RENEWABLE FOR

Journal of Applied Mathematics&DecisionSciences,2(1),^3-21 (1998)

Reprints Available directly from theEditor. Printed inNewZealand.

OPTIMAL INVESTMENT STRATEGIES FOR RENEWABLE FACILITIES

JOE FLOOD

Australian Housing and Urban Research Institute, Level7, 20Queen St Melbourne3000, A USTRALIA.

Abstract. Although a largeliteratureexistsonthe repair and deterioration of machines, the associatedproblem ofmaintenance schedules for deteriorating renewable facilities has beenlittle studied. These facilitiesincludeallthosewhichcan be restored toanear-newstate byrenovation or rebuilding, so that the market value andperformance of the facility depends on thecurrent state of repair rather than on thetime since initial construction. This paper solves the general deterministicproblem offinding the optimal repairstrategyforadepreciating renewable facility.

It isshown that the value of the facility should approach the level whereafunction defined as the"nett internalreturn" isgreatest. Ifthefacility has a finite life before sale ordemolition,an adjustment to repair strategies should be made as the facility approaches thistime, increasing repairs wherethispermits a bettersaleprice to beobtained,or discontinuing repairsifthey are notjustified by scrap or market value. Solutions for arangeof common depreciation functions andfor linearand quadratic repair cost functions are obtained. The optimal life of the facility is determinedat thetime when nett"external" marginal return,which includespotentialcapital gain or loss and opportunity cost of capital, falls to zero.

Keywords: Replacement strategies, renewablefacilities,depreciation, optimal control

1. Introduction

Averylargeliteraturehas developed around the problem of determining optimal

maintenance andreplacementstrategies fordeterioratingfacilities, as thereview articlesby Pierskalla andVoelker(1976)and PintolonandGolders(1992)relate, orasdetailed in texts suchasJardine(1970, 1973),

Rapp

(1974),Gertsbakh(1979),andNewnan (1983).

Arange of techniqueshas beendevelopedfor examiningthe economiclifetimesof machines inparticular, rangingfrom thedeterministic analysesofHotelling(1925), Naslund(1966), Thompson(1968)andHartl (1981), throughstatistical control(Kamien andSchwartz 1971, Virtanen 1982,

Hopp

andWu 1990,Cho 1993,

Tang

1993,Mehrez andBernan 1994),toreliability theory(BarlowandProschan1975, Gertsbakh 1989, Chand and Sethi 1982).

Theproblem of deterioratingbuiltfacilities or infrasmacture israther differentfrom deteriorationofa machine, inthat many facilities may berestoredtonear-newstateby rebuilding orrenovation. The value andperformanceof thesefacilitiesdependon their current stateofrepairratherthan onthe time since initial construction,aswithamachine.

Thosefacilities which have alargenumberofsmallcomponentswhich can beregularly

replacedwithoutaffectingsystem integrity, or^whichhave a basic structure with avery longlife, are mosttypical.

However,

anyfacilitieswhichrequiremaintenance,producean income,andcanbesold onthe marketmightbe consideredusingthistype of analysis;

ranging frommachinesandautomobiles,factories, buildings, rental housing,forestry plantationsandcomputerinstallations tolargescalenetworkedinfrastructure suchas electricity ortelephonenetworks.

Lesse

and

Roy (1987)

treated repair ofdeterioratinginfrastructureas analogousto

operation ofarenewable resource, developingthe system as an autonomous linear control problem,^whilemaking special assumptions abouttheformofthe costand depreciation functions.Theform of solutiondepends^agooddeal, however,onassumptions aboutthe form of the deteriorationfunctionand of the repaircostfunction.Thispaperaddresses the general rangeof functional formswhichareencountered inpracticewhiledevelopingsome alternativesolution methods fortheproblem.

Thepapersets outtodevelop^ageneralsolution to the deterministicproblem ofthe optimalmaintenancescheduleforadeterioratingrenewablefacility. Initiallyallparameters aretakentobe time-invariant andbased onthecurrentconditionofthefacility, andcriteria for optimal repair strategiesare obtainedby standard control theory methods, generalising the workofother authors.

In

the linearcase,further insightis obtainedby usingan alternate solution method of integration by parts. Several common depreciation functions and repaircostfunctionsareconsideredtoshow typical strategies. Then the parameters are considered tovarywith timeaswellascondition, to includethe effectsofcapital gains^or obsolescence.Conditionsfor optimal disposaltimes are established andperfectmarkets considered.Finally somecommentsontheformof functions and thepracticalvalueof the resultsare made.

2. Renewable facilities and optimal repair strategies

Itispresumedthatthe current conditionofadepreciating renewable facilityis determined byasinglevariablex,whichdeterminesitsmarket capitalvalueorscrapvaluepx,with unitprice p,and that thefacilityearns nett rental incomer(x).Thefacility depreciatesata rateD(x)dependent onthecurrentcondition.Investmentormaintenance can beundertaken toimproveor restore the conditionoftheasset ataratem(t)

>

_{0 which is}tobe

determinedatdifferent times, with acostc(x,m)which isdependentbothon theamount ofmaintenance and thecurrentcondition.

