Joint condition-based maintenance and load-sharing optimization for two-unit systems with economic dependency

Joint condition-based maintenance and load-sharing optimization for two-unit systems with

economic dependency

Uit Het Broek, Michiel A.j.; Teunter, Ruud H.; Jonge, Bram De; Veldman, Jasper

Published in:

European Journal of Operational Research

DOI:

10.1016/j.ejor.2021.03.044

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Uit Het Broek, M. A. J., Teunter, R. H., Jonge, B. D., & Veldman, J. (2021). Joint condition-based

maintenance and load-sharing optimization for two-unit systems with economic dependency. European

Journal of Operational Research. https://doi.org/10.1016/j.ejor.2021.03.044

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ContentslistsavailableatScienceDirect

European

Journal

of

Operational

Research

journalhomepage:www.elsevier.com/locate/ejor

Innovative

Applications

of

O.R.

Joint

condition-based

maintenance

and

load-sharing

optimization

for

two-unit

systems

with

economic

dependency

Michiel

A.

J.

Uit

Het

Broek,

Ruud

H.

Teunter,

Bram

de

Jonge

∗

,

Jasper

Veldman

Department of Operations, Faculty of Economics and Business, University of Groningen, Nettelbosje 2, Groningen 9747 AE, the Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 9 March 2020 Accepted 20 March 2021 Available online xxx Keywords: Maintenance Condition-based maintenance Condition-based production Load-sharing Economic dependency

a

b

s

t

r

a

c

t

Manyproductionfacilitiesconsistofmultipleandfunctionallyexchangeableunitsofequipment,suchas pumpsorturbines,thatarejointlyusedto satisfyagivenproductiontarget.Suchsystemsoftenhave toensurehighlevelsofreliabilityand availability.Thedeteriorationratesoftheunitstypicallydepend ontheirproductionrates,implyingthattheoperatorcancontroldeteriorationbydynamically reallocat-ingloadamongunits.Inthisstudy,weexaminethevalueofcondition-basedload-sharingdecisionsfor two-unitsystemswitheconomic dependency.We formulatethe systemas aMarkovdecisionprocess and provideoptimal jointcondition-based maintenanceand productionpolicies. Ournumericalresults showthat,dependentonthesystemcharacteristics,substantialcostsavingsofupto40%canberealized comparedtotheoptimalcondition-basedmaintenancepolicyunderequalload-sharing.Thestructureof theoptimalpolicyparticularlydependsonthemaintenancesetupcostandthepenaltythatisincurred iftheproductiontargetisnotsatisfied.Forsystemswithhighsetupcosts,theclusteringofmaintenance interventionsisimprovedbysynchronizingthedeteriorationoftheunits.Onthecontrary,forlowsetup costs,thedeteriorationlevelsaredesynchronizedandthemaintenanceinterventionsarealternated.

© 2021 The Author(s). Published by Elsevier B.V. ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Many production facilities consist of multiple identical and functionally exchangeable units that are jointly used to satisfy a production target. These units deterioratedue to loadand stress causedbyproductionandeventuallyrequiremaintenanceinorder tokeep thesystemin,orbringitbackto,anoperatingcondition. The resulting maintenance expenses often constitute a substan-tial partofthetotalbudget ofproductionfacilities,andcaneven formup to70percentofthetotalproductioncosts (Bevilacqua& Braglia,2000).Manystudiesaimtoreducethesecostsby develop-ingcondition-basedmaintenancepoliciesandshowthatsuch poli-ciesreducecostswhileimprovingavailabilityandproductivity.

Another option to improve the cost efficiency of production facilities is to control the deterioration of its units by adopt-ing condition-based production policies (Uit Het Broek, Teunter, DeJonge,&Veldman,2020;UitHetBroek,Teunter,DeJonge, Veld-man, & Van Foreest, 2020). Such policies exploit the relation

be-∗_{Corresponding author.}

E-mail addresses: a.j.uit.het.broek@rug.nl (M.A.J. Uit Het Broek),

r.h.teunter@rug.nl (R.H. Teunter), b.de.jonge@rug.nl (B. de Jonge), j.veldman@rug.nl

(J. Veldman).

tween theproduction rateandthedeterioration rateby dynami-callyadjustingtheproductionratebasedonconditioninformation. Althoughothers haveshown theeffectiveness ofcondition-based productionpolicies forsingle-unit systems,there are,to the best ofourknowledge, nostudies devotedtocondition-based produc-tionpoliciesformulti-unitsystemsthatconsiderdynamic realloca-tionofloadamongunits.Optimalmaintenance policiesfor multi-unitsystemsareoftenmoreadvancedthanforsingle-unitsystems because ofthe various typesof dependenciesthat existbetween units(OldeKeizer,Flapper,&Teunter,2017).Itisthereforealso ex-pected that condition-based production policies will be different formulti-unitsystems.

The mostcommonlystudied dependencyis positive economic dependency such as a fixed maintenance setup cost that is in-dependent of the number of units that are maintained. In such cases,clusteringmaintenance interventionsforvariousunitsis of-tenmorecost-efficientthanperformingthemseparately.However, clustering maintenance for units with different degradation lev-elsimpliesthat maintenanceisperformedunnecessarily earlyfor unitswithrelativelylowlevelsofdeterioration.Insuchsituations, an interesting questionis whetherit can be profitableto control thedeteriorationprocessesbyreallocatingloadfromahighly dete-rioratedunittoalowerdeterioratedunit.Herebytheoperatorcan

https://doi.org/10.1016/j.ejor.2021.03.044

0377-2217/© 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

Please cite thisarticle as: M.A.J. Uit HetBroek, R.H.Teunter, B. de Jonge et al., Joint condition-based maintenance andload-sharing optimizationfortwo-unitsystemswitheconomic dependency,European JournalofOperationalResearch,https://doi.org/10.1016/j.ejor. 2021.03.044

activelysynchronizethedeteriorationlevelsoftheunits,which in-troduces opportunities to improve the clustering of maintenance interventions.

In thisstudy, we presentafirst exploration ofthe benefits of condition-basedload-sharingdecisionsformulti-unitsystemswith economicdependency.Aswearethefirsttodoso,werestrictour (numerical)investigationtotwo-unitsystems,whichalsoallowsus topresentmanyinsights intwo-dimensionalgraphs thatareeasy to interpret. Thedeterioration rates ofthe unitsdepend on their respectiveloads,implyingthattheoperatorcancontroltheir dete-riorationbydynamicallyreallocatingloadamongunits.We formu-latetheproblemasaMarkovdecisionprocessandusethisto de-termineoptimalmaintenanceandproductionpolicies.Ourresults showthatcondition-basedload-sharingimprovestheeffectiveness ofcondition-basedmaintenancepolicies,andthatitseffectiveness heavily depends on the degree of overcapacity. Throughout this study,weusethetermovercapacitytorefertosystemswherethe maximumproductioncapacity/rateofall unitscombinedislarger than the target system production rate. Furthermore, by redun-dancywerefertosystemswithsufficientovercapacitytostillreach theproductiontargetifonemachineisnotfunctioning.Substantial cost savings up to20% can be obtainedforsystems with overca-pacity, andthesesavings increaseup to40% forsystemswith re-dundancy.Thesavingsaretheresultoffewerfailures,fewer main-tenanceactionsperunit,improvedmaintenanceclustering,and re-ducedrisksofproductionshortages.

An insightfulobservation is that condition-based load-sharing policies are also effective for systems without economic depen-dency. For such systems, cost savings are possible by actively desynchronizingthedeteriorationlevelsoftheunits.Moreover,for manysystems,therearescenarios inwhichthemostdeteriorated unittakesoverloadfromtheleastdeterioratedunit.Aninteresting sideeffectofadoptingcondition-basedload-sharingpoliciesisthat doing sonotonlyreducestheexpectedcostbutalsoitsvariance, implyinghigherfinancialrobustness.

