Reliability Engineering and System Safety 119 (2013) 76–87

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Reliability Engineering and System Safety

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

Value maximizing maintenance policies under general repair

Karen B. Marais n

School of Aeronautics and Astronautics, Purdue University, USA

a r t i c l e i n f o

Article history:Received 2 October 2012Received in revised form7 May 2013Accepted 13 May 2013Available online 28 May 2013

Keywords:Cost benefit analysisDynamic programmingMaintenanceMarkov processesReliabilityReplacement

20/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.ress.2013.05.015

+1 7654940063.ail address: kmarais@purdue.edu

a b s t r a c t

One class of maintenance optimization problems considers the notion of general repair maintenancepolicies where systems are repaired or replaced on failure. In each case the optimality is based onminimizing the total maintenance cost of the system. These cost-centric optimizations ignore the valuedimension of maintenance and can lead to maintenance strategies that do not maximize system value.This paper applies these ideas to the general repair optimization problem using a semi-Markov decisionprocess, discounted cash flow techniques, and dynamic programming to identify the value-optimalactions for any given time and system condition. The impact of several parameters on maintenancestrategy, such as operating cost and revenue, system failure characteristics, repair and replacement costs,and the planning time horizon, is explored.

This approach provides a quantitative basis on which to base maintenance strategy decisions thatcontribute to system value. These decisions are different from those suggested by traditional cost-basedapproaches. The results show (1) how the optimal action for a given time and condition changes asreplacement and repair costs change, and identifies the point at which these costs become too high forprofitable system operation; (2) that for shorter planning horizons it is better to repair, since there is notime to reap the benefits of increased operating profit and reliability; (3) how the value-optimalmaintenance policy is affected by the system's failure characteristics, and hence whether it is worthwhileto invest in higher reliability; and (4) the impact of the repair level on the optimal maintenance policy.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Significant material and personnel resources are allocated tomaintenance activities in companies—for example over a quarterof the total workforce in the process industry is said to deal withmaintenance work [22]. The importance of maintenance to indus-try is reflected by the extensive and growing literature on optimalmaintenance, devoted to developing methods to ensure that theseconsiderable maintenance resources are allocated and used effi-ciently, as they can be significant drivers of competitiveness—orlack thereof if mismanaged (see the reviews by Pham and Wang[19] and Wang [23]).

1.1. General repair maintenance policies

One class of problems considers the notion of general repairmaintenance policies, where, perhaps in conjunction with apreventive maintenance program, systems are repaired orreplaced on failure. The question investigated in these studiesunder various assumptions is, if the system has failed, when is it

ll rights reserved.

better to replace, and when is it better to repair, and to what level?Minimal repair returns the system to the condition it was inimmediately prior to failure, for example, patching a flat tire. Incontrast, perfect repair returns the system to an as good as newstate. An engine overhaul may be seen as near-perfect repair.(“Worse” repair where the system is in a worse condition afterrepair; for example, if an engine suffers foreign-object damage dueto a lost tool, is also possible, but I do not consider it here) Manyoptimal policies have been proposed, generally in the form of acost, age, or number of failures limit. Hence these policies aretypically referred to as “repair limit” policies.

Wang [23] suggests that repair limit policies were first intro-duced by Gardent and Nonant [4] and Drinkwater and Hastings[3]. Drinkwater and Hastings noted that while many organizationsused repair limits based on the type, age or location of a system,there were no tools available to guide optimal actions. Theyshowed both analytically and through simulation how these limitscould be set to minimize the average repair cost per year of a fleetof vehicles. One problem of their approach is that the decision torepair or replace only depends on that repair, and not on thehistory of repairs of the system. A system could therefore “limpalong” through a series of repairs that fell just below the repaircost limit, even though a replacement was justified when taking alonger view. Beichelt [1] addressed this problem by using a repair

K.B. Marais / Reliability Engineering and System Safety 119 (2013) 76–87 77

cost rate limit, that is, the system is replaced when the repair costper unit time exceeds a fixed value. The repair history can also beincorporated into the problem by considering the number offailures as well as the system age; see Kapur et al. [8], Makis andJardine [13], and Love et al. [12].

The optimizations are usually carried out assuming that repairor replacement is instantaneous—another set of policies is devel-oped by setting a repair time limit rather than a repair cost limit. InNakagawa and Osaka's (1974) approach, a repair is abandoned if itcannot be completed within a predetermined time. Nguyen andMurthy [18] motivate the consideration of repair time by positinga situation where basic (and imperfect) repairs can be completedlocally but more extensive (and perfect) repairs require centralrepair. This situation is readily seen in industry, where, forexample, airlines have small maintenance facilities at mostairports but only a few large maintenance facilities (see also [15]).

While generally not considered as being repair limit policies,policies based on replacing once the system exceeds a certainnumber of failures have been suggested, as well as policies basedon replacing the system once it exceeds some reference operatingtime, for example flight hours or vehicle miles (see Wang [23] for areview).

This paper builds on Kijima [10], Makis and Jardine [13], andLove et al. [12] to develop a stochastic deterioration model of asystem under a general repair policy. Kijima proposed that theeffect of repair could be modeled as reducing the system's virtualage and then used a g-renewal function to determine the optimaltime between replacements [9]. He let Vn be the system's virtualage after the nth repair, Xn the additional age incurred between the(n−1)th and nth repair, and θn the level of repair. In his Type Imodel, the nth repair cannot remove the damages incurred beforethe (n−1)th repair. Thus, after the nth repair the virtual age of thesystem becomes:

Vn ¼ Vn−1 þ θnXn ð1ÞNote that if we start with a new system (and any replacement

systems are also new) at t¼0, the system virtual age will thereforealways be less than or equal to the clock time.