Theframeworkisquitegeneralin that it ispresumedthatrents,capital prices^andcostsof maintenanceactivitymayhaveindependentfunctionalforms, beingdetermined in different markets.The formsofthe functionsmaybe difficulttodetermine and subjectto

OPTIMALINVESTMENTSTRATEGIES FORRENEWABLEFACILITIES

considerable statistical noise.

However,

some general observationsmaybemade asto howeach functionshouldbedefinedinpractice.

RentThefunctionr(x)isrentorincomenettofallfixedcosts notrelatedtomaintenance (egnettofadministration,taxes,financeetcbutnotdepreciation),and evennettof any routine maintenance items which do notaffectthe valueoftheproperty.

Rates

ofrental returnor income areoftenlikeddirectlytocapitalvalue, butcan infact varywith conditionsin thecapital and rental markets; forexample, therateofreturnonresidential rentalproperty(evennettof maintenance) tendstobehigher forcheaper, older dwellings, and tendstovary markedlywith location.

Depreciation ThedepreciationrateD(x)should reflectthemarket price forfacilitiesof differentages,and whichhavebeensubjectonlytoroutine maintenance. The standard forms of depreciation^{in use in}accounting are "prime value"(constant, or straight-line) and"diminishing

cost"

(linear),but inpractice depreciation maytake morecomplex forms,asthepaperdiscusses.Forexample,

automobiles

depreciate rapidlyin the firstfew yearsthenmoreslowly,whilelarge scaleinfrastructure tendstodepreciate slowlyatfirst, thenmorerapidly as reliability becomes anissue.

MaintenancecostMaintenance costc(x,m)dependsbothonthestateof repair and on the rateofmaintenance.

It

shouldincludeall expenditure onmaintenance,replacementor improvement^whichaffectsthe marketvalueofthefacility,alongwithany^costs dependentontherateofmaintenance-includingany expectedlossof rentalincomeat higherratesofmaintenance, whenproduction mighthave to be limitedor

pans

ofthe facility mightnotbeoperational.Maintenancecostshouldgenerally beconsidered relative totheimprovementⁱⁿcapitalvalue itbrings.

The cost function haspartialderivatives

cm>O,

c,

>

O.Thispresumesdiseconomies of scale inrepairs, in that ahigherrateof repair mayresult indisruptionsand a lossofrental income,andmaycause logisticalproblemsintheconductof repairs.

2.1 The optimal repair problem

Therateofchange ofconditionx of a renewablefacility^isdeterminedbythestate equation

. ^{=-D(x) +m}

⁽¹⁾

whereD(x)^{is thenaturalrateof}deteriorationofthe asset.

Theproblemis then tofindthe optimal maintenance schedulem(t)^tomaximisethe expectednettpresentvalueJ(x,m)of the facility overT years,takingintoaccountrentr, repaircostc(x,m),and finalresale valuep.x(T)attime

T,

7"

J = f ^(r(x) c(x,m))e-btdt + p.x(T)e

^-r

0

(2)

where bisthe discountrate.This issubjecttothestateequation(1),andthe constraints

m>_O, x>_O (3)

Insomeproblemstheremayalso beotherconstraints,

XNXo

for example, sothat xmay notexceedits newvalue or condition.

Theusual method ofsolutionofthistype of controlproblemis, using the Pontryagin MaximumPrinciple,toformtheLagrangianofthe system and takepartialderivatives withrespect^tothecontroland state variables, as Pitchford andTurnovsky(1977)and manyothertextsdetail. TheLagrangian ofthissystemis

L

=

((r-c(x,m))+p(-D+m) +

(m-O) ⁺ (x-O))

^e-bt ⁽⁴⁾

where/.t,

, .

are costate variables(shadowpricesatcurrentvalues)

1.

2

Conditions foranoptimal strategyare

= bit- ---, OL .- ^{= 0}

(see, forexample,PitchfordandTurnovsky1977,Chapters ^and2).

so that,

].l (b+D’)

p-

r’ +

c-

⁽⁵⁾

-c,,+p

+

⁰ (6)

with

X

9 non-zero onlywhen therespectiveconstraints areoperational.The shadow

1,

pricevariablepisthe marginalbenefitof conductingextrarepairs, consisting ofboth

These adjustedcostatevariablesareobtainedby multiplyingthe"tree"or discounted costatevariablesbye^b’

OPTIMAL INVESTMENT STRATEGIES FORRENEWABLEFACILITIES

improvedrentsandextracapital value, anditusuallydeterminesboth thelevelof repairs andwhether repairs shouldbeundertakenatall,withrepairsnotundertaken when p<c,,.

Internal stationary solutions

(: = 0, = 0, XI, X2 ⁼

^0)occurwhen

^p

^{c,,m= D(x), so}

that (usingEquations (1), (5)and(6))

(r-(c(x,D(x))’-bc.(x, D(x))

=

0 (7)

Thisequationmaybe solvedtofindx=x*atwhich theoptimum applies.