The remainder of this study is organized as follows. In

Section 2,we discuss theliterature on maintenance and produc-tiondecisionsandspecificallyaddressstudiesthatconsider multi-unitsystemswithdependencybetweentheunits.InSection3,we formally describe the systemthat we consider. TheMarkov deci-sionprocessformulationusedtoobtainoptimalpoliciesisgivenin

Section4.InSections5–7,weexaminethestructureoftheoptimal policiesandtheassociatedcostsavings.We concludeandprovide futureresearchopportunitiesinSection8.

2. Literaturereview

In this study, we introduce condition-based load-sharing de-cisions and combine this with condition-based maintenance, re-dundancy, and economic dependency. For extensive reviews on condition-based maintenance we refer to De Jonge and Scarf (2020) andAlaswadandXiang(2017).Forareview on condition-based maintenance formulti-unit systems withdependencieswe refertoOldeKeizeretal.(2017).Intheremainderofourliterature review, we first discuss studies on condition-based maintenance that also include redundancyor economic dependency. Then we zoomin onstudies withloadsharing,which canbe divided into failure-basedanddegradation-basedloadsharing.Inbothstreams, the loadsharing dynamics are exogenously given andcannot be usedasafeaturetocontrolthedeteriorationofunits.Wealso dis-cussstudies thatexamine condition-basedproductionpolicies for single-unit systems, andwe conclude by discussing a number of papersaboutthe(re)allocationofcomponents/unitstoimprovethe performanceofasystem.

The literature on condition-based maintenance for multi-unit systems is rich, and both redundancy (Lu & Jiang, 2007;

Wang, Zheng,Li, Wang, & Wu, 2009) and economic dependency (Castanier, Grall, & Bérenguer, 2005; De Jonge, Klingenberg, Te-unter, & Tinga,2016; Do, Barros, Bérenguer,Bouvard, & Brissaud, 2013) are addressed in various settings. Also the joint effect of redundancy and economic dependency is studied, including 1-out-of-N systems(Li,Deloux,& Dieulle,2016),k-out-of-N systems (Olde Keizer, Teunter, & Veldman, 2016), and series-parallel sys-tems (Zhou, Zhang, Lin, & Ma, 2013). The aforementioned stud-iesinvestigatecondition-basedmaintenancepoliciesformulti-unit systems with either redundancy, economic dependency, or both, but none of them include the effect of load sharing. The obser-vation that research on the integration ofcondition-based main-tenance withload sharing is lacking is also brought forward by

OldeKeizer etal. (2017) andOlde Keizer,Teunter, Veldman, and Babai(2018).

Others have addressed multi-unit systems with failure-based load sharing and degradation processes that can be monitored. Underfailure-based load sharing,the totalload isequally shared among all functioning units and thus the load faced by a unit canonlychangeuponfailureofanotherunit.Zhang,Wu,Lee,and Ni (2014) and Zhang, Wu, Li, and Lee (2015) investigate main-tenance policies with an opportunistic threshold for preventive maintenance.Theyconsiderasystemwhoseunitsdeterioratewith anominalrateaslongasallunitsarefunctioningandthe deterio-rationrateofallunitsaccelerateonceatleastoneunit hasfailed.

Marseguerra, Zio, and Podofillini (2002) analyze condition-based maintenancepoliciesforseriesandparallelsystems.Theyconsider policiesinwhichthemaintenancedecisionforaunitonlydepends onitsownhealthandnotontheentiresystemstate.OldeKeizer et al.(2018) examine optimal condition-based maintenance poli-cies for1-out-of-N systems with economic dependency andload sharing.They modelthe deterioration rateof unitsas a function of the numberof functioningunits. Their resultsshow that it is important to base decisions on the entiresystem state and that load-sharingeffects shouldnot be ignored in makingthose deci-sions. Theyalsofind that postponingmaintenance of failedunits canbecost-effectiveinordertoimprovetheclusteringof mainte-nancetasks.Zhao,Liu,andLiu(2018)considerthe reliabilityofa multi-unitsystemwhoseunitsdeteriorateaccordingtoaBrownian motion.Inthesestudies,thetotalloadprocessedbythesystemis constant over time andis equally shared among the functioning units.Hence, reallocatingloadistriggered byfailures onlyandis not used asan opportunity todynamically control the deteriora-tionprocessesofunits.

Another research stream that includes load sharing is degradation-based load sharing. In contrast to failure-based load sharing, load is not reallocated upon failure, but the load ofunitsgradually increaseswhen thedeterioration levelofother units increases. Many settings are addressed in this research stream, including settings with condition monitoring and with economic dependency (see, e.g., Do, Assaf, Scarf, & Iung, 2019; Do,Scarf,& Iung,2015;Rasmekomen &Parlikad, 2016;Zhou,Lin, Sun, & Ma, 2016). Studies in this stream clearly differ from our research since, similar to the failure-based load sharing stream, the deterioration processes are not controlled by dynamically reallocatingloadamongunits.

All the above-mentioned studies consider condition-based maintenancepoliciesforsystemswithloadsharing.However,none of them utilize condition information to determine the load ap-plied to a unit by controlling the production rate. We note that thestaticequalload-sharingruleasaddressedbytheabove stud-iesis realisticformanypractical systems.Forinstance,ifone ca-bleofacable-supportedbridgefails,thisincreasestheloadfaced by the other cables andan operator can not dynamically decide whichcableshouldtake overtheload.Inpractice,however,there are also many examples where the operator can determine how

thetotalloadisallocatedamongunits.Thisholdsinparticularfor manufacturing systems,systemsusedintheprocessindustry,and energysystems. Forinstance,forawind farm onlythetotal pro-duction at the system level isrelevant, and not how the load is distributedamongtheindividualturbines.

Uit HetBroek,Teunter, DeJonge, Veldman et al.(2020) study condition-based production rates for a single-unit system for whichthenextmaintenanceinterventionisalreadyscheduled.The productionratedirectlyaffectsthedeteriorationrateandcanthus be used to control the deterioration process. Uit Het Broek, Te-unter, DeJonge and Veldman (2020) extendthisto thejoint op-timization of condition-based maintenance andproduction. Their studyshowsthatcondition-basedproductionandmaintenance de-cisions can complementeach other and that the effectiveness of both strongly depends on various characteristics of the system. There aresomeother studiesoncondition-based production poli-cies,buttheseassumethattheproductionratedoesnotaffectthe deteriorationrateofthesystem(see,e.g.,Iravani&Duenyas,2002; Sloan,2004).Althoughthesestudiesconsidercondition-based pro-duction,noneofthemaddressesthevalueofdynamicallysharing loadbetweenmultipleunits.

We finallydiscussanumberofpapersaboutthe(re)allocation of components/units toimprove the performance ofa system. In manysystems,thepositionsofunitswithinthesystemaffecttheir degradationandtherebythesystemreliabilityandlifetime.The re-latedcomponentallocationproblemhasbeenconsideredbymany authors,andwereferinterestedreaderstoZhu,Fu,Yuan,andWu (2017), who propose a new approach and also review the state-of-the-art. Inrecentyears,some authorshaveconsidered the op-tion to reallocate components, in order to shift workloads and thereby affectthedeteriorationofcomponents.Fu,Yuan, andZhu (2019a) andZhu, Fu,andYuan(2020) deriveoptimalreallocation decisions under the objective to maximize the system lifetime, whilst satisfyingrequirementsforreliabilityandsafety.They con-sidervariousdegradationmodels(linearandexponential)and sys-temstructures(includingparallel,series,andk-out-of-n).Fu,Yuan, andZhu(2019b)andSun,Ye,andZhu (2020)goone stepfurther and consider the joint optimization of reallocation and mainte-nancedecisions.Fuetal.(2019b)considerageneralsystem struc-tureandanalyzeastrategythatperformsperiodicpreventive sys-temreplacements,periodicpreventivecomponentreallocation be-tween systemreplacements,aswell asminimalrepairs for emer-gencyfailures.Sunetal.(2020)limittheirattentiontoseries struc-tures, where the degradation rate of the component installed in slot 1 is the largest and in slot N isthe smallest, butallow the maintenancedecisionstobecondition-based.Inthisrespect,from thepapersoncomponentreallocation,theirsistheclosest tothis paper.However,whereastheyindirectlyaffectdegradationby real-locatingcomponents,wedirectlydosobyalteringtheproduction rates.Anothernote isthat Sunetal.(2020)considermaintenance andreallocationdecisionsthatare fullyspecifiedby thevaluesof a few decision variables,allowing themto consider a continuum of degradation states.Because we use Markovdecisionprocesses todetermineoptimalpolicies,weformulateourproblemina dis-cretetimesettingwithdiscretedegradationstates.