The Type II model allows repair to remove damage caused byprior failures too.

Several other authors have also used a similar simplification ofmodeling repair as being able to only reduce wear since the lastrepair, or being able to reduce wear from the beginning of systemuse. Martorell et al. [17], coming from a proportional hazardsviewpoint, propose the proportional age setback (PAS) and pro-portional age reduction (PAR) models. The PAR model assumesmaintenance proportionally reduces the age gained since theprevious maintenance event, and is therefore similar to the TypeI model. The PAS model shifts the origin of the time from thecomponent age is measured, where the shift is proportional to thedegree of maintenance. It is therefore similar to the Type II model.

Doyen and Gaudoin [11], coming from a failure intensity view-point, model the impact of repair on failure intensity. Theypropose two extremes: repair can at most reduce the increase infailure intensity since the last repair, or, repair can reduce the totalincrease in failure intensity from the beginning of system use.Their models are therefore similar to Type I and Type II respec-tively. The authors add an interesting dimension by then allowinga third model to vary between these two extremes—that is, repaircan reduce aging since the last m repairs, where m is set by themodeler. In a Markov process sense, in other words the memory ofthe process can be one step, m steps, or infinite.

Makis and Jardine [13] used a semi-Markov approach and atwo-dimensional state space defined by the number of failures nand the real age of the system tn to demonstrate that stationaryoptimal policies that minimize the expected average cost per unit

time for Type I systems exist. By formulating their problem as a g-renewal function they were able to find such solutions; however,this approach did not allow them to consider the effect of failurehistory on failure densities.

Accordingly, Love et al. [12] developed a semi-Markov decisionstructure using the (n, tn) state-space and proposed a numericalsearch procedure that could be used to identify repair-cost mini-mizing general repair policies for Type I systems where both therepair cost and the failure rate may depend on the state. Theirpolicy takes the form of a control limit sn that defines themaximum virtual age for a given accumulated number of failuresbeyond which the system should be replaced rather than repaired.

1.2. The value of maintenance

The question of whether the reliability gained through main-tenance is “worth” the cost of maintenance however is usually notaddressed, due, in part, to the difficulty in doing so. Dekker [2] forexample notes “the main question faced by maintenance manage-ment, whether maintenance output is produced effectively, interms of contribution to company profits, […] is very difficult toanswer”. Therefore maintenance planning is usually shifted from avalue maximization problem formulation to a cost minimizationproblem. In short, as noted by Rosqvist et al. [20] a cost-centricmindset prevails in the maintenance literature for which “main-tenance has no intrinsic value”.

In previous work we have proposed an alternative approachusing an objective function related to the value of maintenance[16]. Using a simple preventive maintenance example, we showedhow a maintenance strategy could be developed based on both anassessment of the value of maintenance—how much is it worth tothe system's stakeholders—and an assessment of the costs ofmaintenance.

The purpose of this paper is to show how general repairpolicies that maximize system value can be developed for sto-chastically deteriorating systems. Section 2 qualitatively discusseshow the existing literature on general repair maintenance opti-mization can be leveraged to take a value perspective and thendevelops the quantitative analytical basis, using a semi-Markovdecision process and discounted cash flow techniques. Section 3explores the results and practical implications of the valueperspective and introduces a simple visualization for selectingthe optimal action at each system failure. Finally, Section 4discusses the advantages and limitations of the proposedframework.

Maintenance is often a significant component of an organiza-tion's operating costs. This work offers a way of quantifying thereturn on this investment, or, what I term the value of main-tenance. The analytics developed here allow the identification ofvalue-optimal, or at least value-informed, maintenance policies.

2. Theory

This section first provides a qualitative discussion of theapproach, and then develops the model and optimization.

2.1. The value perspective on general repair policies: a qualitativediscussion

My specific purpose in this paper is to develop a discrete semi-Markov decision structure for a finite horizon problem and then toidentify general repair policies that maximize the net presentvalue generated by the system. While the semi-Markov decisionstructure is based on that proposed by Makis and Jardine [13] and

K.B. Marais / Reliability Engineering and System Safety 119 (2013) 76–8778

Love et al. [12], this approach differs from theirs in twosignificant ways.

First, maintenance is seen as a value-driver, rather than as acost-driver. The present work builds on the premise that engineer-ing systems are value-delivery artifacts that provide a flow ofservices (or products) to stakeholders. When this flow of servicesis “priced” in a market, this pricing or “rent” of these system'sservices allows the assessment of the system's value, as will bediscussed shortly. In other words, the value of an engineeringsystem is determined by the market assessment of the flow ofservices the system provides over its lifetime. For example, thevalue of a commercial passenger aircraft is related to the revenueseat miles provided by the aircraft, while the value of a telecom-munications satellite is related to the bandwidth offered by itstransmitters. This perspective has been developed in severalprevious publications; see for example, Saleh and Marais [21]and Marais and Saleh [16].

Second, the problem is explicitly defined with a finite timehorizon, in contrast to most maintenance optimizations, whichminimize average cost per unit time while explicitly or implicitlyassuming an infinite time horizon. Investment decisions in generalare made with the assumption that the desired return will begenerated within some finite period. For example, an investor in awind farm will typically estimate the value of the investment over15 years. The finite time horizon introduces important dimensionsthat are not visible using an infinite horizon, as will be discussedshortly. See also Mamer [14] and Huang and Guo [7] for treat-ments of the finite horizon maintenance problem andHartman and Murphy [6] for a treatment of the finite horizonreplacement problem.