Whetherx*willprovidea maximum or a minimumreturndepends^{on the}second variationof Equation(2).TheLegendrecondition(Pierre 1969, p.115)statesthat x*will be a maximum if

gxx ^d/dt g, ^{> 0}

everywhere, where

g(x,Sc,t) = (r + c(x, + D))e

^{-t is the}integrandinEquation (2).^Thisreducesto

(r’-c(x,D(x))’-bc,(x,D(x)))’

<= 0 (8)

forall x, when x isstationary.

Equations(7)and(8)maybe combined into asinglecriterion. Maximalreturnsare possibleatmaximaofthe function

l(x) r(x) c(x,D(x)) bC(x) (9)

if this function is concaveto theorigin,

whereC(x)

i ^{cm (x_ D(x_))dx}

^{istheamountofmoneyrequiredtorestore}the facility

X

tovaluexfromsome basestate

xl(eg x=0,

^orX=Xo,the initial

value2),

^withrepairs conductedexactlyat themargintoarrestdepreciation

3.

The functionI(x)canbetaken as the"nettinternal"orrecurrentrateofreturn.Ifxis to be maintainedatsome constantvalue,thetermsofIinclude rent,costofarresting

depreciation,andopportunitycostofcapitalexpenditureonrepairs.

Itissometimes convenient^topresumex1=O, forconvenience incalculating/, asis done inthis

paper,

and sometimestopresume

x

_Xo.so thatC(xo)=Oin Equation

(10)..

The resultswillonly differbya constant.

Althoughthis functionisimportantin thesubsequent analysis, representingthe costof rebuildingafacility usingon-site maintenance asopposedto its marketvalue, itdoes not seemto haveappearedpreviouslyin the literatureoflinearcontrol.

2.2 The linear maintenance cost case and integration by parts

Ofteninpractice^alinearform for thecostof maintenanceisreasonablein thevicinityof the optimum. All other previous authors including Naslund(1966), Thompson(1968), Lesseand

Roy (1987)

and

Tang (1993)

havepresumeda costfunction linearin m, c(x,m)=c(x)m,with cusuallyconstant. The marginal repaircostfunctionc(x)in this case directly represents theextravaluetothe property through repairs,orthe "repair

efficiency"; for example,ifc(x)=2p,thenhalfthe costof repairs^atx canbe recoupedin improved capital value.

In

thislinearcaseabang-bangcontrolresults,andfor a solution to exist, theremustbe anupper^{limit to}therateofmaintenance as wellasalowerlimit. This maximum maintenanceratem,,^canbe takentobethepoint whererents starttobe seriously affectedbymaintenanceand second-ordereffectsor diseconomies ofscale come into

play!

Thenmaintenance eithervariesbetweenminimaleffort(orzero)whenxis toohigh for maximal returns,andmaximaleffort whenit istoolow.

Thesignificance ofI(x)becomesmoreclear in the linear casewhenan alternative solution is obtainedusing integrationby parts. Then,

J(x,c)

T

= r(x) -c(x)(Sc + D))e-b’dt + pxre

^-br

0 T

(r-cD-bC)e-tdt ^{+ (px}

^r

-C(xr)+C(xo))e

^-r

0

(10)

The integrandin the firsttermisI(x). Overan infiniteperiod (orifc(x)=pforall x, so that repaircosts canalwaysbe recoup.ed onsale),thesecond term will vanish.Thenthe problemreducestomaxlmxsmg

l(x)e

dt,whichwill be achievedby moving rapidly towardsa maximumofl(x).

2.3

Finite time horizons and transversality

Foraninfinite-horizon problem(Tvery large)theproblemcanbe solvedby allowingthe condition ofthefacilitytomovetowards a maximumofI.Afinite horizonproblemis

Alternatively,abudgetconstraintmaybeplacedonmaintenance costc(x)m,^which leads to a similar solution.

OPTIMALINVESTMENT STRATEGIESFORRENEWABLE FACILITIES

morecomplex,usually requiting some adjustment of theconditionofthefacility towards the endofits lifetofacilitate resale.

Aswell asEquations(5)and(6),the variable

p

mustsatisfyatransversality equation or final condition

It(T) p.

^Thisfinal condition tendstodeterminethe size

of/.t(t)

forallt, determining^{in turn}thelevel of repairs.Theproblemthen is tofindan initial value and a path for# determinedbythestateequation(5)leadingto finalp

=

p. Repairsare

conducted, accordingtoEquation(6),whenever marginalbenefits

At

of repair^exceed marginal^costs

cm.

From

Equation

(10)

itcan be seen thatif1(x*)^{is at a}maximum,but

I.t*=c(x*)

isnot

equaltop,then a finaladjustment in x should takeplaceⁱⁿordertoreapthefull benefits of sale.If/.t*<p, thenrepairs canbe more than recouped onsale("cosmetic" repairs)and thefacility shouldbeimproved. If p*>p,the facilityisunderpriced, and repairs shouldbe discontinuedpriortosale.

Atypicalsolution,then, istomovethe facility towards^apoint ofmaximum return as expressedby thefunctionl(x), near whereit is sustainedforatime,andthen improvethe facilitytoimproveor allowittorundownfurther priortosale,dependingonmarket conditions.However,this finaladjustmentmayoccupy^muchorall ofthe lifeof the facility, as some ofthefollowingexampleswill show.