We conclude that condition-based maintenance, redundancy, economic dependency,andloadsharingarewell studiedin isola-tion,butarescarcelyjointlyaddressed.Moreover,condition-based productionratedecisionshavereceivedlittleattention,evenin iso-lation of the other effects. To the best of our knowledge, there is no study that examines the value of dynamically redistribut-ing loadamong unitsby directlyaltering production ratesbased on conditioninformation.The aim ofourstudyis to explorethe cost savings potential ofsucha policy.Oursolutionmethodology is the same as in Olde Keizer et al. (2018), Uit Het Broek, Te-unter, DeJonge andVeldman (2020),andUitHetBroek,Teunter,

DeJonge, Veldmanetal.(2020),inthat wemodelour systemas aMarkovdecisionprocessandusevalueiteration(combinedwith policyiteration)toobtainoptimalpolicies.However,differentfrom them,wefocusonloaddistributionoverfunctionallyexchangeable units.

3. Problemdescription

Weconsiderasystemconsistingoftwoidenticaland function-ally exchangeable units. The production rate of each unit is ad-justable over time andaffects the deterioration rateofthat unit. There is an economic dependency between the units as carry-ingoutmaintenance incursafixedsetup cost,independentofthe number ofunits that are maintained. We do not consider struc-tural dependency of the two units,different from for instance a seriessystem ora consecutive-1-out-of-2 system. We modelthis systemin discrete time, withthe time unit normalizedto 1. We consideraninfinitetimehorizon,thatis,analyzethelong-run av-erageperformance.

The set of possible deterioration states for each unit is

{

0,1,...,L

}

, where 0 is the (as good as) new state and Lis the failed state. The state of unit i in period t is denoted by xi

(

t

)

,

and x

(

t

)

=

(

x1

(

t

)

,x2

(

t

))

is the deterioration state (condition) of

theentiresystem.Thesetofpossibleproductionratesforeachunit isU=

{

i/m

|

i=0,1,...,m

}

, where0 isthe idlemode and1the maximumproductionrate. Naturally,unitsinthefailedstate can-notproduceandtheirproductionrateisfixedtozero.Inallother conditionstates,allpossibleproductionratesareallowed.The pro-duction rate of a unit directly affects its deterioration rate. Let Pu

(

x,x

)

denote theprobability to transit fromdeteriorationstate

xtodeteriorationstatexinatimeunit,whentheproductionrate isu.

Theoperatorschedulesmaintenanceinterventionsbasedonthe conditionofthesystem.Onceamaintenanceinterventionhasbeen scheduled,maintenance willbecarriedout aftera fixedplanning timeofstimeunits.Also,whenreachingtheendoftheplanning time,maintenancecannotbefurtherdelayed,andduringthe plan-ningtime noadditionalmaintenance interventionscanbe sched-uled.Themaintenance actionsthemselvesareassumedtorequire anegligibleamountoftime.Thisisoftenrealistic asrepairtimes aretypicallyhourstodayswhereasexpectedlifetimesareoftenin theorder ofyears.Planning maintenance,however, cantake sev-eral months due to lengthy lead times for specialized tools and equipment,andtherefore we doconsider aplanning time inour model.Forthesamereason,wedonotconsiderafasteremergency maintenancerepairoptionincaseaunit hasfailed,since mainte-nancecannotbeperformeduntilalltoolsandequipmentare avail-able. This settingis realistic in manyscenarios. Forinstance, re-placinglargecomponentsofthegearboxofoffshorewindturbines requires specialized equipment such asjack-up vessels. Typically thesevessels haveto be charted months in advance,which pro-hibitsadditionalemergencyrepairsforthesetasks.Ofcourse,there maybepracticalsituationswherethedeliveryoftoolsand/or free-ingup ofequipmentcanbeexpedited,butthatisnot considered inoursetting.Maintenanceactionsareassumedtobeperfect,that is,theyrestoretheconditionofaunittotheas-good-as-new con-dition.

TheorderofeventsinaperiodispresentedinFig.1.Wemodel eachperiodasasequenceofthreeconsecutivestages.Inthefirst stage, the systemstate is observedand we determinewhether a new maintenance intervention will be scheduled. In the second stage, wedetermine whethermaintenance willbe carriedout. In thethird stage, we choosethe productionratesof theunits,and wemodelthedeteriorationofthesystem.

We include costs for maintenance and forloss of production. Thecostofmaintainingaunitdependsonitsconditionatthe

mo-Fig. 1. Order of events in each decision epoch.

mentofmaintenance.Maintainingafunctioningunitisreferredto aspreventive maintenance andcostscpm.Corrective maintenance

isrequiredforaunitthathasfailedandcostsccm≥ cpm.Thefixed

setup costformaintenance isdenoted bycsetup andisincurredat

themomentthatamaintenanceinterventionisscheduled. The costscorresponding totheproductiondecisiondependon the total (achieved)production rate uˆ=u1+u2 and a given

tar-getsystemproductionrate

κ

.Aconstantpenalty

π

˜ isincurredper time periodthatthetargetisnotsatisfied,i.e.,ifuˆ<

κ

.Moreover, there is a variable penalty per time period

π

1 and a bonus per

time period

π

2that areproportional totheshortageand

overpro-duction, respectively. Note that this cost structure isflexible and allows us to study systems with both hard and soft production constraints, andsystemswherefailures aresevere ornot. For in-stance, systems forwhich shortagesmust be avoided atall costs andforwhichthereisnobenefitofoverproduction(e.g.,gas tur-binesthatmustprovideareliablegasflowwithasteadypressure) can be analyzed by using an extremely high constant penalty

π

˜ anda bonusof

π

2=0.Productionfacilities thatpurely maximize

profit(i.e.,production revenuesminusmaintenance costs)can be analyzed by settingthe target systemproduction rate

κ

equalto themaximumproductioncapacity,theconstantpenalty

π

˜ thatis incurred whenthetarget isnot reachedto zero,andthe variable penalty for shortages

π

1 equal to the production value. Systems

that aimtominimizecosts undera givenproductiontargetwhile being able to sell overproduction for lower prices (e.g., offshore windfarms)canbeanalyzedbychoosingpositivevaluesfor

π

˜,

π

1,

and

π

2.

The objective isto determine the jointcondition-based main-tenance and production policy that minimizes the long-run cost rate.Thatis,foreverypossiblecombinationofdeteriorationstates

(

x1,x2

)

oftheunits,wefindtheoptimalproductionratesforboth

units, and determine whether maintenance should be planned. During theplanningtime s theoptimalproductionrates arealso influencedbytheremainingtimeuntilmaintenance.Finally,atthe endoftheplanningtime,wedeterminewhichunit(s)tomaintain basedonthedeteriorationstatesoftheunitsatthattime.