My argument is based on three key components:First, I consider systems that deteriorate stochastically, and I

model their state evolution as a semi-Markov process using athree-dimensional state space defined by the accumulated num-ber of failures, the system's virtual age, and the clock time. Theprobability of failure depends on the system's virtual age. Repairsare Type I and reduce the system's virtual age acquired since thelast failure.

Second, I consider that the system provides a flow of serviceper unit time, and this flow depends on the state of the system.This flow in turn is “priced”. I consider both constant revenue withdeterioration, and decreasing revenue using a family of revenuecurves. Similarly, the operating cost of the system can be eitherconstant, or increasing using a family of cost curves. I thencalculate a discounted cash flow incorporating maintenance costsresulting in a Present Value (PV) for each possible evolution of thesystem, or “value trajectory” of the system. In the semi-Markovenvironment, the possible trajectories are defined by the stochas-tic failures and the general repair decisions.

Third, I formulate a dynamic programming approach and use itto assess the optimal expected value of the system for each stateand point in time. These values can then be used to determine theoptimal action (repair/replace) for a failure occurring in a givenstate and point in time. Several interesting findings result, forexample, a system that is worthwhile replacing towards thebeginning of the time period may not even be worth repairingtowards the end of the time period.

In the following section, I set up the analytical framework thatcorresponds to this qualitative discussion.

2.2. An analytical model of the value of systems under general repairpolicies

In developing the value model of maintenance, I make anumber of assumptions to keep the focus on the main argument

of this work. These assumptions will be progressively relaxed infuture work.

The assumptions are the following:

1.

The system is modeled as a semi-Markov decision process. 2. Repairs are Type I [10] and the repair level θ is constant. When

a system is replaced, its virtual age resets to zero.

3. The failure intensity of the system is solely defined by its

virtual age.

4. The failure instants k are decision epochs, and at each failure

instant there are two possible actions: repair (a¼1) andreplace (a¼0). That is, the system state is observed only atfailure events.

5.

Time at failure instant k is denoted by tk is discretized intoslices ik using a scaling parameter ξ such that ik/ξ≤tko(ik+1)/ξ.Thus failures are assumed to occur precisely at ik [12]. Forexample, if the time unit is hours, and the first failure occursafter 3:25 h, i1¼3. The time slices are assumed to be muchsmaller than the intervals between failures.

6.

The state of the system is described by (nk, vk, ik) where nk isthe number of failures, vk is the discretized virtual age and ik isthe discretized time at the kth decision epoch. The index k onthe failure count is necessary to account for systems that arereplaced. Where possible without causing ambiguity, I willsuppress the subscript k and refer the state is (n, v, i).

7.

The one-time replacement cost is fixed at C0 and the systemhas no salvage value.

8.

One-time repair costs, C1, are a stationary bounded non-decreasing function of the number of failures that haveoccurred and the system's virtual age (C1(n, v)≤K1, n≥1, v≥0).

9.

Operating costs C2, are also a stationary bounded non-decreasing function of the number of failures that haveoccurred and the system's virtual age (C2(n, v)≤K2, n≥1, v≥0).

10.

Operating revenues C3, are a stationary bounded non-increasing function of the number of failures that haveoccurred and the system's virtual age (C3(n, v)≤K3, n≥1, v≥0).Both operating costs and revenues are associated with specifictime slices and may change to reflect for example increasingfuel costs or decreased demand for the service.

By definition the problem time horizon is finite, so i≤imax whereimax is defined by the user. Since the virtual age cannot exceed thetotal time that has passed, we also have v≤imax. In the semi-Markov formulation, the maximum number of failures in the finitetime horizon is bounded by the time discretization. Thus, the statespace is finite and discrete, I¼{(n, v, i)|0≤n≤N, 0≤v≤imax, 0≤i≤imax}.

2.2.1. Stochastic deterioration modelThere are two possible actions: repair (a¼1), and replace

(a¼0). A system with virtual age vk has a discretized time tofailure xk. After failure nk+1, but before repair, the virtual age is:

v−kþ1 ¼ vk þ xk

where the – denotes the virtual age prior to repair.The Type I repair reduces the virtual age gained since the last

sojourn, so that after repair the virtual age is:

vþkþ1 ¼ vk þ θxk

Thus repair on failure nk+1 takes the system from (nk, vk, ik) to(nk+1, vk+θxk, ik+xk) at the next decision epoch, where xk is thediscretized time to failure given the virtual age after repair vk.Replacement takes the system from (nk, vk, ik) to (1, x0, ik+x0) at thenext decision epoch, where x0 is the discretized time to failure of anew system.

The discrete sojourn time xk between the nth and (n+1)thfailure is assumed to depend only on the failure rate function and

K.B. Marais / Reliability Engineering and System Safety 119 (2013) 76–87 79

the virtual age, though it can be extended to include dependenceon the accumulated number of failures. In Type I imperfect repair,the repair can at best reduce the effective aging incurred since thelast failure, so the probability density function (pdf) of xk is thetime to first failure conditioned on the virtual age vk at the kthfailure.