3. Some examples

Some examplesare now considered whichshowtheutility of employingthe internal returnfunctionl(x)todetermine solutionsoftheproblem fordifferentforms of the depreciationfunction oroftherentor repaircostfunctions. Example(1)isessentiallythe machineproblemofThompson (1968)and others, whileExample(4)istheproblem of Lesseand

Roy

(1979). Intheseexamples,theinternalreturnfunction provides an immediate solutionfor theinfinite time horizoncase, and providesamoregeneral solution thanthose ofearlierauthors.Examples (2)and(3)are newproblems; Example (2)introduces aquadraticrepair function,tobettersimulate the diseconomiesofscale in conducting repairs^atafasterrate.Thecontrol solution isthen showntobea continuous

"smoothed"approximation^totheusualbang-banglinearrepairstrategy.Example3 considers aconvex quadratic depreciationfunction,representingfacilitiesthatdepreciate morerapidlyinpricewhen new.Inthiscasean internal solution isgenerallyvalid,and onceagain^the"bang-bang" repairstrategy can be smoothedby adoptingaquadratic repair costfunction.

in thelinearcontrol case,buttheresultholds moregenerallywhen

l.t*=c,,,(x*,D(x*))T.

0.18 0.16"

0.14"

0.12 0.1 D(x)^o.8

0.06 0.04 0.02 0

0 0.2 0.4 0.6 0.8

qua,dr,

^ati

Figure1.Depreciation^functions.

Thedifferentforms of depreciation functionsused intheexamplesareshown in Figure1.

Example 1. ,Linear^costs,straightlinedepreciation..,

Ifdepreciationis constantD(x)=d,and allotherfunctions are linear, so thatr(x)=rpx, c(x,m)=cpm, forconstantsr,c, this is the(time-invariant)classic machineproblemof NaslundandThompson.

In

^thiscase,l(x) px(r-cb)-cd,whichtakesa maximumonly^at maximumor minimum valuesofx. Therefore,when time horizons arelong,thefacility shouldbeimprovedor maintained in original^conditionifr>cb, thatis, ifrentis sufficiently hightocovertheopportunity^costof capital expenditure^onrepairs;and allowedtodepreciateotherwise.Towardstheend of the period,^anadjustmentupwards or downwardstoreflect salevalues isrequired, dependingwhether c<l or c> 1.

Inthisparticular case,the time when the strategy istobeadjustedcan beeasily determined.

In

Thompson’s(1968)originalsolution,Equation(5)becomes

b/t-

^{r (5’)}

Thissimplelinear d.c. issolvedtogive

12 = 12 * +(p --/.t*)e -b(r-’,

wherethesteady state

value/.t* =

^r/b.

Then/.t

^isstrictly increasing or decreasingwith timeaccordingas

OPTIMALINVESTMENT STRATEGIES FOI:t I:tENEWABLE FACILITIES 11

whetherris greaterthan or lessthanbp

’.

The repair strategywill reverse atthe single point oftime where

p

c.This time is calculatedfrom(T-t)^=-l/b

In

((c-p*)/(p-p*)).

Figure 2a shows the optimumsolutionfor xand#^{over a}20 year period, when r=0.18, d=0.06, b=0.1, c=1.6,Xo=^1,p= 1,

Cmax=0.15. ^In

^{curve (1),thefacilityrises}in value while

I.t>c

^{for 6years,}after which it isallowedtodeclinebecause repairsarenotjustifiedbythe sale value.Inthe case(2)that xisnotallowed to rise above its initial conditionXo,then the turning pointremains at the sametime,butforthe first6years,_{#=c and x}is preservedinits initial condition.

InFigure 2b, where r=0.06 and c=0.8,withotherparametersunchanged,the rental return isinsufficienttojustify repairs; however after 14yearsthe prospect of sale causes

"cosmetic" repairstobeundertaken, lifting the salevalueofthefacility.

Exa .m.

ple.2.

Ouadrati

^cots,

linear

depreciation

Ifcisquadratic.in m, so thatc(x,m) _clm +c2m

2,

forconstantscl.^cz andr(x) rpx,D(x) dx (linear^or"diminishing balance" depreciation),then

cm

cl+

2crn

^andp*=r/(b+d).

In

this case, internal solutionsx* may^exist.The stationaryvaluex*canbecalculated from the maximumof l(x)

=

^rx-(b+d)

cx

^(b+d)

c2d

^x

,

or by notingc,,,(dx*)=#*,sox*=(bt*-

cl)/2cfl

^Itfollowsthat the maximum valuex*will liebetween 0andXo^wheneverc,,,(O)

<

l.t* <- cm(dxo).

^Otherwise,

if/*<c

the facility should be allowedtodecay;or if

c+2cflxo<-

^l.t*,thebestreturns areobtainablebyimprovingthefacility aboveits initial value(orpreservingit in newcondition ifthis is notpossible).

