

$ Thi

2008 co� Corr

E-m

journal homepage: www.elsevier.com/locate/ress

Beyond its cost, the value of maintenance: An analytical framework forcapturing its net present value$

Karen B. Marais a,�, Joseph H. Saleh b

a School of Aeronautics and Astronautics, Purdue University, USAb School of Aerospace Engineering, Georgia Institute of Technology, USA

a r t i c l e i n f o

Article history:

Received 27 February 2008

Received in revised form

1 July 2008

Accepted 2 July 2008Available online 15 July 2008

Keywords:

Maintenance

Present value

Directed graph

Value trajectory

Multi-state failure

20/$ - see front matter & 2008 Elsevier Ltd. A

016/j.ress.2008.07.004

s research builds on and extends previous w

nference.

esponding author. Tel.: +1765 494 5117; fax:

ail address: [email protected] (K.B.

a b s t r a c t

Maintenance planning and activities have grown dramatically in importance across many industries

and are increasingly recognized as drivers of competitiveness if managed appropriately. Correlated with

this observation is the proliferation of maintenance optimization techniques in the technical literature.

But while all these models deal with the cost of maintenance (as an objective function or a constraint),

only a handful addresses the notion of value of maintenance, and seldom in an analytical or quantitative

way.

In this paper, we propose that maintenance has intrinsic value and argue that existing cost-centric

models ignore an important dimension of maintenance, namely its value, and in so doing, they can lead

to sub-optimal maintenance strategies. We develop a framework for capturing and quantifying the

value of maintenance activities. Our framework is based on four key components. First, we consider

systems that deteriorate stochastically and exhibit multi-state failures, and model their state evolution

using Markov chains and directed graphs. Second, we consider that the system provides a flow of service

per unit time. This flow in turn is ‘‘priced’’ and a discounted cash flow is calculated resulting in a present

value (PV) for each branch of the graph—or ‘‘value trajectory’’ of the system. Third as the system ages or

deteriorates, it migrates towards lower PV branches of the graph, or lower value trajectories. Fourth, we

conceptualize maintenance as an operator (in a mathematical sense) that raises the system to a higher

PV branch in the graph. We refer to the value of maintenance as the incremental PV between the pre-

and post-maintenance branches of the graphs minus the cost of maintenance. The framework presented

here offers rich possibilities for future work in benchmarking existing maintenance strategies based on

their value implications, and in deriving new maintenance strategies that are ‘‘value-optimized.’’

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Maintenance planning and activities have grown dramaticallyin importance across many industries. This importance ismanifested by both the significant material resources allocatedto maintenance departments as well as by the substantial numberof personnel involved in maintenance activities in companies—forexample over a quarter of the total workforce in the processindustry is said to deal with maintenance work [1]. This situation,coupled with an increasingly competitive environment, createseconomic pressures and a heightened need to ensure that theseconsiderable maintenance resources are allocated and usedappropriately, as they can be significant drivers of competitive-ness—or lack thereof if mismanaged.

ll rights reserved.

ork prepared for the ESREL

+1765 494 0307.

Marais).

In response to these pressures, the notion of ‘‘optimality’’ andthe mathematical tools of optimization and operations research(OR) have seeped into maintenance planning, and resulted in theproliferation of ‘‘optimal’’ maintenance models (see the reviewsby Pham and Wang [2] and Wang [3], for example). In each‘‘optimal’’ maintenance model developed, an objective function isfirst posited, then analytical tools are used to derive a main-tenance policy that maximizes or minimizes this objectivefunction subject to some constraints. For example, the objectivefunction can be the minimization of cost (cost rate, or life cyclecost) of maintenance given a system reliability and/or availabilityconstraint; conversely, the objective function can be the max-imization of reliability or availability, given a cost constraint. Inaddition to varying the objective function, different ‘‘optimal’’maintenance models are obtained by: (1) varying for example thesystem configuration (e.g., single-unit systems versus multi-unitsystems, k-out-of-n systems); (2) by including several degrees ofmaintenance (e.g., minimal, imperfect, perfect); (3) by varying theplanning horizon; (4) by using different analytical tools; or (5) by

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positing different types of dependencies between the variousunits in a multi-unit system.

Yet, while all these models deal with the cost of maintenance(as an objective function or a constraint), only a handful of modelstouches on the notion of value of maintenance, and seldom in ananalytical or quantitative way (e.g., [24]). Wang [3] highlights acritical idea for the development of a value-based perspective onmaintenance when he suggests that the cost of maintenance aswell as the resulting system reliability should be consideredtogether when developing optimal maintenance strategies. Un-fortunately, where the benefits of maintenance are considered, itis usually in the sense of avoiding the costs of failure. Interest-ingly, it is only within the civil engineering community that thebenefits in the sense of service delivery are considered and cost-benefit considerations explicitly taken into account in thedevelopment of maintenance strategies (e.g., [24]).

The argument for dismissing or not focusing on the value ofmaintenance, when it is made, goes along these lines: while it iseasy to quantify the (direct) cost of maintenance, it is difficult toquantify its benefits. Other authors wishing to consider the valueof maintenance lament the difficulties in quantifying the benefitsof maintenance. Dekker [4] for example notes ‘‘the main questionfaced by maintenance management, whether maintenance outputis produced effectively, in terms of contribution to companyprofits, [y] is very difficult to answer’’. Therefore maintenanceplanning is usually shifted from a value maximization problemformulation to a cost minimization problem (see [5,6] for adiscussion of why these two problems are not the same and donot lead to similar decisions in system design and operation).Incidentally, in many organizations, maintenance is seen as a costfunction, and maintenance departments are considered costcenters whose resources are to be ‘‘optimized’’ or minimized. Inshort, as noted by Rosqvist et al. [7] cost-centric mindset prevailsin the maintenance literature for which ‘‘maintenance has nointrinsic value’’.