Assuming a repair was performed at (n, v, i), the discretizedconditional pdf of x is

f vðxÞ ¼f ððxþ vÞ=ξÞ1−Fðv=ξÞ ¼ f ððxþ vÞ=ξÞ

Fðv=ξÞ ð2Þ

where f(x) is the pdf of the first time to failure. Here the time indexis not necessary because the transition probabilities do not dependon clock time.

Now the transition probabilities can be determined. First,consider the transition from a new system to the first failure attime x1:

Pð0;0Þð1;x1Þ ¼Z x1þ1

x1f ðxÞdx ð3Þ

The integration begins at x1 since by definition the failure couldnot occur before x1 and ends at (x1+1) to account for thediscretization (see assumption 5).

Alternatively, after a repair after the kth failure, the systemtransitions to the next failure at time (vk+xk):

Pðn;vkÞðnþ1;vkþxkÞ ¼Z xkþ1

xkf ðvÞðxÞdx ð4Þ

The expected mean durations for repair and replacement are:

τðn;0Þða¼ 0Þ ¼Z ∞

0xf ðxÞdx

210Time

w(0,0,0)

w(0,0,1)

w

w(1,θ,1)

w

w(0,1,1)

w

w

w

w

Repair

Replace

Fig. 1. State transitions and net rev

and

τ n;vð Þ a¼ 1ð Þ ¼Z ∞

0xf v xð Þdx

2.2.2. Cost and revenue flowFig. 1 shows the possible state transitions for a four-step

problem. The problem is formulated as a semi-Markov decisionprocess, but since it must be discretized to be solved on acomputer, I consider the possible events at each time step. Timeis shown on the horizontal axis while the vertical axis shows thevirtual age. Define wðn; v; iÞ as the net value (revenue – costs)generated over one time step by a system with n accumulatedfailures and virtual age v, at time i. At time i¼0 the system is newand the associated value of the revenue flow to the next time stepis

wð0;0;0Þ ¼ −C0−C2ð0Þ þ C3ð0Þ ð5Þwhere C0 is cost of acquiring the new system.

The first possible failure opportunity is at i¼1 (see assumption4). If the system does not fail, its virtual age increases by one timestep and the value of the revenue in time step 1 is w(0,1,1). If thesystem fails it can either be repaired, as indicated by the arrow andthe dotted line, resulting in revenue in time step 1 of

wð1; θ;1Þ ¼−C1−C2ðθÞ þ C3ðθÞ ð6Þwhere θ is the virtual age of the system after repair and C2 and C3are the operating cost and revenues of a system of virtual age θ.For simplicity the repair cost here is shown as independent of thestate; this dependency can easily be incorporated into the analysis.

Alternatively, if the system fails, it can be replaced, as indicatedby the diamond and the dotted line, resulting in revenue in time

43

0

1

2

3

4

Virt

ual A

ge(0,0,2)

w(0,0,3)

(1,1+θ,2)

w(1,2+θ,3)

(0,2,2)

w(0,3,3)

(2,2θ,2)

w(3,3θ,3)

(0,1,2)

w(0,2,3)

w(0,1,3)

w(1,θ,3)

w(1,1+θ,3)

w(2,2θ,3)

w(2,1+2θ,3)

(1,θ,2)

enues for a four-step problem.

K.B. Marais / Reliability Engineering and System Safety 119 (2013) 76–8780

step 1 of

wð0;0;1Þ ¼−C0−C2ð0Þ þ C3ð0Þ ð7Þ

where the virtual age has now reset to zero.In a similar manner the net revenue at each time step can be

assessed depending on whether or not a failure has occurred andon which action (repair/replace) was taken

wðn; v; iÞ ¼−C2ðvÞ þ C3ðvÞ no failure

−C1−C2ðv′Þ þ C3ðv′Þ repair−C0−C2ð0Þ þ C3ð0Þ replace

8><>: ð8Þ

where the time intervals are assumed to be sufficiently small thatany changes in cost and revenues are negligible.

2.2.3. A dynamic programming formulationThe optimal action for failures occurring for each possible

combination of state and time can be determined using a back-wards recursion from the time horizon as follows. Define W(n, v, i)as the optimal expected net present value looking forward fromtime step i to time step imax for a system with n failures and virtualage v. At each failure instance we then seek a general repair policya such that (c.f. [12])

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

Virtual Age [years]

Ope

ratin

g C

ost a

nd R

even

ue

Operating CostRevenue

Fig. 2. Nominal cost and revenue.

Wðn; v; iÞ���f¼ max

a ¼ 0;1

−C0 þ PVðC3−C2; τðn;0Þ; βÞ þ βR imax−ix ¼ 0 Pð0;0Þ;ð1;xÞWð1; x; iþ xÞdx

−C1 þ PV ðC3−C2; τðn; v0Þ; βÞ þ βR imax−ix ¼ 0 Pðn;vÞðnþ1;v0þxÞWðnþ 1; v0 þ x; iþ xÞdj

8<:

9=; ð9Þ

where v0 is the virtual age after the repair action and PVð⋅Þ is thepresent value of the cost and revenue stream for the expectedmean duration τ and discount factor, β, scaled to the timeinterval size.