Since#

=

^c foraninternal solution, maintenance rn

=

(p-

cl)/2cz

Also,

la (b+d)

t-t-

^r (5")

sop g* + (p-l.t*)e

"a/bxr’’),

similarly asthepreviousexample,andsolving for xin Equation(1),

x ke^"at+

ke

^{a/’)r’ +x*}

(p-p*)

forconstants

kl ⁼

2c(2d+b)

^{k Xo -x*}

^ke

^a/b)r.

Thompson consideredonlytheprofitablefacilitycase

r>

bp,andpresumedthat maximumrepairs couldnotfullyarrestdepreciation.

r=0.18, c=1.6 r=O.06, c=0.8

1.2 0.8 0.6 0.4 0.2 0

0 5 10 15 20

0.8 0.6 0.4 0.2

0 5 10 15 2O

mu mu ^0.4

0.2

0 5 10 ,15 20

Figure2a. Profitablefacility. Figure2b.Unprofitable facility,cosmetic repair.

Soxdecays towards x*, then adjusts towards/.t=p^attime

T.

If

T

islarge or l.t*=p, maintenanceshouldbe conductedatthe constant levelm*

=

(At*- c)/2c2^whicharrests depreciationwhenx=x*. Otherwise

both/

^andmriseor fall consistently duringthe time period, accordingasp>/.t*orp</*.

Analternativeand possibly more easilymanagedstrategytothe optimal strategy of conducting repairsattheconstant ratem*is similar to thatofExample 1,^sothatno maintenance isundertakenuntilthefacility fallsto thevaluex*,where it is maintained with maintenancem*. Thereis not agreatdifference inlong-term

remm

betweenthe two

OPTIMALINVESTMENT STRATEGIESFORRENEWABLE FACILITIES 13

r=0.2, d=0.25, c1=c2=0.5

1,2-

0.8"

0.6"

0.4- 0.2"

0

0 5 10 15 20

0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82

r=0.4, c1=0.9

0 5 10 15 20

Figure 3a. Unprofitable facility,^cosmetic repairs

Figure 3b.Profitablefacility

Figure 3ashows an"unprofitable" facilitywhere cosmeticrepairsarepossible,with r=0.2, d=0.25, b=0.1,ci=c2=0.5, xo=l,p=l, b=0.1. Then#*=0.57,x*=0.29. The repair raterisesduringthelifeof thefacilityfrom virtually zero initiallyto 15%atYear20, bringingthefacility backtoits initial conditionforsale.

However,

ifr=0.4,

c1=0.9

^asin

Figure 3b,then#*=1.14,^{x*=0.97, and}repairlevels fallfrom0.24 initially,almost arresting depreciation,^to0.11at

Year

20.

Example4. Quadratic.concave deprecial;i.0n

Thespecial case considered by

Lesse

and

Roy

(1987)isoflinearearningsr(x) rx, unit priceandrepaircostsp=c=1, andquadratic depreciationD(x)

dx(1-x/xo),

^forconstanth

and initial(new)value x.

Lesse

and

Roy

foundthat in thiscaseno internalbeststrategy existed.Inthis case, the

acility

shouldeitherbe maintained in its initial condition, or allowedtodeteriorate withoutmaintenance, dependingon the relative sizes ofratesof return,repaircosts,andeconomic discountrates.

strategies, andinfactitmaytake avery longtime torecoupthe costsof early repairs through improved rentalsifthe optimal strategy^isused inpreference.

These results can beshownrapidlyusingtheanalysis of^Section2.Equation (9)becomes l(x) (r-b)x-

dx(1-x/x^),

^{witha minimumatx*}

(d-b-r)xo/2d.

^{Maximaoccuronlyatthe}

endpoints x*=x and

xg=O,

so that as inExample 1,thefacility shouldeitherbe maintainedinnewconditionorallowedtodepreciatewithminimalrepair.Asinthat example,whichofthetwo extremesolutionsshould bepreferred depends^ontheinternal returns

I(xo)= ^(r-b)

^{andI(0)=0.Ifthe rentisbetter}than the discount rate, then thefacility should definitely bemaintained in thebest possiblecondition. If therentislessthan the discountrate,itwould beexpectedthatthefacility should be allowedto deterioratetox=O.

However,if therent isonly slightly lower than thereturnonan equivalent investment,^it maybe betterto maintain thefacilityinnewcondition rather thanto takeevenlower returnswhilethe facility depreciates.

This istypical of thesituation wherel(x)has severallocal orextrememaxima.It maybe preferable^{to move to}some suboptimalmaximumratherthan incurextralosses of proceedingtoaglobalmaximum.

Usingaquadraticcost functionc(x,m)

cm

⁺c2m doesnotalterthe solutiongreatly, althoughnow if d issmallit ispossible forthere to be a small local maximumofInear 0, and if_{c2 is}large,thismaybetheglobalmaximum, asFigure4shows.However, approachingthis maximum involves substantiallosses, and it isbettertokeepthe facility in new condition.

d=O. 1, cl0.8,c2=22,r=0.3

(x) 0.01

0 -0.01 -0.02 -0.03 -0.04 -0.05

Figure4.Internalreturn atdifferentvalues ofx,concave depreciation.