Westressthatalthoughwemodelourunitsinaccordancewith

UitHetBroek,Teunter,DeJongeandVeldman (2020)andUitHet Broek,Teunter,DeJonge,Veldmanetal.(2020),especiallywith re-specttohowproductionaffectsdeterioration,weaimforvery dif-ferent strategic insights. Whereas theexisting papers are limited tosingle-unitsystems,ourmainaimistodiscovertowhatextent andinwhatwayloadsharingbetweenunitscanhelp toimprove overallsystemperformance.

4. Markovdecisionprocessformulation

We formulatethe systemasaMarkovdecisionprocess (MDP) in order to determine optimal policies. An MDP is defined by a set of decision epochs, a finite set of system states, a finite set of admissible actions per state, and state- and action-dependent transition probabilities andimmediate costs. Inthe remainder of this section, we first introduce the states and the corresponding admissible actions. Thereafter,we give an overviewof each deci-sionepochandthecorrespondingBellmanequations.Weendthe

sectionwiththealgorithmthatweusetodetermineoptimal poli-cies.

4.1. Thevaluefunctions

Thestateofthesystemisdescribedbythedeteriorationlevels x₌

(

x1,x2

)

ofthe two units andthe remaining planning

τ

until

thenext scheduledmaintenance interventions(whichtakesvalue ‘ns’ ifno maintenance iscurrently scheduled). Thestate spaceis givenby

S=

{

(

x,

τ

)

=

((

x1,x2

)

,

τ

)

:x1,x2∈

{

0,...,L

}

,

τ

∈

{

0,...,s,ns

}

}

,

and the number of states equals

|

S

|

=

(

L+1

)

2

₍

_s+2

₎

_{. We let}

_v

_,

w1, andw2 denote the value functions at the start of the three

stagesofaperiod(asdiscussedinSection3),respectively.Inwhat follows,we discussthestagesin moredetailandwe provide ex-plicitformulationsforthethreevaluefunctions.

Stage1:Observestateandschedulemaintenance

At thestart of each period,the deterioration levels x andthe remainingplanningtime

τ

∈

{

1,...,s,ns

}

areobserved.Weremark that

τ

=0isnot possibleatthisstage aswillbe explained later. Also,pleasenotethatthemaintenanceplanningtimesisconstant, andso the time until the next scheduledmaintenance operation cannotbemorethans.

Whenthe next maintenance interventionis alreadyscheduled (i.e.,

τ

=ns), there isno decisionto be made andthe remaining planningtimeisreducedbyone,thus

v

(

x,

τ |

τ

=ns

)

=w1

(

x,

τ

−

1

)

. When maintenance has not been scheduled yet(i.e.,

τ

=ns), the operator has to decide whetheran intervention to carry out maintenanceafterstimeperiodswillbescheduledornot.Incase no new intervention is scheduled, both the remaining planning time

τ

and the deterioration state x remain unaltered. In case maintenance willbescheduled,themaintenance setup costcsetup

is incurredand the remaining planning time is setto

τ

=s. The valuefunctionthusequals

v

(

x_,

τ |

τ

₌ns

)

₌min

{

w1

(

x,ns

)

,csetup+

w1

(

x,s

)

}

.

Summarizing,thevaluefunction

v

equals

v

(

x,

τ

)

=



min

{

w1

(

x,ns

)

, csetup+w1

(

x,s

)

}

if

τ

=ns,

w1

(

x,

τ

− 1

)

otherwise. (1)

Stage2:Carryoutmaintenance

Thesecond stepisto determinewhethertocarry out mainte-nance,whichisonlypossibleattheendoftheplanningtime. Re-callthat w1

(

x,

τ

)

representsthevalue function afterthedecision

hasbeenmadewhethertoscheduleanewmaintenance interven-tion.Maintenancethat hasbeenplannedwillbecarriedafterthe planningtime of stime units, implyingthat there is nodecision in this stage as long as

τ

_=0. It follows that w1

(

x,

τ |

τ

=0

)

=

w2

(

x,

τ

)

.

When

τ

=0, the operator decides which units to maintain. We denotethe maintenance decisionasr=

(

r1,r2

)

, whereri=1

if unit i is maintained and ri=0 if not. The set R=

{

0,1

}

2

de-notes the setof all possible maintenance decisions.Maintenance restoresaunittotheas-good-as-newconditionandthusthe post-maintenanceconditionforunitiequals

x_i

(

xi,ri

)

=



xi ifri=0, 0 ifri=1.

Wedenotethedeteriorationlevelsofthewholesystemafter main-tenance actionr as x

(

x,r

)

. Furthermore,regardlessofdecision r, theremainingplanningtimeisresetto

τ

=nstoindicatethatthe next maintenanceintervention isnot scheduledyet.Thisalso ex-plainswhystage1canneverstartwith

τ

=0.Thedirectcosts in-curredbyperformingmaintenanceactionr dependonthesystem

conditionxandequals

ϕ

1

(

x,r

)

=r1c˜

(

x1

)

+r2c˜

(

x2

)

,

where c˜

(

xi

)

=cpm if xi<L and c˜

(

xi

)

=ccm if xi=L. The value

function w1 given that

τ

=0 thus equals w1

(

x,

τ |

τ

=0

)

=

minr∈R



ϕ

1

(

x,r

)

+w2

(

x

(

x,r

)

,ns

)



.

Summarizing,thevaluefunctionw1 equals

w1

(

x,

τ

)

=



w2

(

x,

τ

)

if

τ

=0, minr∈R

{

ϕ

1

(

x,r

)

+w2

(

x

(

x,r

)

,ns

)

}

if

τ

=0. (2)

Stage3:Productiondecisionanddeterioration

In thefinal stage,the operator selectsthe productionrates of thefunctioningunitswhiletheproductionratesofthefailedunits arefixedtozero.Thefunctionw2

(

x,

τ

)

representsthevalue

func-tion of the post-decision state after maintenance has been per-formed.Remarkthattheremainingplanningtime

τ

isonly decre-mented inthefirststage andthatthesecond stageresetsittons attheendoftheplanningtime;hence,

τ

=0isnotpossibleatthe startofthisstage.

We let U

(

x

)

denotethe set ofall admissibleproduction deci-sionsgiventhatthesystemisindeteriorationconditionx.The pro-ductiondecisionu_∈U

(

x

)

affectsboththedirectcost

ϕ

2

(

u

)

andthe

expecteddeteriorationincrements.

Thedirectcostsconsistofapossiblefixedandvariablepenalty ifthetargetsystemproductionrateisnotsatisfiedandabonusin caseofoverproduction.Todefinethedirectcostfunction,weletIA

bean indicatorfunctionthatequalsoneifconditionAistrueand zerootherwise.Recallthatuˆ₌u1+u2 equalsthetotalproduction

rateofthesystemasdefinedinSection3.Nowwehave

ϕ

2

(

uˆ

)

=Iuˆ<κ



π

˜ +

(

κ

− ˆu

)

π

1



+Iuˆ>κ

(

uˆ−

κ

)

π

2.

Welet_X

(

x

)

₌

{

(

x_1,x₂

)

|

x_{i≤ x}_{i≤ L}

}

denotethesetofall reach-able deteriorationstatesfromstatex.Notethat, althoughthe de-teriorationincrementprobabilitiesdependontheselected produc-tion ratesu,thesetof reachablestatesonlydependsonthe cur-rentstatex.Thevaluefunctionw2equals

w2

(

x,

τ

)

= min u∈U(x)



ϕ

2

(

u1+u2

)

+  x_∈X₍x) Pu

(

x,x

)

v

(

x,

τ

)

. (3)

4.2. Modifiedpolicyiteration

We use modifiedpolicy iteration, an algorithm that combines value iteration with policy iteration, to find stationary



-optimal policies for the value functions given in Section 4.1. In general, policy iteration spends most of the time in exactly solving the value functionsforagivenpolicy,whereasvalueiterationis com-putationallyexpensivebecauseitconsidersallpossiblepoliciesin each iteration andtypically requires manyiterationsto converge.