Since for simulation purposes W is needed at each time step,when no failure has occurred W is updated according to

Wðn; v; iÞjf ¼ PVðC3−C2; τðn; vÞ; βÞ

þβ

Z imax−i

x ¼ 0P n;vð Þ nþ1;vþxð ÞWðnþ 1; vþ x; iþ xÞdj ð10Þ

The optimal policy is found by setting W to zero for i≥imax andthen working backwards to i¼0 as follows. Begin at time stepi¼ imax−1; in the example shown in Fig. 1 it would be time 3. Foreach virtual age, calculate the value looking forward, assumingthat the system has failed, for the case where maintenance isperformed (triangle node), the case where the system is replaced(square node), and the case where nothing is done (circular node).The virtual ages depend on the chosen repair level. Select theoption that gives the maximum value. This calculation is shown byEq. (9), where the next step W is set to zero. Also calculate thevalue looking forward assuming that the system has not failed;this calculation is shown by Eq. (10), where the next step W is setto zero. The probability of the system failing for each time andvirtual age combination is given by Eqs. (3) and (4). Thus theexpected value of each node is given by

Wðn; v; iÞ ¼ pfailedWðn; v; iÞjf þ ð1−pfailedÞWðn; v; iÞjf ð11Þ

Now, step back one more time step (in the example shown inFig. 1, to time 2). Repeat the previous calculations for the failedand functioning cases, using the next step W's just calculated.Repeat the process until the first time step is reached.

3. Results and discussion

This section illustrates the concepts introduced in the previoussection using a hypothetical system, and shows how changes in

the system and market assumptions affect the optimal mainte-nance policy.

3.1. The nominal case

First, the results are discussed for a set of nominal parameters.For clarity in the graphical presentation of results, the parametersvary only with virtual age and not with the accumulated numberof failures. Also, for comparison with Makis and Jardine [13],Kijima [10], and Love et al. [12], I model the lifetime of a newsystem using a Gamma distribution with density f ðtÞ ¼ λαtα−1=ΓðαÞexpð−λtÞ and first mean passage time (or expected time tofailure) α=λ where the shape parameter α is set equal to the scaleparameter λ. Operating costs and revenues increase and decreasewith system virtual age as follows:

C2ðv=ξÞ ¼ a2bv=ξ2

C3ðv=ξÞ ¼ a3b−ðv=ξÞ=c33

ð12Þ

Fig. 2 shows the costs and revenues using the nominal valuesassumed in the simulation.

Table 1 summarizes the remaining nominal values.

Fig. 3 shows the repair/replace decision that maximizes the netpresent value of the system for each possible virtual age andcalendar time combination. The figure reads as follows. The x-axisshows calendar time and the y-axis shows the virtual age of thesystem. The virtual age of the system cannot exceed the calendartime, but may be as low as zero if the system has been replaced. Inthe black region of the plot the system should be repaired if afailure occurs, while in the gray region the system should bereplaced. For example, if a failure occurs at time four years to asystem with virtual age 1 year (as indicated by the black ellipse onthe figure), it is best to replace the system. However, 6 monthslater at time 4.5 years (the white rectangle on the figure), it isbetter to repair the system. Thus the decision to repair or replacethe system depends on both the system's condition as representedby its virtual age (i.e., failure probability and operating profit) andon the time remaining.

The figure can be constructed by tracking the repair/replacedecision the optimization makes for each time and virtual agecombination. In this case, I used Matlab and created a two-dimensional matrix where the rows represent time and the columns

Time [years]

Virt

ual a

ge [y

ears

]

Revenue = 0Black = RepairGray = Replace

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Fig. 4. Optimal repair/replace decisions for the nominal problem when revenue isnot considered.

Table 1Nominal parameters.

Parameter Nominal value Remarks

α 3 Gamma distributionλ 3C0 10 Replacement costC1 5 Repair costa2, b2 1,1.15 Operating cost parametersa3, b3, c3 20, 1.2, 4 Revenue parameterstmax 5 years Time horizonξ 30 Time slices per yearθ 0.8 Repair levelr 5% Annual interest rate

Time [years]

Virt

ual a

ge [y

ears

]

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Gray = ReplaceBlack = Repair

Fig. 3. Optimal repair/replace decisions as a function of virtual age and calendartime for the nominal problem.

K.B. Marais / Reliability Engineering and System Safety 119 (2013) 76–87 81

virtual age. Each matrix element is then set to zero (repair), or to one(replace), or to undefined (impossible time and virtual age combina-tions). The matrix is then plotted such that the value one correspondsto gray, zero to repair, and undefined to no color.

The lack of definition at the border between the repair andreplacement areas in the graph can be addressed by decreasing thetime step size, which results however in very long run times. Forthis paper the time step is therefore kept at one thirtieth of a year,which yields sufficiently clear results while keeping run time at areasonable time.

It is best to repair the system in two situations: (1) when thevirtual age is low and the failure probability is low and operatingprofit high; or (2) when the system is close to the time horizonand there is not enough time to recoup the investment in a newsystem. Contrary to the cost-centric viewpoint, the system doesnot have a maximum virtual age beyond which it is always betterto replace—this finding arises because the finite time horizonmeans that late investments in new systems cannot be recouped.While incorporating a salvage value may shift the curve somewhatin favor of replacement, it is unlikely to result in a maximum agebecause older systems will have lower salvage values.

Conversely, it is best to replace the system when the failureprobability is high, the operating profit is low (high virtual age),and there is sufficient remaining time horizon to recoup theinvestment in a new system.

Fig. 4 shows the optimal decisions when revenue is set to zeroand the other parameters are kept at the nominal values. In this case,the value maximization becomes the standard cost minimization,since revenues are zero. Because the revenue benefits of replacedsystems with lower virtual ages are not considered, and becausereplacement is more costly, replacement becomes less attractive.