In this paper, we argue that maintenance has intrinsic valueand argue that existing cost-centric optimizations ignore animportant dimension of maintenance, namely its value, and inso doing, they can lead to sub-optimal maintenance strategies. Wetherefore develop a framework for capturing and quantifying oneimportant aspect of the value of maintenance activities, theirimpact on revenue-generation capability, by connecting anengineering and OR concept, system state, with a financial andmanagerial concept, the net present value (NPV).1 Here the systemstate refers to the condition of the system and hence its ability toperform and thereby provide a flow of service (hence generaterevenue, or ‘‘quasi-rent’’). In order to build this connection, wefirst explore the impact of a system’s state on the flow of servicethe system can provide over time—for a commercial system, thistranslates into the system’s revenue-generating capability. Nextwe consider the impact of maintenance on system state evolutionand hence value generation capability over time. We then usetraditional discounted cash flow techniques to capture the impactof system state evolution with and without maintenance on itsfinancial worth, or NPV. For simplification, we call the results ofour calculations the ‘value of maintenance’. Finally, we discuss theadvantages and limitations of our framework. This work offersrich possibilities for assessing and benchmarking the valueimplications of existing maintenance policies, and deriving new

1 Note that we consider ‘‘value’’ as the net revenue generated by the system

over a given planning horizon. We do not consider additional dimensions of value

such as the potential positive effects of maintenance on environmental or health

impacts. Such effects can be incorporated in future work, see, for example, Marais

et al. [25] for a discussion of the quantification of environmental and health

impacts of aviation.

policies based on maximizing value, instead of minimizing cost ofmaintenance.

2. Background

This section provides a brief overview of various maintenancemodels. The purpose of this section is to provide context andbackground to the model assumptions and analytics we developin Sections 3 and 4. The reader interested in extensive reviews ofthe subject is referred to the survey papers by Dekker [4], Phamand Wang [2] and Wang [3]. In the following, we discuss(1) maintenance classification, (2) maintenance models, and(3) maintenance policies.

2.1. Types and degrees of maintenance

Maintenance refers to the set of all technical and adminis-trative actions intended to maintain a system in or restore it to astate in which it can perform at least part of its intendedfunction(s) [4]. Fig. 1 provides a simple (not comprehensive)classification scheme of maintenance along three axes: (1) thetype of maintenance; (2) the degree of maintenance; and (3) typeof system to be maintained (system configuration can beconceived of as a subset of this axis).

Maintenance type can be classified into two main categories:corrective maintenance and preventive maintenance (PM) [2]. CM,also referred to as repair or run-to-failure (RTF), refers tomaintenance activities performed after a system has failed inorder to restore its functionality.

PM refers to planned maintenance activities performed whilethe system is still operational. Its aim is to retain the system insome desired operational condition by preventing (or delaying)failures. PM is further sub-divided into clock-based, age-based,and condition-based. These sub-divisions refer to what triggersmaintenance activities [8].

�
Clock-based maintenance is scheduled at specific calendartimes; its periodicity is preset irrespective of the system’scondition (e.g., every Tuesday). � Age-based maintenance is performed at operating time
intervals or operating cycles of the system (e.g., every 500on/off cycles, or every 4000 h of flight).
� Condition-based maintenance is triggered when the measure-
ment of a condition or state of the system reaches a thresholdthat reflects some degradation and loss of performance of asystem (but not yet a failure). Condition-based maintenance isalso referred to as predictive maintenance.

Opportunistic maintenance encompasses both corrective andPM and is relevant for multi-unit systems with economic andfunctional dependencies in which the failure of one unit, andhence its corrective maintenance, offers an opportunity toperform PM on other still functional units.

Each type of maintenance can be further classified according tothe degree to which it restores the system [2]. At one end of thespectrum, perfect maintenance restores the system to its initialoperating condition or renders it ‘‘as good as new’’. At the otherend of the spectrum, minimal repair returns the system to thecondition it was in immediately prior to failing (in the case ofcorrective maintenance), or ‘‘as bad as old’’. In between theseextremes lies imperfect maintenance, which in effect returns thesystem to a condition somewhere in between as good as new andas bad as old. Finally, there is also the possibility that maintenanceleaves the system in a worse condition than before the failure,

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Preventive Corrective

Opportunistic (relevant only to multi-unit systems)

Minimal

Imperfect

Perfect

Degree of maintenance

Type of system (tobe maintained)

Multi-unit systems

Preventive Corrective

Type ofmaintenance

Single-unit systems

Fig. 1. Classification of maintenance strategies.

K.B. Marais, J.H. Saleh / Reliability Engineering and System Safety 94 (2009) 644–657646

through, for example, erroneous actions such as damagingadjacent parts while replacing a faulty unit.

2.2. Maintenance models

Models used to derive optimal maintenance policies generallycover four main aspects [4]: (1) a description of the system beingmaintained; (2) a model of how the system deteriorates and theconsequences thereof; (3) a description of the available informa-tion on the system and the available response options; and (4) anobjective function and an analytical framework (or tools) accord-ing to which the optimal maintenance policy is to be derived. Thissection reviews the four main classes of maintenance models,following the reviews in Refs. [2,3,9,10].

The first class of models developed considered the possibilityonly for perfect or minimal repair [2,11,12]. Thus, followingmaintenance, the system is returned to as good as new withsome repair probability p, or to as bad as old with probability(1�p). This basic concept is then expanded to take into accounttime-dependent repair probabilities, the possibility that main-tenance causes the system to be scrapped or to transition to someintermediate state, and non-negligible repair times (and hencenon-negligible downtime losses).

The second class of models considers maintenance as improv-ing the failure rate or intensity, and thus allows the possibility ofimperfect maintenance [2,13]. It is assumed that maintenanceprovides a fixed reduction in failure rate, or a proportionalreduction, or that it returns the system to the failure rate curve atsome time prior to the maintenance activity. Perfect maintenancereturns the failure rate to that of a new system, while minimalmaintenance returns it to that of the system immediately prior tothe failure. The degree of improvement of failure rate is referred toas the improvement factor. The improvement factor is determinedbased on historical data, experiment, expert judgment, or byassuming it correlates with maintenance cost, system age, or thenumber of prior maintenance activities [2,14].

The third class of models views maintenance as reducing thevirtual age of the system [15]. It is assumed that maintenancereduces the age of the system by some proportion (assumingincreasing failure rate, which implies among other things that thesystem exhibits no infant mortality). Perfect maintenance returnsthe system virtual age to zero, while minimal maintenancereturns the virtual age to the age immediately prior to the failure.Kijima et al.’s [15] original model allowed only a reduction to thevirtual age of the system following the previous repair effort,though larger reductions in virtual age can be seen as resultingfrom more extensive maintenance efforts. Pham and Wang [2]consider the reduction in virtual age as decreasing over time—inother words, repairs become successively less effective over time.They further assume that maintenance time increases withsubsequent repairs.