OPTIMAL INVESTMENT STRATEGIES FOR RENEWABLE FACILITIES 15

Whilethelogistic functionmaybe areasonable representation of reliabilityover time for adepreciatingfacility,it isnotveryusefulinthe presentcontext.This functionimplies there isnocostinkeepingafacilityinnewcondition(andsothistendstobe the optimum forevenquite modestrentlevels),when infact maintainingperfectcondition wouldnormally be expectedtorequireconsiderableoutlays.

Example^4. Quadratic. convex depreciation

In

contrast to thepreviousexample,thevalueof capitalitemsmight beexpectedto depreciatemostrapidly initially, so thatD’<O, Dn<O,andfor theseconvexfunctions, internalpointsofmaximal returnmaybe found asinExample2,even whencostsare linear.

In

the linear costcase, taking

D dlx+d:zd,

^forconstants

^dl,d,

and other functions asin Example 1,l(x)hasa maximumatx*=

(r-c(b+d))/2cd2.

AsFigure 5 shows,thefacility shouldbeallowed todeterioratetothispoint,sustainedthere, and then allowedtoimprove or deterioratepriortosale dependingon whetherc<porc>p.

0.8 0.6

x 0.4 0.2

o

r=O.33,d1=O.07, d2=0.05, c=1.4 r=0.18, c=0.8

0 5 10 15 20

0.8 0.6

0 5 10 15 20

Figure 5. Facilitywith linearrepaircostsandconvex depreciation.

Althoughthis strategy iseasytoapply,in fact the"bang-bang" solution isproblematic^at the stationarypoint,since either maximum or zerorepairsarerequiredwhen x drifts from x*.Inpractice, repairs^areactuallyarrestedatx*withm*=D(x*).

Thisproblem^canbe partially resolved usingaquadraticcostfunctionas inExample2.

Then asbefore,there is asingleequilibrating(x*,

At*)

wherel(x*)is maximum(orwhich canalsobefound by solving c,,(D(x*))=p* and

(b+D’(x*))bt*

=riteratively).

0.8 0.6 X ^0.4 0.2

c1=0.8,r=0.2 c2=0.5,d1=0.075, d2=0.075

0 5 10 15 20

c1=1.2,r=0.3

0.8 0.6 X ^0.4 0.2

0 5 10 15 20

0.25 0.2 0.15 0.1 0.05 0

0 5 10 15 20

0.05 0.04 0.03

m ^0.02 0.01

0 5 10 15 20

Figure6. Quadraticcostand depreciation: xandmfordifferent parametervalues.

AsFigure 6^shows,the resultsaresimilartothe linear case.

In

thefirstexample,with lowrepaircosts,repairsareinitiallynil, then areclosetom*inthemiddleof theperiod whenx is closetox*, andfinallyareaccelerated priortosale.

In

thesecondcase with high repair^costs,repairsareundertaken only in themiddleoftheperiod.

4. Time-dependent functions and obsolescence

There areseveralcircumstances indealingwitharenewable facilitywhen functionsmay betime-basedas well as functionsofthe valueorconditionoftheutility. Thefirstoccurs when theunderlyingstructuremaybesubjecttoslow deterioration that results in a

OPTIMAL INVESTMENTSTRATEGIESFORRENEWABLEFACILITIES

17

gradualrundown ofthefacilitynomatterwhatrepairs^areundertaken. The second concerns obsolescence.

Oftenaprimeconsideration in economicreturnproblemsisnotso much the technical conditionof the facilityasthe existence of other facilities using improvedtechnology.If there iscompetition,this willlower boththereturnand themarketvalueof the facility over time.

Thiscan beaccommodated in thepresentframeworkby assuming that depreciation,rental returnand repaircostmaybe functions oftimeaswellascondition

s.

4.1

Time-dependent control

Inthiscase,Equations(1)to(6)stillhold.

In

thelinearcontrol case,anexplicitsolution canbe obtained as before. Integrationbyparts yields

J (r-cD-bC+C,)e-’dt ^{+ (PXr} -C(xr,r)+C(x,O))e

^{-r ()}

whereas beforeC(x,t)

= c

_m

(, D(.),t)dx

x

Thenettinternal

return

isthen

I(x,t) r-cD-bC + C_t (12)

wherethe last termallowsfor possible savings through avoiding future increasesincosts.

The linear control solution is,again,tomovetowardsa maximumof 1,which willoccur whenI_x=0or

r

x-

^(cD)

x-

^{bc +c 0 (13)}

If theLHSisnotindependent ofx, thisprovides possible optimal paths for xovertime, above andbelow which minimal and maximaleffortare to beapplied.This is shown in Figure7,withatypical time-dependent optimal pathandadjustments priortosale.

Notethat Naslund

(1966)

andThompson (1968)^arespecial cases where functions^are onlydependenton andnotx.

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

5 10 15 20

Years

Figure 7.Repairstrategywithobsolescence.

Theslope oftheoptimal pathwillhavethe samesignas

Ixt,

^{which isthechangein}

marginalnettreturnwith time.Itwillbe negativewhenrentsarefallingover time, and whenrepaircostsarerising^nofaster thanthe discountrate.