Figs. 3 and 4 show graphically how bringing system revenue intoconsideration alters the optimal policy. For the nominal case, apply-ing the maintenance policy based on minimizing cost rather than thepolicy based on maximizing value results in a 20% penalty on theexpected net present system value. While the specific numbers arefor illustrative purposes only, it is important to note the significantpenalty that results from using a cost minimization approach.

The representation of the optimal repair/replace decisionsshown in Fig. 3 offers a convenient graphical summary of the bestaction to take in response to a failure at any time and virtual age. Itcan be prepared in advance for a particular system and marketconditions. Where there is uncertainty about future market con-ditions (e.g., future operating revenue), several scenarios can beexamined, as shown in the next section.

3.2. Variation in optimal maintenance strategy

This section explores the impact of varying the followingparameters on the optimal maintenance policy relative to thenominal values given in Table 1: replacement and repair costs;operating costs and revenues; time horizon; failure characteristics;and the repair level.

3.2.1. Replacement and repair costsReplacement, while more expensive, offers benefits in terms of

both reduced probability of failure and increased operating profit. Iconsider here how the relative value of repair and replacementchanges as their costs are varied. Fig. 5 shows the change in theexpected maximum net present value (NPV) as the replacementcost is increased relative to the nominal repair cost. As the ratio rof replacement to repair cost increases, the present valuedecreases, to the point r0 where it becomes negative. The variationin the crossover value r0 as a function of the problem parameters isleft as a subject for future work. For a given set of operating cost

K.B. Marais / Reliability Engineering and System Safety 119 (2013) 76–8782

and revenues, such a system is always unprofitable, regardless ofmaintenance policy. Similarly, increasing both repair and replace-ment costs will also eventually result in a system that is alwaysunprofitable to operate, regardless of maintenance policy.

Fig. 6 shows the repair/replace decision plots when the repairand replacement costs are equal, and when replacement is muchmore costly than repair, relative to the nominal case shown inFig. 3. When the replacement cost is equal to (or less than) therepair cost, there is nothing gained by not replacing and thedecision is always to replace. As the replacement cost increasesrelative to the repair cost, the net value of replacement decreases.Finally, once the replacement cost becomes too high, it is alwaysbetter to repair.

In the nominal case the repair costs are constant with virtualage, limiting the cost associated with an older system. However, ifrepair costs increase with virtual age, repair becomes less attrac-tive because the immediate benefit of lower repair/replacementcosts is smaller and the reliability and operating profit of an olderrepaired system are lower. Consider for example variable costs ofthe following form, as also shown in Fig. 7 (cf. [12]):

C1ðv=ξÞ ¼ pðv=ξÞ1:5 þ 1 ð13ÞSetting p¼0 results in constant repair cost of 1.Fig. 8 shows the resulting optimal repair/replace decisions for

these three curves, together with a constant cost curve (p¼0,C1¼1). As repair cost increases more rapidly with virtual age(larger p), the relative value of replacement increases, because(1) repair approaches the cost of replacement and (2) the expectedcost of future repairs on a repaired system is higher. Thus systemswith rapidly increasing repair costs will result in more frequent

Time [years]

Virtu

al a

ge [y

ears

]

0 1 2 3 4 50

1

2

3

4

5C0 = C1 = 5Gray = Replace

Fig. 6. Repair/replace decision when replaceme

Fig. 5. Impact of replacement and repair costs.

replacements, and, in the extreme case where repair costsapproach replacement costs, replacement will always be theoptimal decision.

3.2.2. Operating costs and revenuesThe value-centric model proposed here allows the impact of

operating revenues on maintenance decisions to be captured, incontrast to cost-centric models, which do not consider this aspect.In this model the decision to repair or replace a system is drivenboth by the maintenance and operating costs and by the systemrevenues. The impact of varying repair and replacement costs hasbeen explored above; here, I consider the impact of varyingoperating costs and revenues. System operating profit can beincreased in two ways: by decreasing operating costs, or byincreasing operating revenue. While the numerical impact of thesechanges may be the same, they are achieved in different ways.Operating costs depend primarily on the system design and thestructure of the organization. Operating revenues, in contrast,depend primarily on the market conditions (e.g., total market size,market share). Stated simplistically, operators have two levers toincrease profit: a design lever to decrease operating cost, and amarketing lever to increase revenues. In turn, the operating profitaffects the optimal maintenance strategy, and the maintenancestrategy can be used to increase the total value under a given set ofmarket conditions.

To make the importance of operating cost and profit salient,consider first the somewhat stylized situation where operatingcost and revenue are independent of system condition. Here weexpect that since the profit does not decrease with systemdeterioration, investments in replacement do not yield much

Time [years]

Virtu

al a

ge [y

ears

]

0 1 2 3 4 50

1

2

3

4

5C0 = 10C1 = 50Black = Repair

nt costs are varied relative to repair costs.

1 1.5 2 2.5 3 3.5 4 4.5 51

2

3

4

5

6

7

Virtual Age [years]

Rep

air C

ost

p = 0.05

p = 0.2

p = 0.5

Fig. 7. Variable repair cost curves.

Time [years]

Virt

ual a

ge [y

ears

]

p = 0Black = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Time [years]

Virt

ual a

ge [y

ears

]

p = 0.05Black = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Time [years]

Virt

ual a

ge [y

ears

]

p = 0.2Black = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Time [years]

Virt

ual a

ge [y

ears

]

p = 0.5Black = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Fig. 8. Repair/replace decision when repair cost increases with virtual age.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

Virtual Age [years]

Ope

ratin

g C

ost a

nd R

even

ue

Operating CostRevenue

Fig. 9. Operating cost and revenue variation (b2¼1.5; c3¼1). The black curvesshow the nominal case, the gray curves show the more rapidly changing case.