The fourth class of models considers system failures asmanifesting as some level of damage or degradation in responseto a shock. These models are therefore referred to as shockmodels. Perfect maintenance then reduces the damage to zero,minimal maintenance returns the damage level to that immedi-ately prior to the failure, and imperfect damage reduces thedamage by some factor greater than 0 and less than 100%. Thesemodels also allow the possibility for less-effective and moreexpensive repairs over time by making the reduction in damage adecreasing function of time and by successively increasing theduration of maintenance activities over time [16,17].

In each case these models have been used to derivemaintenance policies that minimize cost or downtime, orthat maximize system availability, as we discuss in the nextsubsection.

2.3. Maintenance policies

Maintenance policies describe what types of maintenance(repair, replacement, etc.) are considered in response towhat types of events (failure, calendar time, machine cycles,

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etc.). In the following, we confine our discussion to maintenancepolicies for single-unit systems with increasing failure rates(IFRs).

One popular maintenance policy is age-dependent PM where asystem is repaired or replaced at a pre-determined ‘‘age’’ [3]. Thetriggering of maintenance in this case may be pre-determinedbased on machine time (e.g., every 10,000 cycles) or on timeelapsed since the last maintenance activity. Under a random age-dependent maintenance policy, maintenance is performed basedon age and system availability. This policy takes account of thefact that systems may not be available for maintenance in themiddle of a production run, for example. A further extension ofage-dependent replacement is failure-dependent replacementwhere the system is repaired in response to failures and replacedwhen a given number of failures has occurred, or at a given time,whichever occurs first [18]. Many other variations on the theme ofage-dependent maintenance have been proposed; see [3] for anextensive review.

An alternative family of maintenance policies, referred to asperiodic PM, is based on calendar time. Here maintenance occurson failure and periodically regardless of the failure or operatinghistory of the system [3]. Variations on this theme are developedby selecting degrees of repair from minimal to perfect at specifictimes or in response to failures. Further variations are developedby incorporating the failure or operating history of the system. Forexample, the level of maintenance may be dependent on thenumber of previous repairs [19].

Sequential PM can be seen as a variation of periodic PM wherethe interval between PM activities is not constant. For example,the PM interval may be decreased as the system ages, so that thesystem does not exceed a certain operating time withoutmaintenance [3,20].

This brief overview of maintenance, as mentioned previously,is intended to provide context and background to our proposedperspective on maintenance as well as to the model assumptionsand analytics we develop in the following sections.

3. The value perspective in design, operations and maintenance:a qualitative discussion

The present work builds on the premise that engineeringsystems are value-delivery artifacts that provide a flow of services(or products) to stakeholders. When this flow of services is‘‘priced’’ in a market, this pricing or ‘‘rent’’ of these system’sservices allows the assessment of the system’s value, as will bediscussed shortly.2 In other words, the value of an engineeringsystem is determined by the market assessment of the flow ofservices the system provides over its lifetime. We have developedthis perspective in a number of previous publications; for furtherdetails, the interested reader is referred to [5,6,22,23].

In this paper, we extend our value-centric perspective ondesign to the case of maintenance. Our argument is based on fourkey components:

(i)

2

‘‘pric

head

syste

that

First, we consider systems that deteriorate stochastically andexhibit multi-state failures, and we model their stateevolution using Markov chains and directed graphs.

(ii)
Second, we consider that the system provides a flow ofservice per unit time. This flow in turn is ‘‘priced’’ and a
There are other cases when the flow of services provided by the system is not

ed’’ in a market. This situation is treated in the economic literature under the

ing of ‘‘un-priced values’’ (see for example [21]). While the value of the

m can still be assessed in this case, its treatment raises additional subtleties

are beyond the scope of the present work.

3

probl

discounted cash flow is calculated resulting in a PV for eachbranch of the graph—or ‘‘value trajectory’’ of the system.

(iii)
Third, given our previous two points, it is straightforward toconceive of the following: as the system ages or deteriorates,it migrates towards lower PV branches of the graph, or lowervalue trajectories.
(iv)
Finally, we conceptualize maintenance as an operator (in amathematical sense) that raises the system to a higher PVbranch in the graph, or to higher-value trajectory. We refer tothe value of maintenance, or more specifically the NPV ofmaintenance, as the incremental PV between the pre- andpost-maintenance branches of the graphs minus the cost ofmaintenance.
In the following section, we set up the analytical frameworkthat corresponds to this qualitative discussion.

4. Maintenance and present value branches

In developing our value model of maintenance, we make anumber of simplifying assumptions to keep the focus on the mainargument of this work. These assumptions affect the particularmechanics of our calculations but bear no impact on the mainresults, as will be shown shortly. Our assumptions are thefollowing:

(i)

e

We consider only the impact of maintenance on revenue-generating capability.

(ii)
We restrict ourselves to the case of perfect maintenance; inaddition we assume that maintenance does not change thesystem’s deterioration mechanism.
(iii)
We restrict ourselves to the case of single-unit systems. (iv) We only consider systems that exhibit an increasing failure
rate. In other words, as our systems age, they become morelikely to deteriorate in the absence of perfect maintenance.

(v)
The systems in our model can be in a finite number ofdiscrete states, and the current state depends only on theprior state, though the state transition probabilities may betime-dependent. This assumption allows us to model thestate evolution of the system as a Markov process.
(vi)
The systems in our model have no salvage value atreplacement or end of life.
(vii)
Finally, for simulation purposes, we consider discrete-timesteps, and assume that the duration of maintenanceactivities is negligible compared with the size of these timesteps.
These assumptions will be relaxed in future work. In thefollowing, we consider first how a system deteriorates under no-maintenance and introduce the concept of ‘‘value trajectories’’ ofthe system. Next, we show how maintenance moves the systemonto higher PV trajectories. Finally, we develop expressions for theNPV of maintenance. We illustrate the framework by means ofnumerical examples.