Puterman(1994,p.386)describesamodifiedpolicyiteration algo-rithm toacceleratethe convergenceratebycombiningboth algo-rithms.Theintuitionbehindthisapproachistoapplypolicy itera-tionbutinsteadofsolvingtheexactvaluesfor

v

,w1,andw2,the

valuesareapproximatedbyvalueiterationwhilethepolicyiskept fixedforanumberofsuccessiveiterations.

The modified policy iteration that we use is provided in the Appendix in Algorithm3. Welet

v

¯ denote thevalue function af-ter an iteration that starts with value function

v

. The algorithm starts withinitializing

v

(

x,

τ

)

=0for all x and

τ

, anditeratively updatesthe bestactions andcorrespondingvaluesforeach state. Thedifferencewiththedefaultvalueiterationalgorithmisthatnot alladmissibleactionsareconsideredineachiteration.Instead,the currentbestpolicyisfixedforanumberofiterations,followedby asingleiterationthatconsidersallpolicies.Thealgorithmstopsif thespan,definedassp

(

w

)

=maxx,τw

(

x,

τ

)

− minx,τw

(

x,

τ

)

where

w=

v

¯−

v

,issmallerthanagivenpositivenumber



>0.The op-timallong-runcostrateg∗isthenestimatedas

g=

(

min

{

v

¯−

v

}

+max

{

v

¯−

v

}

)

/2, (4)

forwhichholdsthat

|

g− g∗

_|

_<

_

_/2. 5. Setupnumericalexperiments

Weexamine thevalue ofdynamicallyreallocatingloadamong units based on condition information by comparing the optimal joint condition-based load-sharing and maintenance policy to a policy that only uses condition information to schedule mainte-nanceand that equally shares loadamong the functioningunits. We refer to the former asthe condition-based load-sharingpolicy andtothelatterastheequalload-sharingpolicy.Wenotethatthis benchmarkpolicyequalstheoptimalcondition-basedmaintenance policystudied byOldeKeizeretal.(2018),whichtheyshowedto bemuch moreeffectiveforsystemswithloadsharingthanother commonlyappliedmaintenancepolicies.

In Section 5.1, we introduce the discretized gamma process that we use to model deterioration of the units. Thereafter, in

Section5.2,weintroducetwobasesystemsthat arecharacterized by their production contractsthat prescribe a target system pro-duction rateandtheassociated penalties ifthe target isnot sat-isfied.Thefirstcontract typemodels asystemwithsome overca-pacity anda smallpenalty if the fixed target system production rateisnotreached.Thesecondcontracttypemodelsasystemthat primarilyfocusesonreliability,whichisdonebyincluding redun-dancyandincurringanextremelyhighpenalty ifthetargetisnot met.

The structureof the optimalpolicy underboth contract types andthe corresponding cost savings compared to the equal load-sharingpolicywillbediscussedinSections6and7.Inthese sec-tions,wewillprovidemanyillustrationsoftheoptimalpoliciesin ordertogiveaclearinsightintohowtheoptimalpolicyisaffected bythevarioussystemparameters.

5.1. Deteriorationprocess

We use discretized (to be explained later) stationary gamma processes to model deterioration as these are suitable to model monotonicallyincreasingdeteriorationprocessessuchaswear, ero-sion,andfatigue(VanNoortwijk,2009).Moreover,thegamma pro-cessisflexibleandallowstoexaminedeteriorationprocesseswith differentcharacteristicsasits rateandvolatilitycanbecontrolled by two parameters. A gamma process consist of independently gammadistributedincrements.Weusethesameparametricform asDeJonge,Teunter,andTinga(2017),implyingthatdeterioration incrementspertimeunitaregammadistributedwithashape pa-rameter

α

andascaleparameter

β

.Denoting suchadeterioration incrementbyY,wehaveE[Y]=

αβ

andVar

(

Y

)

=

αβ

2_.

In accordance with Uit Het Broek, Teunter, De Jonge and Veldman (2020) and Uit Het Broek, Teunter, De Jonge, Veldman etal. (2020), we usea function g that describesthe production-deteriorationrelation(pd-relationinshort).Whenaunitproduces atrateu, itsdeteriorationrateequals g

(

u

)

.Moreover,we assume that unitsdeteriorate fasterwhen producing athigher rates,and thus the pd-relation g is an increasing function. For clarity, we denotetheminimumandmaximumdeterioration rateby

μ

min=

g

(

0

)

and

μ

max=g

(

1

)

,respectively.

Welet the productionrateaffectthedeterioration increments ofaunitinsuchawaythattheexpecteddeteriorationincrement per time unit equals E[Y

|

u]=g

(

u

)

, the variance of the deterio-ration increments while producing at the maximum rate equals Var

(

Y

|

u=1

)

=

σ

2

max,andthe coefficientofvariation is

Table 1

Parameter values for the base case excluding those for the production contract. Parameter Value Interpretation

μmin 0.5 Deterioration rate when idle

μmax 5.0 Deterioration rate at maximum production rate σmax 6.0 St. dev. deterioration increments at maximum rate γ 2.0 Shape pd-relation

s 1.0 Planning time for maintenance csetup 3.0 Maintenance set-up costs cpm 5.0 Preventive maintenance costs ccm 20.0 Corrective maintenance costs L 100 Failure level

η 20 Number of positive production rates Table 2

Parameter values for the two contract types. Parameter Type I Type II Interpretation

κ 1.6 1.0 Target system production rate ˜

π 10.0 10 6 _{Fixed penalty for production shortages} π1 1.0 1.0 Variable penalty for production shortages π2 0.0 0.0 Bonus for producing more than κ

by setting the parameters ofthe gammadeterioration process to

α

=

μ

2

max/

σ

max2 and

β

(

u

)

=g

(

u

)

σ

max2 /

μ

2max, i.e., thescale

param-eter dependson the production rate. If verifying this, recall that g

(

1

)

=

μ

max.

We set the fixed deterioration level of thegamma process at which failure occursequal to theindexL ofthe failedstate, and we discretizethegammaprocessby roundingdeterioration incre-ments (andvalues)totheirnearest integers.WeletFu denotethe

distribution functionofthegamma distributedincrementsduring a timeunit,givenaproductionrateu.Forthediscretized gamma process, the probability Pu

(

k,k+i

)

to transit from deterioration

level kto deterioration levelk+i whenproducing atrateuthen equals Pu

(

k,k+i

)

=

⎧

⎪

⎨

⎪

⎩

0 ifi<0, Fu

(

0.5

)

ifi=0, Fu

(

i+0.5

)

− Fu

(

i− 0.5

)

if0<i<L− k, 1− Fu

(

i− 0.5

)

ifi=L− k. 5.2. Basesystems

Theparametervaluesforthebasecaseconsideredinthisstudy are listed inTables1 and2.Wemodel thepd-relationby g

(

u

)

=

μ

min+

(

μ

max−

μ

min

)

uγ, which allows to address concave (0<

γ

<1),linear(

γ

=1),andconvex(

γ

>1)relations.The deteriora-tionrateequals

μ

min=0.5foridleunitsand

μ

max=5.0forunits

that produceatthemaximumrate.Thus,a unitalsoslowly dete-riorateswhilebeingidle.Inpracticethishappens,forinstancedue tocorrosion,bearingsthat becomeslightlyunbalancedasaresult of one-sided pressure, or externally caused load due to weather conditions.Moreover,wefocusonconvexpd-relationsastheseare mostconceivableforreal-lifesystems.Aconvexpd-relationimplies anincentivetoshareloadequallyamongunitsbecausethisresults inthelowestaveragedeteriorationrateatthesystemlevel.