K.B. Marais / Reliability Engineering and System Safety 119 (2013) 76–87 83

value. The only benefit of replacement lies in the reduced failureprobability; the additional benefit of increased profit at low virtualages is lost.

In contrast, if operating profit decreases rapidly as the systemdeteriorates because for example the system uses more fuel or isless attractive to customers, replacement becomes more attractive.In this case, replacement offers both reduced failure probabilityand increased operating profit. To investigate this case, I consid-ered the more rapidly changing cost and revenue curves shown ingray in Fig. 9.

Fig. 10 shows the resulting optimal repair/replace decisionswhen profit is constant, and when profit decreases rapidly withsystem condition. When profits are constant, replacement is lessattractive, as expected. However, when profits decrease morerapidly, replacement becomes more attractive. Replacementallows the higher operating profits at lower virtual ages to berealized more often, and offers the added benefit of lower prob-ability of failure in the future.

Next, consider the case where the operating profit is lower orhigher than the nominal case across all system states. A higherprofit may for example correspond to a market in which thesystem's revenues have increased (e.g., increased demand forairline tickets over the summer), or to a system or organizationaldesign with lower operating costs (e.g., a more fuel efficientaircraft, or a new labor agreement). The optimal strategy is notobvious: repair may be more attractive because profits are

increased in all system conditions, or, replacement may be moreattractive because the investment is easily recouped.

Fig. 11 shows the impact on the optimal maintenance decisionof varying the operating profit by increasing the operating revenuerelative to the operating cost. When the operating profit is low,replacement is less attractive because the return on investment ina new system is low. As the profit increases, the investment inreplacement becomes more valuable because of the benefits ofincreased revenue and reduced probability of failure. Therefore the

Time [years]

Virt

ual a

ge [y

ears

]

Constant ProfitBlack = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Time [years]

Virt

ual a

ge [y

ears

]

More RapidBlack = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Fig. 10. Repair/replace decision when operating profit is (a) constant, and (b) decreases rapidly.

Time [years]

Virt

ual a

ge [y

ears

]

0 1 2 3 4 50

1

2

3

4

5C3 = 0.5 NominalGray = ReplaceBlack = Repair

Time [years]

Virt

ual a

ge [y

ears

]C3 = 2*NominalGray = Replace Black = Repair

0 1 2 3 4 50

1

2

3

4

5

Fig. 11. Repair/replace decision when operating revenue is varied relative to operating cost.

Time [years]

Virt

ual a

ge [y

ears

]

Black = Repair

0 0.5 10

0.2

0.4

0.6

0.8

1

Time [years]

Virt

ual a

ge [y

ears

]

Black = RepairGray = Replace

0 2 4 6 8 100

2

4

6

8

10

Fig. 12. Repair/replace decision when the time horizon is (a) decreased, and (b) increased.

K.B. Marais / Reliability Engineering and System Safety 119 (2013) 76–8784

maintenance strategy should depend not only on the systemcharacteristics as represented by operating and maintenance costs,but also on the market conditions as represented by the operatingrevenue.

3.2.3. Time horizonBecause the optimal decision depends also on the time remain-

ing, the maintenance strategy also depends on the planninghorizon. Fig. 12 shows the optimal repair/replace decisions for

Time [years]

Virt

ual a

ge [y

ears

]

= = 1Black = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Time [years]

Virt

ual a

ge [y

ears

]

= = 10Black = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Fig. 13. Repair/replace decision for changing failure characteristics.

Time [years]

Virt

ual a

ge [y

ears

]

Zero revenue = = 1

Black = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Time [years]

Virt

ual a

ge [y

ears

]

Zero revenue = = 10

Black = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Fig. 14. Repair/replace decision for changing failure characteristics when revenue is not considered.

K.B. Marais / Reliability Engineering and System Safety 119 (2013) 76–87 85

the nominal system under two different planning horizons. Whenthe planning horizon is short, it is better to repair, because(1) there is little time to recoup the investment in a new system,and (2) the increase in reliability is small since the system's virtualage is always low. In contrast, as the planning horizon increases,replacement becomes more valuable. It is never advisable toreplace near the end of the planning horizon, unless this actionextends the planning horizon. Thus for example an organizationthat has been maintaining a wind turbine under a maintenancecontract or warranty would choose to repair the system as thecontract or warranty approaches expiration, whereas the turbineowner might choose to replace the system in the hope ofextending the wind farm's lifetime.

3.2.4. Failure characteristicsIf costs and revenues are kept constant, increasing the system

reliability increases the system expected value (See Saleh andMarais [21] for a discussion of why the cost of additional reliabilitymay not always be recouped.). The optimal maintenance strategyalso changes in interesting ways, as shown in Fig. 13. When theprobability of failure is low and does not increase significantly asthe system ages (i.e., low gamma parameters, α and λ), repair isrelatively more attractive, since the benefits of increased reliabilityoffered by replacement are small. As the gamma parameters, α and

λ, are increased, the probability that the system will fail increasesas the system ages (as does its variance). In this case, replacementbecomes more valuable since it returns the system to a lowervirtual age and hence lower probability of failure.