4.1. Deterioration under no-maintenance, and value trajectories

We consider a k-state discrete-time Markov deterioratingsystem3 with time-dependent transition probabilities as shown inFig. 2, for the no-maintenance case with three states. The states arenumbered from 1 through k in ascending order of deterioration

Markov models are frequently used to analyse reliability and maintenance

ms (see, for example [10]).

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New

Deteriorated

Failed

1

pnn(i)

pnd(i)

pdf(i)

pdd(i)pnf(i)

Fig. 2. Three-state model of system with no maintenance.

p11(0)

p13(0)

p12(0)

p11(1)

p13(1)

p12(1)

p23(1)

p22(1)

p11(2)

p13(2)p12(2)

p22(2)

p23(2)

p22(2)

p23(2)

New

Deteriorated

Failed

B1

Bworst

Fig. 3. Three-state system evolution over time with no maintenance.


where state 1 is the ‘‘new’’ state and state k is the failed state. Thetime-dependence allows us to take account of the fact that a new(or deteriorated) system will become more likely to transition tothe deteriorated (or failed) state as it ages.4 With no maintenancethe failed state is an absorbing state whence it is not possible totransition to any of the other states. Further, it is not possible totransition from the deteriorated state to the new state withoutperforming maintenance. In other words, the system can onlytransition in one direction, from new to failed, perhaps via thedeteriorated state (but the system has no self-healing properties).

The transition matrix for a system with k states and no self-healing is given by

PðiÞ ¼

p11ðiÞ p12ðiÞ � � � p1kðiÞ

0 p22ðiÞ � � � p2kðiÞ

0 ... . .

. ...

0 0 � � � 1

266664

377775 (1)

where i is the index of the time step considered, and P(i) is ineffect P(ti) in which ti ¼ iDT. For simplification purposes, weretain only the index i in our notation.

Most work in estimating transition probabilities has been donein civil engineering. Macke and Higuchi [24] demonstrate howtransition probabilities can be derived from failure rates, and alsoprovide references to several publications demonstrating the useof fatigue data, condition rating, analytical models of fatigueaccumulation on the structural component level, and globaldamage indices to estimate transition probabilities in the civilengineering domain. Macke and Higuchi’s approach can also beapplied to systems other than civil engineering structuresassuming that appropriate failure histories are available. For

4 Here the ‘‘time-dependence’’ implies dependence on the virtual age of the

system.

example, mining companies compile detailed failure logs on largeequipment such as drills and heavy trucks.

We represent the evolution of the system over time using adirected graph, as shown in Fig. 3 for a three-state system. Thisrepresentation expands on Fig. 2 and allows in effect an easy readof the time-dependent transition probabilities, which is difficultto visualize using the traditional Markov chain representation(Fig. 2).

We assume that the probability of transitioning to a lowerstate increases over time, and correspondingly that the probabilityof remaining in a given state (other than the failed state)decreases over time:

p11ð0ÞXp11 ið ÞXp11ðiþ jÞ

pmnð0ÞppmnðiÞppmnðiþ jÞ

i; jX1

1pmonpk(2)

If we define p0 to be the initial probability distribution of thesystem, the probability distribution after j state transitions is

pj ¼ Pj . . .P2P1p0 (3)

For convenience we assume that the system is initially in thenew state, i.e.,

p0 ¼ ½1 0 � � � 0 � (4)

Next, we consider that the system can generate um(t) revenueper unit time when it is in state m; a degraded system havinglower capacity to provide services (hence generate revenues) thana fully functional system. This um(t) is the expected utility modelof the system or the price of the flow of service it can provide overtime. We discretize time into small DT bins over which um(t) canbe considered constant.5 Therefore

umðiÞ ¼ umðiDTÞ � um½ðiþ 1ÞDT� (5)

To simplify the indexing, we consider that the revenues thesystem can generate between (i–1)DT and iDT are equal toum(i)DT.

5 The time step can be set for example according to the organization’s

accounting period. For instance, if accounts are paid, or revenue is accounted for,

on a quarterly basis, the time step can be set to 90 days.

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Table 1Branches for a three-state system evolution over four periods

Branch Transitions Comment

B1 {1, 1, 1, 1, 1} The system starts in state 1 and remains in this state

throughout the four periods

B4 {1, 1, 1, 2, 2} The system starts in state 1; it remains in state 1 for

two periods, then transitions to the degraded state 2

in the third period and remains in this state 2

B8 {1, 1, 2, 2, 3} The system starts in state 1; it remains in state 1 for

the first period, then transitions to the degraded

state 2 in the second period; it remains in this

degraded state for the third period, then transitions

to the failed state 3 in the fourth period

6 Given that the duration of maintenance is much smaller than the duration of

the time step considered, we can assume that maintenance occurs quasi-

instantaneously during the transition from the end of one time step to the

beginning of the next time step.


Each branch of the graph represents a particular valuetrajectory of the system, as discussed below.

Each time step, the system can remain fully operational, or itcan transition to a degraded or failed state. A branch in the graphis characterized by the set of states the system can ‘‘visit’’ for alltime periods considered. For example, the branch B1 ¼ {1, 1, 1,1,y,1} represents the system remaining in state 1 throughout theN periods considered, whereas Bj ¼ {1, 1, 1, 2, 2,y,2} represents asystem starting in state 1 and remaining in this state for the firsttwo periods, then transitioning to a degraded state (here state 2)at the third period and remaining in this particular degraded state.Notice that the branch Bworst ¼ {1, k, k, y, k} represents a newsystem transitioning to the failed state at the first transition.

Since each state has an associated utility um(i), a PV can becalculated for each branch over N periods as follows:

PVðN;BjÞ ¼

PNi¼1uBj

ðiÞ

ð1þ rDT Þi

(6)

where uB(i) is shorthand for the revenues along the branch B. rDT isthe discount rate adjusted for the time interval DT. Notice thatBranch 1 has the highest PV since the system remains fullyfunctional throughout the N periods (in state 1), and branch Bworst

has the lowest PV (PVworst ¼ 0) since the system starts its servicelife by failing!