Thetwobasesystemsshareallparametervaluesexceptforthe onesthat describe theproductioncontracts. Bothsystemsconsist oftwounitsandthustheirtotalcapacityequals2.ContracttypeI representsaproductionfacilitythathassomeovercapacitybutno redundancy andthat aims to meet the target system production rate, although not at any cost. We model this systemby setting a targetbelowthemaximumproductioncapacity

κ

=1.6,afixed penalty

π

˜ =10,avariablepenalty

π

1=1,andnobonusfor

over-production, i.e.,

π

2=0. Contracttype II represents a systemthat

primarily focuseson a reliable production output. This base sys-temhasa redundantunitandan extremepenalty ifthetarget is

not met.The redundant unit is modeled by setting the target to

κ

=1andthefixedpenaltyissetto

π

˜=106_{.Thereisnobenefitof}

producingmorethanthetargetandthus

π

2=0.Onecouldargue

thatoverproductionisevendiscouragedorimpossibleinsuch sys-temsandthusthat weshould have

π

2<0.Althoughthisistrue,

by choosing

π

2=0there is noadvantage ofproducing athigher

rateswhilethesystemwilldeterioratefaster,andthustheoptimal policies for

π

2<0 and

π

2=0are the same.Moreover, the fixed

penalty issubstantialandthus theoptimalpolicy alwaysaims to avoid production shortages. For numerical reasons, however, we still set a smallpositive variablepenalty

π

1=1, whichdoes not

affecttheobservedoptimalpolicyanditscorrespondingcosts. Ifaunitcontinuouslyproducesatrateu,thenitsexpected life-timeapproximatelyequalsL/g

(

u

)

timeunits.Thus,ifaunitwould always produce atfull speed, itsexpected lifetimeapproximately equals20 timeunits.Toprovidesome intuitionon the deteriora-tionprocessofthebasesystem, Fig.2depicts25samplepathsof thedeteriorationprocessfordifferentproductionrates.Weclearly seethat producing atlower rates increasesthe expectedlifetime andresultsinmorestabledeteriorationpertimeunit.

6. ResultscontracttypeI

In this section, we consider contract type I, which has some overcapacity, but no redundancy, and with a fixed penalty in casethetarget systemproduction rateisnot met.In Section6.1, we zoom in on the optimal decisions for both the equal load-sharing andthe condition-based load-sharingpolicies. Thereafter, in Section 6.4, we examine how the policies and their perfor-mancesare affectedby themaintenance setup cost,the volatility ofthedeteriorationprocess,andthedegreeofovercapacity.In do-ingso,wedefinethegap=

|

x1− x2

|

astheabsolutedifference

be-tweenthedeteriorationlevelsofthetwounits. 6.1. Optimalpolicyforthebasesystem

Fig.3showstheoptimaldecisionsundertheequalload-sharing (left) and the condition-based load-sharing (right)policy for the basesystemdescribedinSection5.2.Theproductionrateofunit1 is indicated by gray scale, ranging from idle (black) to produc-ingatthemaximumrate(white). Theremaining areasatthetop andrightsideindicate(inbothtextandcolor)whenmaintenance is scheduled, where the three subareas indicate which units are maintainedattheendoftheplanningtime. Theoptimal produc-tionrateofunit2immediatelyfollowsfromthatofunit1because theoptimalpolicyexactlymeetsthetargetsystemproductionrate wheneverpossible.This isintuitive since thereisno incentiveto produce morethan thetarget as

π

2=0 whilethere is apenalty

˜

π

=10ifthetargetisnotmet.

Intheconsideredsystem,thereisamaintenancesetupcostand thus there is an incentive to cluster the maintenance actions of bothunits.However,deteriorationisstochasticandthusclustering maintenanceimpliesthatmaintenanceiseitherperformed unnec-essarilyearly forone unit or too late for the other.From a first inspection oftheoptimalpoliciesprovided inFig.3,we immedi-atelyseethatthemaintenancedecisionsarefairlysimilarforboth policies,whereastheirproductiondecisionsdifferalot.

Fig. 4depicts the long-run stationary state distribution under theoptimalpolicies.Suchdistributionsshowtheprobabilitytobe in a certain state at an arbitrary moment in time, thereby pro-viding insights on how the deterioration processes are expected to behave over time and on the expected gap.We see that un-der condition-based load-sharing, the deterioration processes are expectedto move close along the diagonal, that is,the expected gap remains small when the units become further deteriorated. On the contrary, under equal load-sharing, it is likely that the

Fig. 2. Effect of the production rate on the deterioration process in the base case.

Fig. 3. Optimal decisions for the base case under equal load-sharing (left) and condition-based load-sharing (right). Gray scale indicates the production rate of unit 1, ranging from idle (black) to the maximum rate (white). In the remaining areas, a maintenance intervention is scheduled. In these areas, the units continue producing until the end of the planning time, however, for clarity these production rates are not shown.

Fig. 4. Heat map of the long-run stationary state distribution under both policies.

gapbecomeslargerover time.Itfollowsthat thecondition-based load-sharing policy uses the adjustable production rate to retain a smallgapsuch thatthe maintenanceinterventionscanbe clus-teredwithoutwastingremainingusefullife.

6.2. Observationsontheproductiondecisions

The production decisions under the condition-based load-sharingpolicycanbecharacterizedasfollows.Loadisonly reallo-catedwhenthegapexceedsacertainthreshold,andthisthreshold becomessmallerwhentheunitsgetfurtherdeteriorated. Further-more, the larger the current gap,the more skewed load will be shared amongthe units.This structure stems from the fact that sharingloadunequallyamongunitsimpliesahigheraverage dete-rioration rate. Forsmallgaps,itis quitelikelythat the deteriora-tionprocesseswillsynchronizewithoutintervening.Consequently, it isbettertocontinueproducingatthe mostefficientloads,that is,equally sharingloadamongunits.Ifthe processesdonot syn-chronize,thentheoperatorcanstillinterveneatalaterstage.

Anexception to theabove isa situationwitha healthy anda highlydeteriorated unit. In thiscase, the deteriorated unit takes over load from the healthy unit and synchronization is reached byonlymaintainingthedeterioratedunit.Performingmaintenance immediatelywouldwasteremainingusefullifeofthedeteriorated unit,whereaspostponingitimpliesalargergapafterthe mainte-nanceactionbecausethehealthyunitalsocontinuestodeteriorate. Byreallocatingload,maintenancecanbepostponeduntilthe dete-rioratedunithasdepleteditsremainingusefullifewhiletheother unitcanretainitshealth.Wenotethatthisscenarioisunlikelyto occurbecauselargegapsaregenerallycorrectedinanearlierstage.

6.3. Observationsonthemaintenancedecisions

Themaintenancedecisionsunderbothpoliciesarelargely sim-ilar.Maintenanceisclusteredifbothunitsarehighlydeteriorated whereas onlythe mostdeterioratedunit ismaintained ifthe de-terioration levels differtoo much. Furthermore,for a given

dete-rioration levelofthehealthiestunit, theotherunit ismaintained accordingtoathresholdpolicy.

A particularobservation isthat thisthreshold isfirst decreas-ingandthenincreasinginthedeteriorationleveloftheotherunit. Thethresholdisnon-constantbecauseoftwoopposingincentives. Ontheonehand,maintenanceforthedeterioratedunitshouldbe performed early because this synchronizes the deterioration lev-els. Ontheotherhand,postponingthemaintenance actionbetter utilizestheusefullife ofthedeterioratedunit.Ifthedeterioration levelofthehealthyunitisverylow,thenmaintenance ofthe de-terioratedunit canbe postponed withoutcausingatoolarge gap afterthemaintenanceaction.Thehigherthedeteriorationlevelof thehealthyunit,theearliermaintenanceforthedeterioratedunit should be performed inorder to avoid a too large gap. This ex-plains whythethresholdfirstdecreasesasfunctionofthe deteri-oration level ofthe healthy unit. Ifthe deterioration level ofthe healthy unit increases further,it becomesmore likelythat main-tenance can be clustered in this cycle, which explains why the threshold eventually increases.We note that thiseffect,although notmentionedbyothers,issolelycausedbytheeconomic depen-dencyandnotbyloadsharingdynamics.