TheLHS of Equation (13)^willbeindependentofx only in the specialcaseconsideredby Thompson

(1968),

whererand

D

are linear in x, and c is a functionof only (itwillalso beindependentinaperfectmarket,as Section4.3shows).Thecontrolisthen independent ofinitialvalues of x,and isappliedattimes determinedbysolving thedifferential equation(5)_{for #.}

4.2 Optimal time horizon

Undermanyconditions anoptimaltimeforretainingthefacility^canbe determined. This canbedoneusing another transversalitycondition, orby observingthatthe original problem^{can be}transformedas

7"

J(x,m,T)

(r ^c + (px)-bpx)e

^"t

dt +

o

p.x

_o (14)

OPTIMAL INVESTMENTSTRATEGIESFORRENEWABLE FACILITIES

19

eliminatingtheunknownfinalvalueofx.Clearly,if theintegrandbecomes andremains negative as

T

increases, the integralwilldecrease, so thatanecessaryconditionforan optimal Twill be

E(x,m,T)

= r(Xr) c(xz, ^mr)

⁺

⁽¹³ rxr ⁾ _bPrX

r

=

0 (15) Thisfunction

E

iswhat isusuallyknown as thenettmarginalrateofreturn.Itincludes rentpluscapitalgainsless repaircostsandopportunitycostof capital.Equation(15)is then arestatementof the standard resultthat thefacilityishelduntilsuch time asnett marginalreturnon capital,including capitalgain or loss, isequaltothediscount rate.

4.3. Perfect capital market

One particularlyimportantexampleofadepreciationfunctionoccursunderperfectmarket assumptions.^In,aperfectmarketwithperfectforesight, the marginalreturnonany investment isthe discountrate,so that,for allt,

(px)

+ r(x)- c(x,m) bpx

=

E(x,m,O

=

0

(16)

calculated underanoptimalrepair strategym.Priceintheperfectmarket adjuststo prevent any advantagebeinggainedatdifferenttimes,andtokeepreturnsequaltothe discount rate. The nettmarginalrateofreturniszeroatalltimes,and notimefor saleis preferred^overanyother.

Ifp(T)x(T) ^isknownscrapvalueatsomefuturetimeT,the marketvaluep(t)x(t)atany timet is then

T

p(t)x(O

= e ’ j(r(x,’r)- ^{c(x,m,’f ))e-bd’ +} eb(r-t) p(T)x(T)

orthepresent value ofits nettfurore earnings aftert,(asinHotelling’s

(1925)

original paper).Thisequation maybe used in determininga’nominal’market price or market depreciationforan’unpriced’ facility such asapowergenerating plant,where nocapital marketexistsbutnettincomemaybe estimated.

Alternatively, ratherthanthecapital price adjustingtomaintain return,the rentalmarket may adjusttoreflectmarginalcosts.

In

thiscase,rentals willbesetaccordingtoEquation (16)tocovermarginal costs,including opportunitycostsof capital,nettof capital gains.

Onceagain, anysale time will beoptimal.

5. Conclusions and applications

The

general

problemof determining the optimalrepairstrategy foradepreciating renewable facility hasbeensolved, generalising theresultsofother authors who have lookedonly^attime-dependent systems^oratparticular examplesofdepreciation^functions.

A^betterinsightintothequalitativenatureofthe solutionhasbeen obtainedby using several differentsolutionmethods,including optimalcontrol andintegration byparts.

Ithasbeen shown that the facility should move towards^apointorpathwhere the"nett internal

return"

isgreatest, and shouldthenberepaired^orallowedtodeclinepriortosale, dependingon whether thecostofrepairscan berecoupedin improved capital value.

Typicalsolutionsfor severalcostanddepreciationfunctionshavebeen obtained. The time whenthefacility should be soldisthepointwhere "external"marginalreturn(which includescapital gain)isequaltothediscount rate.

In

aperfectcapitalor rentalmarket, this willhold everywhere.

Thegeneralstrategiesdevelopedin thispapercan beappliedtoany facility aslongasthe appropriatefunctions areknown.

However,

applyingtheresultsinpractice impliesa deterministicknowledge of depreciation,costs,rentsand priceswhichisunlikelytobethe case inpractice.The discountrateisapartial proxyfor uncertainty,but thissingle parameter may be inadequatetoallow for changes in futureconditions indetermining optimalstrategies.Strategiesare quite sensitivetothe values and form of these parameters andinpractice^aregularreviewof optimalstrategieswouldbenecessary.

Astatistical approachtooptimalmaintenancewould beanaturalextensionofthiswork.

Anumber of authors havegeneralisedthe deterministic workon machine maintenanceby Naslund(1966)andThompson(1968),to take intoaccounttherandom natureof

breakdowns^orof future prices,and thiscould also be donein thepresent^case.

However,

as hasoccurredwith stochasticapproachestomachinemaintenance, aknowledge ofthe deterministicsolution isthe firststep in examiningstochasticstrategies. Also,the deterministicsolutionprovides^thenecessaryframeworkfor determiningthecomparative staticeffects ofgovernmentmeasures suchastaxation and depreciation allowancesupon

mai,ntenance

and retirementstrategies, and the companionpaperFlood(1997)showsthat these.canbevery significant.