The likelihood of the system following a particular branch overN steps is given by the products of the transition probabilitiesalong that branch:

pðBjÞ ¼YNi¼1

piðBjÞ (7)

The right-hand side of Eq. (7) is shorthand for the product of thetransition probabilities along the branch Bj. Here the assumptionthat the probability of transition to the next state depends only onthe current state allows us simply to multiply the probabilities toobtain the probability of the branch.

Finally, the expected PV of the system over all the branches iscalculated by weighting the PV of each branch by its likelihood:

hPVðNÞi ¼X

all branches

pðBjÞPVðN;BjÞ (8)

Using the Markov chain terminology, this PV can also beexpressed as

hPVðNÞi ¼XN

i¼1

Xk

m¼1

umðiÞpiðmÞ

ð1þ rDT Þi

(9)

The following simple numerical example will help clarify thisdiscussion and equations.

Numerical example: Consider a three-state system with thefollowing transition matrix:

P ¼

:95 :04 :01

0 :9 :1

0 0 1

264

375

The system has a constant revenue model over N periods, andcan generate $100,000/year in state 1, $60,000/year in thedegraded State 2, and $0 in the failed state 3. Assume a yearlydiscount rate of 10%. No maintenance is performed on the system.Next, consider the branches of the system evolution defined bythe transitions in Table 1.

Applying Eq. (5) to this simple case, we obtain the valuetrajectories and probabilities shown in Fig. 4.

The expected PV of this system across all branches is given byEq. (7):

hPVðNÞi ¼ $296;672

In the next subsection, we show how maintenance moves thesystem onto higher-value trajectories and therefore increases theexpected PV of the system.

4.2. Deterioration, maintenance, and present value

Maintenance compensates for the lack of self-healing in thesystem, and, from a modeling perspective, given a three-statesystem, maintenance allows the system to move back, (1) from afailed state to a deteriorated state (imperfect corrective main-tenance), (2) from a failed state to a new state (perfect correctivemaintenance), and (3) from a deteriorated state to a new state(perfect PM). These maintenance-enabled transitions are shownin Fig. 5.

In addition to returning the system to a higher-functional state,maintenance provides another advantage: it modifies the time-dependent transition probabilities. In particular, the transitionprobabilities from the new state after perfect maintenance areequal to those of a new system. That is, performing perfectmaintenance returns the system to the initial transition prob-abilities pij(0). Seen from a reliability viewpoint, perfect main-tenance returns the system to the initial reliability curve, asshown in Fig. 6. In the remainder of this work, we focus only onthe case of perfect maintenance.

Qualitatively, we can already foresee the two-sided impactof maintenance: not only does maintenance raise the systemto a higher-value trajectory, it also increases the likelihoodthat the system will remain on higher-value trajectories thanwithout maintenance (through its impact on the transitionprobabilities). These two effects are captured quantitatively inthe following sub-section, and they help us determine the PV ofmaintenance.

Fig. 7 shows the Markov model for our simple system underthe option of perfect maintenance. The dotted lines indicate the‘‘new’’ transition probabilities assuming that perfect maintenanceoccurred at the end of the previous time step.6

Between maintenance activities the system deterioratesaccording to:

pj ¼ Pj � � �P2P1p0 (10)

where pj is the vector of probabilities of being in states 1through k.

The effect of perfect maintenance is to return the system to theinitial probability distribution p0 and to the initial transitionprobability matrix P. Thus we can model the transition assuming

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$-

$50,000

$100,000

$150,000

$200,000

$250,000

$300,000

$350,000

Period 1 Period 2 Period 3 Period 4

Pre

sent

Val

ue

PV of Branch 1PV of Branch 4PV of Branch 8

p(B1)=81.4%

p(B4)=3.2%

p(B8)=0.3%

Fig. 4. Illustrative value trajectories for a three-state system (branches defined in Table 1).

New

Deteriorated

FailedImperfectmaintenance

Perfectmaintenance

Perfectmaintenance

Perfectmaintenance

Fig. 5. Performing perfect maintenance returns the system to the new state.

Rel

iabi

lity

2

Perfect maintenanceshifts the reliabilitycurve to the right

t1 3 4 5 6

Fig. 6. Impact of maintenance on reliability.

p11(0)

New

Deteriorated

Failed

1

p11(i)

p12(i)

p23(i)

p22(i)

p12(0)

p13(0) p13(i)

Fig. 7. Performing maintenance introduces new transition possibilities.


maintenance at time tm4t0 as

pm�1 ¼ Pk � � �P2P1p0

pm ¼ p0

pmþk ¼ Pk � � �P2P1p0 (11)

In short, maintenance acts on two levers of value: (1) it lifts thesystem to a higher-value trajectory, and (2) in the case of perfect

maintenance, it restores the initial (more favorable) transitionprobabilities, which in effect ensures that the system is morelikely to remain in the ‘‘new’’ state post-maintenance than pre-maintenance. The two effects of maintenance on the value of asystem are shown in Fig. 8 for a three-state system. The followingsimple examples will help further clarify this discussion.

Example 1: Perfect maintenance and the restoration of the initial

transition probabilities. Consider Branch B1 in the state evolution ofthe system (see Fig. 3 and Table 1); B1 as defined previouslycorresponds to a system remaining in state 1 (or the new state)throughout the time periods considered. Under no maintenance,as the system ages, the likelihood that it will remain on B1

decreases. For example, a system that remained on B1 until theend of the second period, as shown in Fig. 9, will find its likelihoodof subsequently remaining on B1 to be:

pnoM_2ðB1Þ ¼ p11ð2Þp11ð1Þ . . . p11ðnÞ (12)

whereas if perfect maintenance were performed on the system atthe end of the second period, the initial transition probabilities arerestored, and the likelihood that the system will remain for the

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p11(0)

p13(0)

p12(0)

p11(0)

p13(0)

p12(0)

p23(1)

p22(1)

p11(1)

p13(1)p12(1)

p22(1)

p23(1)

p22(2)

p23(2)

mai

nten

ance

mai

nten

ance

New

Deteriorated

Failed

Maintenance

Fig. 8. Maintenance moves the system to a higher PV branch and, in the case of

perfect maintenance, restores the initial transition probabilities.