The optimal decision determined using the value approach issignificantly different from that obtained when revenue is notconsidered, that is, when cost is minimized, as shown in Fig. 14.For both the low and high failure probability scenarios, ignoringrevenues results in fewer replacements. When revenue is notconsidered, only the reduced failure probability and operating costbenefits offered by newer systems are considered. Therefore thebenefits of new systems are underestimated, resulting in a valuesub-optimal strategy, as discussed earlier.

3.2.5. Repair levelThe repair level indicates how much the condition of the

system is improved by a repair. How does the optimal main-tenance strategy change as repair is made more or less extensive?Recall that by convention low values of the repair level θ meanthat the repair level is high (reduction in virtual age is large), andhigh levels mean that the repair level is low. As θ is decreased, therepair approaches perfect maintenance. When θ¼0, repair andreplacement are equivalent.

Time [years]

Virt

ual a

ge [y

ears

]

= 0.98 Black = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Time [years]

Virt

ual a

ge [y

ears

]

0 1 2 3 4 50

1

2

3

4

5

= 0.8Gray = ReplaceBlack = Repair

Time [years]

Virt

ual a

ge [y

ears

]

= 0.4Black = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Time [years]

Virt

ual a

ge [y

ears

] = 0.2

Black = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Fig. 15. Repair/replace decision for decreasing repair levels.

Time [years]

Virt

ual a

ge [y

ears

]

= 0.98 C1 = 4.75Black = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Time [years]

Virt

ual a

ge [y

ears

]

= 0.2C1 = 7Black = RepairGray = Replace

0 1 2 3 4 50

1

2

3

4

5

Fig. 16. Repair/replace decision for decreasing repair levels with repair cost adjustment.

K.B. Marais / Reliability Engineering and System Safety 119 (2013) 76–8786

Fig. 15 shows the optimal repair/replace decisions for thenominal system as the repair goes from near-minimal (θ¼0.98)to near-perfect (θ¼0.2). Here the nominal constant repair costs areassumed. When repair is near-minimal, it offers little benefit interms of reduced failure probability or operating profit.

Replacement is therefore more attractive and there is a cleardivision between repair and replacement, as shown in the firstpart of the figure. As repair becomes more extensive, it becomesmore like replacement and offers greater benefits in terms ofreduced failure probability or operating profit. Repair therefore

K.B. Marais / Reliability Engineering and System Safety 119 (2013) 76–87 87

becomes more attractive, and the division between repair andreplacement becomes less defined. Stated differently, for low θ theselection of repair/replace is quite sensitive to time and virtual age—this sensitivity occurs because high repair levels have similareffects on reliability and operating profit to replacement. Finally,when repair is near-perfect, there is little distinction betweenrepair and replacement.

Note that this trend persists even when the repair cost isincreased as the repair becomes more extensive, as shown inFigs. 16 for θ¼0.98 and θ¼0.2.

This section has explored the impact on the optimal main-tenance policy of varying the replacement and repair costs; theoperating costs and revenues; the time horizon; the failurecharacteristics; and the repair level.

4. Conclusion

In previous work we have argued that while maintenance istraditionally seen as a cost-driver, this view is limited and ignoresthe contribution of maintenance to the value of a system. Thispaper shows how the value view of maintenance can be applied tothe familiar general repair problem. I used a semi-Markov decisionprocess coupled with a discounted cash flow techniques toestimate the net present value of a system under differentresponses to failure, and then used a dynamic programmingapproach to identify the optimal actions for any given time andsystem condition, represented here by the system's virtual age.

The analysis showed that the value perspective results indifferent decisions and that ignoring system revenue results invalue sub-optimal strategies that decrease the net value of thesystem.

This approach provides a quantitative basis on which to basemaintenance decisions and thus ensure maximum expectedvalue. In particular, the results show:

1.

The optimal action for a given time and condition changes asreplacement and repair costs change, and identifies the point atwhich these costs become too high for profitable systemoperation. This approach can therefore be used to identify“lemon” designs that cannot be rescued through carefulmaintenance.

2.

The impact of planning horizon on the optimal action. Forshorter planning horizons it is better to repair, since there is notime to reap the benefits of increased operating profit andreliability. As the planning horizon grows, replacementbecomes more attractive.

3.

The impact on the optimal maintenance policy of the system'sfailure characteristics. In particular, it is better to replacesystems where the probability of failure increases rapidly withdeterioration. This approach can therefore be used to assess thevalue of investing in higher reliability, either through inher-ently more reliable systems, or through preventivemaintenance.

4.

The impact on the optimal maintenance policy of the repairlevel. As the repair level is decreased, the relative value ofreplacement increases, because lower repair results in lowerreliability gains. This approach can therefore be used todetermine the optimal repair level.

This work opens several interesting avenues for future work.While virtual age and number of failures are useful proxies for thecondition of many systems (e.g., mileage and failure count for avehicle), it would also be useful to consider more direct measuresof system state such as those offered by condition monitoringsystems. Another extension here is to allow a range of failures asdefined by repair cost and time to occur in each state.

A second area for future work revolves around more detailedmodeling of the market and operating environment. Earlier Ialluded to the idea of using different scenarios to identify optimalmaintenance policies, another approach is to model the marketstochastically (cf. [5]). The effect of relaxing the assumption thatrepair and replacement times are negligible should also beinvestigated.

Finally, while this approach allows the optimal maintenancepolicy to be determined ahead of time, it is computationallyintensive, and the problem dimension increases rapidly as theplanning horizon extends. Computationally efficient approaches tosolving the problem would be a fruitful venue for future work.

Acknowledgments

This work was partially funded through a Purdue ResearchFoundation Summer Faculty Fellowship.

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