6. References

1. Barlow,R. and Proschan,F. StatisticalTheory

of

Reliability andLife^Testing: Probability Models. NewYork: Holt, Rinehart and Winston. 1975.

OPTIMALINVESTMENTSTRATEGIES FORRENEWABLE FACILITIES 21

2. Chand,S. and Sethi, S.P.Planninghorizon proceduresfor machinereplacementmodels with severalpossible alternatives. Naval Research Logistics Quarterly 29, 483-493, 1982.

3. Cho, D.Optimalproduction andmaintenancedecisions when a system experiencestime- dependent deterioration. Optimal Control 14,pp 153-167, 1993.

4. Flood,J. Optimal replacementand taxation.To appear.

5. ^Gertsbakh, I.B.Models

of

Preventive Maintenance. Amsterdam, North-Holland, 1979.

6. Gertsbakh, I.B. StatisticalReliability Theory. NewYork: M Decker, 1989.

7. Hartl,R. A mixed linear-non-linearoptimizationmodel ofproductionand maintenancefor a machine, inFeichtingerG. (ed). Optimal Control Theory andEconomicAnalysis.

Amsterdam:NewHolland, 1981.

8. Hopp, W.J. ^andWu, S.C. Machine maintenancewith multiple maintenance^choices, liE Trans 22, pp 226-233, 1990.

9. Hotelling,H. A generalmathematical theoryofdepreciation. Journal

of

^theAmerican

StatisticalAssociation (20), pp 340-353, 1925.

10. Jardine,A.K.S. OperationalResearch in Maintenance. Manchester: Manchester University Press, 1970.

11. Jardine, A.K.S. Maintenance, Replacement and Reliability. London: Pitman, 1973.

12. Kamien,M.I.and Schwartz,N.L. Optimalmaintenanceand saleage forasmachinesubject tofailure. ManagementScience (17), pp B495-B504. 1971.

13. Lesse, P.F. ^andRoy, J.R.Optimalreplacementandmaintenance of urbaninfrastructure.

Environmentand Planning A (19), pp 1115-1121, 1987.

14. Mehrez, A. andBernan,W. Maintenance optimal control:three-machinereplacement modelundertechnological breakthrough expectations. Journal

of

Optimization Theory and Applications($1), pp 591-618, 1994.

15. Naslund,B.Simultaneousdeterminationof optimal repairpolicy and service life. Swedish Journal

of

^{Economics (68), pp 63-73, 1966.}

16. Newnan,D.G. Engineering Economic Analysis. SanJose:EngineeringPress Inc, 1983.

17. ^Pierre,D.A. Optimization Theory withApplications. NewYork: Wiley, 1969.

18. Pierskalla,W.P. andVoelker,J.A. Asurveyofmaintenancemodels: the controlof deteriorating systems. NavalResearch Logistics Quarterly(23), pp 353-388, 1976.

19. Pintolon, L.M. andGolders, L.F. Maintenancemanagementdecisionmaking.European Journal

of

Operational^Research (58), pp301-317, 1992.

20. Pitchford,J. andTurnovsky, S.J. (eds.). Applications

of

Control Theoryto Economic Analysis. Amsterdam: North Holland, 1977.

21. Rapp, B. Models

for

^{OptimalInvestmentand}MaintenanceDecisions. NewYork: Wiley, 1974.

22. Tang, M.G.Astochasticmachinemaintenance and saleproblem resultswith different production functions. Naval Research Logistics (40), pp 677-696, 1993.

23. Thompson,G.L. Optimalmaintenancepolicy and saledateof amachine.Management

Science (14), pp 543-550, 1968.

24. ^Virtanen,I. Optimalmaintenancepolicy andplanned sale date for amachine subjectto deterioration andrandom failure.EuropeanJournal

of

Operational^Research(9), pp33-40, 1982.

Special Issue on

Intelligent Computational Methods for Financial Engineering

Call for Papers

As a multidisciplinary field, financial engineering is becom- ing increasingly important in today’s economic and financial world, especially in areas such as portfolio management, as- set valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the re- cently approved Basel II guidelines advise financial institu- tions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to im- prove the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.

However, more and more researchers have found that the financial environment is not ruled by mathematical distribu- tions or statistical models. In such situations, some attempts have also been made to develop financial engineering mod- els using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estima- tion technique which does not make any distributional as- sumptions regarding the underlying asset. Instead, ANN ap- proach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting pa- rameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior perfor- mance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should es- pecially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelli- gent, easy-to-use, and/or comprehensible computational sys- tems (e.g., decision support systems, agent-based system, and web-based systems)

This special issue will include (but not be limited to) the following topics:

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• Application fields: asset valuation and prediction, as- set allocation and portfolio selection, bankruptcy pre- diction, fraud detection, credit risk management

• Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, imple- mentation

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Guest Editors

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Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong;

yulean@amss.ac.cn

Shouyang Wang,Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; sywang@amss.ac.cn

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