7 Here the subscript ‘‘m1’’ represents the transition from state m to the new

state (state 1) brought about by perfect maintenance.


third period onwards on B1 increases and is given by

pM_2ðB1Þ ¼ p11ð0Þp11ð1Þ . . . p11ðn� 2Þ

and

pM_2ðB1Þ4pnoM_2ðB1Þ (13)

This result is easily extended for perfect maintenance occurringat the end of any period j:

pM_jðB1Þ ¼ p11ð0Þp11ð1Þ . . . p11ðn� jÞ

and

pM_jðB1Þ4pnoM_jðB1Þ (14)

Fig. 9 illustrates the system evolution over time under themaintenance and no-maintenance policies.

Example 2: The PV of maintenance. Consider the three-statesystem example discussed in Section 4.1, and assume that thesystem transitions to a degraded state at the end of the firstperiod. If no maintenance is performed, the system will remain onthe lower value branches shown in Fig. 10 (lower box). Theexpected PV of these lower branches, calculated at the end of thefirst period, is given by Eq. (8). Using the numerical valuesprovided in the previous example, we find:

hPVino_maintenance_1 ¼ $165;600

Next, assume that perfect maintenance is carried out at the endof period 1 after the system has degraded to State 2. Perfectmaintenance lifts the system up to higher-value trajectories,shown in the upper box in Fig. 10. The expected PV of these upperbranches, calculated at the end of the first period, after theoccurrence of perfect maintenance, is given by Eq. (8). Using thenumerical values provided in the previous example, we find:

hPVimaintenance_1 ¼ $242;500

Finally, consider the difference between the expected PV of thebranches enabled by maintenance and the branches withoutmaintenance:

DPV ¼ hPVimaintenance � hPVino_maintenance (15)

This incremental PV, which in this case it is calculated at theend of period 1, is what we define as the PV of maintenance:

DPV1 ¼ hPVimaintenance_1 � hPVino_maintenance_1 ¼ $76;900

Since maintenance is the only cause of the system accessingthese higher-value trajectories after it has degraded, it isappropriate to ascribe this incremental PV to maintenance andterm it ‘‘the PV of maintenance’’.

In the next subsection, we account for the cost maintenance,and conceptually, by subtracting it from the PV of maintenance,we finally obtain what we set out to derive in this work, namelythe NPV of maintenance.

4.3. Net present value of maintenance

We model the cost of maintenance and the revenue-generationcapability of the system as being state and time-dependent. Themodel can easily be extended to be dependent on the number ofprevious repairs. The cost of perfect maintenance is representedby cm1(i).7 Utility, as discussed earlier, is represented by um(i)where m is the state of the system. The cost of maintenance isassumed to increase as deterioration increases, and the revenue-generation capability decreases as deterioration increases:

c1ðiÞocm1ðiÞock1ðiÞ

u1ðiÞ4umðiÞ4ukðiÞ ¼ 02pmok (16)

We assume that the only costs of maintenance are those ofperforming the maintenance. We do not account for costsassociated with the system failure, such as damage to otherequipment. The lost revenue caused by downtime is alsoneglected since we assume that downtime is negligible withrespect to our time interval.

The expected PV of the system under no maintenance over N

time steps is given by Eq. (9), repeated here for convenience:

hPVðNÞi ¼XN

i¼1

Xk

m¼1

umðiÞpiðmÞ

ð1þ rDT Þi¼XN

i¼1

UðiÞ � pi

ð1þ rDT Þi

(17)

This PV of an engineering system under no maintenance policy(Eq. (13)) will be used as the benchmark against which theincremental PV of maintenance minus its cost will be compared.When this value increment is positive, we consider that main-tenance added net value to the system (or provided a positiveNPV). Once this framework is established, it is easy to conceive ofmaintenance models that are designed to ‘‘optimize’’ this NPV ofmaintenance.

4.4. Numerical examples

In the following, we consider two numerical examples that willhelp clarify the previous discussion, and unambiguously capture—

and we hope convince the reader of—the value of maintenance.We start with a simple case of a single PM intervention: First, we

capture the value of this PM intervention. Second, by varying thetiming of this intervention, we determine different values ofmaintenance, which in turn allows us to identify the optimal timingthat maximizes the value of PM. This analysis provides the analyticalgroundwork for developing value-optimal clock- and age-based PM.

In the second example, we consider the more reactive case ofcorrective maintenance (i.e., no maintenance planning involved)and demonstrate the value of corrective maintenance and itsdependence on different parameters.

Example 1: Single PM intervention. Consider the simple casewhere we allow a single maintenance intervention, at time stepNj. We assume the system is operated for Nf steps. In this case thesystem deteriorates according to its transition matrix until step Nj,

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Time

Branch 1without maintenance

Period 1

p11(0)

p11(1)p11(2) p11(3)

Period n

p11(n)

Branch 1with perfectmaintenance

p11(0)

p11(1)p11(0) p11(1) p11(n-2)

Perfectmaintenanceoccurs here

Period 2 Period 3 Period 4

Fig. 9. Perfect maintenance and the restoration of the initial transition probabilities.

PV

Time

PV no_maintenance_1

PV maintenance_1

Maintenance lifts adegraded system tohigher value branches

Higher value (and morelikely) branches for themaintained system

Lower value branches ofthe degraded system withno maintenance

ΔPVprovided bymaintenance

Period 1 Period 2 Period 3 Period 4

Fig. 10. The incremental present value provided by maintenance.


where we perform perfect maintenance, regardless of the statethe system is in. The PV of the system up to step Nj�1 is given byrunning Eq. (13) over (Nj–1) steps:

hPVðNj � 1Þi ¼XNj�1

i¼1

Xk

m¼1

umðiÞpiðmÞ

ð1þ rDT Þi¼XNj�1

i¼1

UðiÞ � pi

ð1þ rDT Þi

(18)

The expected cost of maintenance in step Nm is given by

hCm1ðjÞi ¼Xk

m¼2

cm1ðNjÞpN1ðmÞ

ð1þ rDT ÞNj

(19)

Perfect maintenance returns the system to the transitionprobabilities of a new system. The PV of the system from step Nj

to step Nf is therefore given by

hPVðNj;Nf Þi ¼u1ðNjÞ

ð1þ rDT ÞNjþXNf�Nj

i¼1

Xk

m¼1

umðNj þ iÞpiðmÞ

ð1þ rDT ÞNjþi

¼u1ðNjÞ


i¼1

UðNj þ iÞ � pi

ð1þ rDT ÞNjþi

(20)

Combining the above three equations yields the PV of thesystem up to step Nf:

hPVðNf Þi ¼XNj�1

i¼1

UðiÞ � pi

ð1þ rDT Þiþ

u1ðNjÞ


i¼1

UðNj þ iÞ � pi

ð1þ rDT ÞNjþi

�Xk

m¼2

cm1ðNjÞpN1ðmÞ

ð1þ rDT ÞNj

(21)

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Fig. 11. Value of a single maintenance intervention.