We also observe two structural differences between the two policies. Firstly, condition-based load-sharing allows to schedule maintenance interventionsathigherdeteriorationlevels. The rea-son is thatlower productionrates not onlyreduce the deteriora-tionratesbutalsothevolatilityofthedeteriorationincrementper period. With condition-based load-sharing, the mostdeteriorated unittypicallyproducesatalowerspeed,therebyreducingtherisk offailure.

Secondly, because the equal load-sharing policy can only use maintenancetosynchronizedeteriorationlevels,itclusters mainte-nanceforconsiderablymorestatesthanthecondition-based load-sharing policy. For instance, if unit 1 is in the highly deterio-rated state x1=90, then the equal load-sharing and

condition-based load-sharingpolicies opportunisticallymaintainthe second unit fordeteriorationlevels above46and55,respectively.To un-derstandthisdynamic,letusconsiderthesituationthatthe deteri-orationlevelofthesecond unitliesbetweenthesethresholds,e.g,

(

x1,x2

)

=

(

90,50

)

.The secondunit clearlyhasno needfor

main-tenance whereas maintenance for the first unit cannot be post-poned. By only maintaining the first unit, the system moves to state

(

x1,x2

)

=

(

0,50

)

.Underequalload-sharingthisimpliesthat

the next maintenance actions are again unlikely to be clustered, and thus it is better to synchronize their deterioration by main-taining both units, thereby wasting a substantial remaining use-fullifeofthehealthyunit.Onthecontrary,undercondition-based load-sharing, the resulting gapcan easily be synchronizedbefore thenextmaintenanceintervention,andthusitisnotnecessaryto wastetheremainingusefullifeofthehealthyunit.

From the above effects, it follows that both policies use the maintenancedecisiontosynchronizethedeteriorationlevelsofthe units(e.g.,byperformingmaintenance foradeterioratedunit ear-lierthanactuallynecessaryforthissingleunit).However,such in-terventions waste remaining useful life of units andis therefore significantly moreexpensivethanusingthemoresubtleoptionto synchronize the deterioration levels by reallocating load. We in-deed observethat the condition-basedload-sharing policyuses a maintenance intervention substantially less often to synchronize thedeteriorationlevels.

6.4. Parametersensitivity

We continuebyexamining theeffectsofchanging various pa-rametervalues onthestructure oftheoptimalpolicy andonthe corresponding cost savings of condition-based load-sharing

com-paredtoequalload-sharing.Theresultsareobtainedbytakingthe basesystemandadjustingtheparametervaluesonebyone. 6.4.1. Effectofthemaintenancesetupcost

Fig.5showstheoptimalpoliciesforvariousmaintenancesetup costs for the equal load-sharing policy (top) and the condition-basedproductionpolicy(bottom).Underequalload-sharingwe ob-servethat1)theareainwhichthehealthyunitisopportunistically maintaineddecreasesinsizeifthesetupcostdecreases,and2)for verylow setupcosts themaintenance decisionsforthe twounits areindependentofeachother.

Now considerthecondition-based load-sharingpolicy.As long asthesetupcostsaresubstantial(saycsetup=2),themaintenance

decisionsare insensitivetoan increaseofthe setupcost whereas the productiondecisions are affected.If thesetup cost increases, clustering becomes more important and the optimal policy as-signsmoreloadtothehealthyunit.Forinstance,supposewehave x1=10andx2=70.Then,forcsetup=2 thepolicy doesnotfully

reallocatetheloadtothehealthyunit(u1=90% andu2=70%)in

order to produce ata more efficient rate, whereas for csetup=3

theloadisfullyreallocated(u1=100%andu2=60%).Further

in-creasingthesetupcosthasalmostnoeffectontheoptimalpolicy becausethemaintenanceactionsarealreadyvirtuallyalways clus-tered.

Fig.6 (left)showshow the costsaving ofadopting condition-basedload-sharingisaffectedbythe maintenancesetupcost. We indeed see that the cost saving first increases in the setup cost and then stabilizes. An interesting observation is that without a setup cost, the optimalproduction andmaintenance decisions of the unitsare still dependent,and costsavings around 5% are re-alized. In this case, the deterioration levels of the units are ac-tivelydesynchronized andtheir maintenanceinterventionsare al-ternated.Hereby,theusefullifeoftheunitscanbebetterutilized by slowingdown themostdeteriorated unit whenit reachesthe failurelevel.

6.4.2. Effectofthetargetsystemproductionrate

Fig.7showsoptimalcondition-basedload-sharingdecisionsfor different target system production rates (increasing from left to right) forboth stable (bottom)and volatiledeterioration (top). A lower target impliesmore overcapacity,which givesthe operator moreflexibilitytoreallocateloadamongtheunits,resultingintwo benefits.Firstly,becauseitiseasiertosynchronizelargegaps,the optimalpolicyallowsforlargergapsbeforeloadisreallocated. Sec-ondly,the loadofthemostdeterioratedunit canbe reduced fur-ther,resultinginaconsiderablylessconservativemaintenance pol-icythatutilizestheusefullifeofunitsmoreeffectively.

InFig.6(middle), we seethat thecost savingincreaseswhen thetargetdecreases,andthatthere isnocostsavingifthetarget equalsthemaximumproductioncapacity.Moreover,thecost sav-ingsareverysensitivetotheproductiontargetifthereisonlylittle overcapacity,anditbecomeslesssensitiveiftheproductiontarget comes closerto1. However, ifthetarget drops below1,the cost savingbecomes moresensitive againbecausethe redundantunit providesnewoperationaloptions(seealsoSection7).

6.4.3. Effectofthevolatilityofthedeteriorationprocess

Now we consider the effectof the volatility of the deteriora-tion process by comparing the policiespresented in thetop row ofFig. 7 tothose presented inthe bottom row. We seethat, re-gardlessofthetargetsystemproductionrate,theoptimal produc-tiondecisionsclosetothediagonalarenotaffectedbythe volatil-ity ofthe deterioration process. The main difference is observed forlarge gaps that are notsynchronized before thenext mainte-nanceintervention(i.e.,topleftandbottomrightareasinthe fig-ures).Forstabledeterioration, theloadisgraduallyshifted tothe

Fig. 5. Effect of the maintenance setup cost under the equal load-sharing (top) and condition-based load-sharing (bottom) policy. Gray scale indicates the production rate of unit 1, ranging from idle (black) to the maximum rate (white). In the remaining areas, a maintenance intervention is scheduled.

Fig. 6. Relative cost savings of condition-based production decisions compared to equal load-sharing as function of the maintenance setup cost c setup (left), the production

target κ(middle), and the volatility of the deterioration process σmax (right).

Fig. 7. Effect of the production target and of the volatility of the deterioration process. Gray scale indicates the production rate of unit 1, ranging from idle (black) to the maximum rate (white). In the remaining areas, a maintenance intervention is scheduled.

Fig. 8. Effect of the volatility of the deterioration process on the optimal production and maintenance decisions (top) and the corresponding long-run state distribution (bottom).

Fig. 9. Effect of the maintenance setup cost on the cost savings compared to the equal load-sharing policy.

healthyunitifthegapincreases.Ifthegapbecomestoolarge,the mostdeterioratedunit suddenlytakesoverloadfromthehealthy unit. Formorevolatiledeterioration,thistransitionislesssudden and the area in which loadis shifted to the deteriorated unit is smaller.Thisisthecasebecausethelikelihoodofsynchronization by chance is higher,and becauseacceleratinga deteriorated unit toomuchresultsinunacceptablefailurerisks.