Fig. 11 shows the expected PV of our example system (as definedby Eq. (21)) for the case where maintenance is performed2 years into the system’s operating lifetime. The cost of amaintenance intervention is set in our numerical exampleto be equal to the revenue generated during a single time step.One can clearly see on Fig. 11 the effect of maintenance, whichlifts the system onto a higher-value trajectory. Furthermore,the impact of the cost of maintenance—in this case, it is setequal to the revenues generated by system during one timestep—is indicated on Fig. 11 by the leveling of the value trajectoryat year 2.

Using Eq. (21), we can determine the NPV-optimal pointat which to perform a single maintenance interventionfor a system with a given set of transition probabilities andutility profile u(t). We varied the timing of the maintenanceintervention, and performed a Monte Carlo simulation of theexpected PV of the system in each case. Fig. 12 shows therelationship between the timing of the single maintenanceintervention and the expected PV of the system for differentinitial probabilities of deterioration to a lower state. In each case,we identify an optimum time at which maintenance should beperformed. The value-optimal timing of PM is shown to increaseas the probability of deterioration decreases; this can beinterpreted intuitively by noticing that the lower the transitionprobability to a lower state, the more likely the system willremain on a higher-value trajectory and for a longer period oftime. Thus, it is not ‘‘optimal’’ to have a maintenance interventionearly on that can only lift the system onto a slightly higher-valuetrajectory.

Having determined the optimal timing of a single maintenanceintervention, we propose to explore in future work, the value-optimal interval(s) for PM.

In the next example, we turn our attention to the more reactivecase of corrective maintenance and demonstrate its value.

Example 2: Perfect corrective maintenance following any dete-

rioration. We calculate the expected NPV of running the systemwith perfect corrective maintenance by determining the cumula-tive PV of the maintenance strategy in response to differentfailure sequences. The expected value of the maintenance strategyis then found by determining the probability of each failuresequence and summing over the probability-weighted PV of all

the failure sequences. The result is shown in Fig. 13, and thefollowing discussion provides the analytics and assumptions forour result.

Begin with a system which has not experienced deteriorationup to step (N1�1) and which transitions to a deteriorated or failedstate in step N1. We refer to this trajectory as branch Ba.Maintenance is accordingly performed and the system is returnedto the new state. The PV of the system up to and including step N1

is found by subtracting the cost of returning the system to thenew state from the revenue generated by the fully functionalsystem:

hNPVðN1;BaÞi ¼XN1

i¼1

u1ðiÞ

ð1þ rDT Þi�Xk

m¼2

cm1ðN1ÞpN1ðmÞ

ð1þ rDT ÞN1

(22)

To determine the expected PV of all the possible valuetrajectories, we must determine the probability of being onany given value trajectory. The general expression of thisprobability is given by Eq. (7). We take advantage of the Markovnotation to express this probability for the particular case of nodeterioration for (N1�1) steps, followed by deterioration in theN1th step as:

pðdeteriorate at N1Þ ¼ pðN1Þ

¼ pN1�1ð1ÞXk

m¼2

pN1ðmÞ

" #

¼ pN1�1ð1Þð1� pN1ð1ÞÞ (23)

Therefore the probability-weighted NPV for the system up toand including step N1 is

hNPVðN1Þi ¼XN1�1

i¼1

u1ðiÞ

ð1þ rDT Þi� pðN1Þ

Xk

m¼2

cm1ðN1ÞpN1ðmÞ

ð1þ rDT ÞN1

(24)

We apply the probability only to the second term since underour assumptions of perfect maintenance and negligible main-tenance interval the system is instantaneously (relative to thelength of the time step) returned to its new state revenue-generating capability. Following perfect maintenance the systemtransition matrix returns to the initial value P1.

The system now remains in the new state for another (N2�1)steps, before again deteriorating.

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Fig. 12. Normalized value of maintenance relative to timing of maintenance intervention.

Fig. 13. Value of perfect corrective maintenance.


The probability-weighted PV for the system up to andincluding step N1+N2 is given by

hNPVðN1 þ N2Þi ¼XN1þN2

i¼1

u1ðiÞ

ð1þ rDT Þi�Xk

m¼2

pðN1Þcm1ðN1ÞpN1

ðmÞ

ð1þ rÞN1

�

þpðN2Þcm1ðN1 þ N2ÞpN2

ðmÞ

ð1þ rÞN1þN2

�(25)

For a series of failures (N1, N2, y, Nn) the probability-weightedPV under corrective perfect maintenance follows:

NPVXn

j¼1

Nj

0@

1A ¼ XPn

j¼1Nj

i¼0

u1ðiÞ

ð1þ rDT Þi�Xk

m¼2

pðN1Þcm1ðN1ÞpN1

ðmÞ

ð1þ rDT ÞN1þ � � � þ

24

pðNnÞcm1ð

Pnj¼1NjÞpNn

ðmÞ

ð1þ rDT Þ

Pn

j¼1Nj

35 (26)

Eq. (18) allows us to calculate the probability-weightedPV of corrective maintenance for a given series of failureevents. The expected value of corrective maintenance over N timesteps is then found by averaging over the possible failuresequences.

We illustrate our argument using a simple hypotheticalexample of a single-unit system. The system deterioratesprogressively through four states, from new (state 1) to failed(state 4). For each state the probability of transitioning to the nextlower state is identical, and given by pdet. The transition matrix istherefore given by

P ¼

1� pdet pdet 0 0

0 1� pdet pdet 0

0 0 1� pdet pdet

0 0 0 1

26664

37775 (27)

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To model system ageing despite maintenance, we allow pdet toincrease over time so that by the end of the planning horizonthere is probability of .5 of deteriorating to a lower state, even ifperfect maintenance has been performed in the previous timestep. We consider those failure sequences where failure occurs on10% of the state transitions. We further assume that revenue andmaintenance costs are constant over time and use a time step of 6months. For the nominal case we set the initial probability ofdeterioration to 5%.

Assume the system generates $5000 per time step in the newstate, and revenues decrease quadratically through the lowerstates, with the system generating no revenue in the failed state.For the nominal case, maintenance costs eight times themaximum revenue-generating capability of the system. We alsoassess the impact of these assumptions on the value ofmaintenance. Since the system proceeds probabilistically along

Fig. 14. Normalized NPV o

Fig. 15. Value of maintenance versu

the value trajectories we used a Monte Carlo approach to samplethe trajectory space.

Fig. 13 shows the cumulative PV of a policy of perfectmaintenance in response to system deterioration compared tothat of a policy of no maintenance for the nominal case. The uppercurve shows the net benefit of the perfect maintenance policy,that is, the discounted revenues minus the discounted main-tenance costs. The lower curve shows the expected PV of a zeromaintenance policy. The difference between the curves is thevalue of corrective maintenance. Maintenance increases the PV ofthe system, and delays the time at which cumulative NPV stopincreasing. The PV of maintenance for a given design lifetime canbe read off as the difference between the two curves. For example,after 5 years of operation, the NPV of maintenance is approxi-mately $13,500, in our example, as indicated by the arrow in thefigure.

f maintenance policy.

s probability of deterioration.

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Fig. 16. Impact of maintenance cost on value of maintenance. The x-axis shows the ratio of the cost of maintenance to the revenue generated by the system per time step.


Fig. 14 shows the ‘‘normalized NPV’’ of maintenance, defined asthe quotient of the NPV of the maintenance and PV of the no-maintenance policy:

norm_npv ¼NPVmaintenance

hPVinomaintenance(28)

Consider next the drivers of the value of maintenance in ourexample. Fig. 15 shows the relationship between the net value ofmaintenance and the probability of deterioration. In this case weassume that the probability of deterioration does not increaseover time. As the probability of deterioration increases, the valueof maintenance increases, achieving a maximum value at aprobability of approximately .84. Why is there a maximum value?As the probability of deterioration increases, and maintenancemust be performed more often, the total cost of maintenanceincreases. Eventually, the total cost of maintenance exceeds therevenues generated by the system, at which point the systemshould be written off as it will no longer be profitable to operate.

Fig. 16 shows the impact of the cost of maintenance on thenormalized NPV of maintenance. The x-axis shows the ratio of thecost of maintenance to the revenue generated by the system pertime step. As expected, the normalized NPV decreases as the costof maintenance increases, but with a slope of more than �1. Inother words, the value of maintenance decreases more slowlythan the cost increases.

Thus the NPV of maintenance increases as the probability ofdeterioration increases, and decreases as the cost of maintenanceincreases relative to the revenue generated by the system.

This example quantifies the value of corrective, and illustratesthe functional dependence of this value on various characteristicsof the system (e.g., probability of deterioration, and cost ofmaintenance relative to system income).

5. Conclusions

While maintenance optimization techniques abound in theliterature, most of these models focus on minimizing the cost ofmaintenance or maximizing system availability, and seldom isthere mention or discussion of the value of maintenance.

In this paper, we argued that maintenance has value andargued that existing cost-centric models ignore an importantdimension of maintenance, namely its value, and in so doing, theycan lead to sub-optimal maintenance strategies. We argued thatthe process for determining a maintenance strategy shouldinvolve both an assessment of the value of maintenance—howmuch is it worth to the system’s stakeholders—and an assessmentof the costs of maintenance.

We considered ‘‘value’’ as the net revenue generated by thesystem over a given planning horizon. We did not includeadditional dimensions of value such as the potential positiveeffects of maintenance on environmental or health impacts. Sucheffects can be incorporated in future work. Further, we did notinclude factors such as safety or regulatory requirements. Suchfactors can be easily included as constraints on the optimizationof value in future work.

We developed a framework for capturing and quantifying thevalue of maintenance activities by connecting an engineering andOR concept, system state, with a financial and managerial concept,the NPV.

We explored the impact of a system’s state on the flow ofservice the system can provide over time. Next we considered theimpact of maintenance on system state evolution and hence valuegeneration capability over time. We used traditional discountedcash flow techniques to determine the ‘value of maintenance’,which we defined as the difference between the expected NPV of asystem with and without maintenance.

By identifying the impact of system state or condition on NPV,the framework and analyses developed in this paper provide(financial) information for decision-makers to support in part themaintenance strategy development. Maintenance, as a conse-quence, should not be conceived as ‘‘just an operational matter’’and guided by purely operational matters but by multi-disciplin-ary considerations involving the marketing, finance, and opera-tions functions within a company.

We made several simplifying assumptions in order to advanceour main argument. In future work these assumptions will bevalidated or relaxed. Our assumption that downtime is negligiblecan also be relaxed to incorporate the impact of downtime on forexample production levels. The work can be used as the basis of

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the development of optimal maintenance policies in the presenceof constraints (safety, maintenance resources, etc.).

In particular, the work will be expanded to consider multi-unitsystems, in which the possibility of opportunistic maintenancearises. For example, airlines perform additional PM once specificpanels on aircraft have been opened for corrective maintenance,or when delays result in the aircraft being on the groundfor longer periods than originally planned. This work can beused to (1) value opportunistic maintenance interventions and (2)identify optimal approaches to selecting targets for opportunisticmaintenance.

Is it possible to take limitations in maintenance resources intoaccount (in the context of multi-unit systems this is relevant)? Isopportunistic maintenance possible to be taken into account inthe method? How?

One important implication of this work is that the main-tenance strategy should be tied to market conditions and theexpected utility profile (core revenue-generating capability of thesystem). In other words, a value-optimal maintenance strategy isdynamic and changes not only in response to environmental andsystem conditions but also in response to market conditions.

Finally, we believe that the framework presented here offersrich possibilities for future work in benchmarking existingmaintenance strategies based on their value implications, and inderiving new maintenance strategies that are ‘‘value-optimized.’’

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