Fig.6(right)depictstheeffectofthevolatilityonthecost sav-ings. For stabledeterioration, the cost savings are smallbecause the deterioration levels ofthe units are not expected to diverge. Ifthevolatilityincreases,theexpectedgapatthe endofthe life-time of the unitsincreasestoo. Bothpolicies still usea high de-greeofclustering,butthecondition-basedload-sharingpolicy bet-terutilizestheusefullifeoftheunitsbysynchronizingtheir dete-riorationlevels.Consequently,thebenefitofcondition-based load-sharing increasesifthevolatility increases.Finally,ifthe

deterio-rationprocessbecomeshighlyvolatile,thenlargegapsthatcannot be correctedforbyreallocating loadbecomemorelikely, andwe indeedseethatthecostsavingsstarttodecline.

7. ResultscontracttypeII

Wecontinuewiththesecondcontracttypethatappliesto pro-ductionfacilitiesthatmustprovideaconstantandreliable produc-tionoutput.Thekey priorityforsuch systemsisto avoid produc-tionshortages,whereasminimizingoperationalcostsisonlya sec-ondary objective.We model thisby setting the fixed penalty for shortagesto

π

˜₌106_{.Moreover,inpractice,thereliabilityofsuch}

systems is often improved by including a redundant unit, which we model by setting the target system production rate equal to thecapacityofasingleunit

κ

=1.0.

Mostinteractionsforthiscontracttype aresimilartothosefor contract type I as discussed in Section 6 and are therefore not repeated here. We do, however, observe different effects of the volatilityofthedeteriorationprocessandofthemaintenancesetup cost,whichweaddressinthissection.

7.1. Effectofthevolatilitydeteriorationprocess

Compared to the base case, we lower the maintenance setup costtocsetup=1inthissection becausethisgivesmoreclear-cut

policieswhileitdoesnotaffectthestructuralinsightsthatwe ob-tain. InSection 7.2,we show that other maintenance setupcosts resultinsimilarinsights.

Fig. 8 shows the optimal condition-based load-sharing pol-icy (top) and the corresponding long-run state distribution (bot-tom)forstable(left),mediumvolatile(middle),andhighlyvolatile (right) deterioration. For stable deterioration, the risk that both units fail simultaneously is negligible. As a result, their main-tenance can be clustered without risking excessive penalties for shortages. For medium volatile deterioration, having two units withintermediate orhighdeterioration levels becomestoo risky and the focus lies on minimizing the risk of production short-ages.The deteriorationlevels ofthe unitsareactively desynchro-nized such that the gap is around 45. Note that, because of the redundancy, failure of one unit is allowed, and consequently the

Fig. 10. Optimal condition-based load-sharing policy and the corresponding long-run state distribution if c setup = 3 and σmax = 6 .

maintenance policy itself is actually less conservative than in all previously considered cases with the same volatility. For highly volatile deterioration,not onlythe maintenance interventionsare alternatedbutalsotheusageoftheunits.Thereby,thesystem pro-ducesata lessefficientrate, butalsoalways keepsoneunit ina goodcondition.Thisreducestheriskofshortagesiftheotherunit failsunexpectedly.Noticethatmaintenanceisvirtuallynever clus-tered,notevenifbothunitsarehighlydeteriorated.Insuchcases, onlyoneunitismaintainedtodesynchronizethedeterioration lev-els.

7.2. Effectofthemaintenancesetupcost

Fig. 9 showsthe effect of the maintenance setup cost on the cost savings compared to the equal load-sharing policy. Higher maintenancesetupcostshavealmostnoeffectontheoptimal pol-icyforstabledeterioration(

σ

max=3)asforthosemaintenance

ac-tions arealways clustered, andnot onthat forhighlyvolatile de-terioration (

σ

max=9)asfor thosemaintenance actions arenever

clustered.Correspondingly,thereisalsonosignificanteffectonthe potentialcostsavings.

However, for medium volatile deterioration (

σ

max=6), the

structure doeschangeifwe increase themaintenance setupcost, ascanbeseeninFig.10.Forlowsetupcosts,theoperatorfocuses oneliminatingtheriskofshortagesbyalternatingthemaintenance interventions.Forhighersetupcosts,totalcostscanbereducedby synchronizingthedeteriorationlevelsoftheunitsaslongasthese are ina goodconditionsuchthat their maintenance canbe clus-tered. When theunits arehighly deteriorated,their deterioration levelsaredesynchronizedagaintoreducetheriskofsimultaneous failure. The equal load-sharingpolicy canonly reduce the risk of shortages by alternating the maintenance interventionsand thus cannot share the maintenance setup cost among the units. This also explains thesignificant increase incost savings that we ob-serveinFig.9.

8. Conclusion

We have investigated joint condition-based production and maintenance policies fortwo-unitsystems witheconomic depen-dencyandwhoseunitshaveadjustableproductionrates.The pro-duction rateof a unit affectsits deterioration rate, implying that condition-basedproductionpoliciescanbeusedtocontrolthe de-terioration of the units. A production target at the system level is adopted and a penalty is incurred if this target is not satis-fied. Condition-basedproductiondecisionsenablethe operatorto (de)synchronize the deteriorationlevels of theunits, thereby

im-proving the clustering of maintenance interventions or reducing theriskthatbothunitsfailsimultaneously.

We haveformulated the systemasa Markovdecisionprocess andusedthis todetermine cost-minimizingjointproductionand maintenancepolicies.Thebenefitsofdynamicallyreallocatingload amongunits isexamined by comparing thenewly proposed pol-icytoapolicythatcombinescondition-basedmaintenancewitha staticproduction policy that shares loadequally among all func-tioningunits.Resultsshowthatcostsavingsupto20%canbe ob-tainedforsystemswithovercapacitybutnoredundancy,andthat these savings increase to 40% for systems with redundancy. The costsavingsoriginatefromfewerfailures,reducedrisksof produc-tionshortages,improvedclusteringopportunities,andfewer main-tenance interventions per unit. Another promising observation is thatadoptingcondition-basedproductionpoliciesnotonlyreduces expectedcostsbutalsoitsvariance.

Forsufficientlyhighmaintenance setupcosts, theoptimal pol-icyaimstosynchronizethedeteriorationlevelsoftheunitsby as-signingmoreloadtotheleastdeterioratedunit.Thelargerthe dif-ference indeterioration,themore loadisassignedtothe healthy unit.Moreover,forlowdeteriorationlevels,theoptimalpolicydoes not immediately adjust the productionrates as the deterioration levelsmaysynchronizethemselvesandotherwisethereisstill suf-ficient time left to correctthe gap ata later stage. Interestingly, when the deterioration levels are far apart, the operator should nottrytosynchronizethembeforethenextmaintenance interven-tionandshouldevenacceleratethemostdeterioratedunit.Atthe nextmaintenanceintervention,maintenancewillthenonlybe car-riedoutforthisunit,resultinginbetter-synchronizeddeterioration levelsafterthismaintenanceintervention.Postponingthe mainte-nanceinterventionimpliesalargergapaftermaintenancebecause thehealthyunitalsocontinuestodeteriorate,whereasperforming itimmediatelyresultsinwastingremainingusefullifeofthemost deterioratedunit.Thesetwoaspectsarebetterbalancedby reallo-catingloadtothemostdeterioratedone.

Another insightful result is that even without maintenance setup costs, the optimal production and maintenance decisions oftheunitsarestill dependent.The condition-basedload-sharing policyactivelyalternatestheirmaintenanceinterventions.Thereby, the most deteriorated unit can decelerate when its deterioration levelapproachesthefailurelevel.Thisresultsinbetterutilization oftheusefullife oftheunits,whichcanresultincostsavings up to10%.

Condition-basedload-sharingdecisionsseemto beparticularly useful for systems with redundancy and severe consequences if thetargetsystemproductionrateisnotsatisfied.Examplesare fa-cilities that must provide a reliable productionflow such as gas