Optimal overhauls and replacements of deteriorating and stochastically failing equipment

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MAL OVERHAULS AND REPLACEMENTS OE

DETERIORATING AND STOCHASTICALLY

RAILING ECJUIPMENT

A THESIS

SUBMITTED TO THE DEPARTMENT OF INDUSTRIAL

ENGINEERING

AND THE INSTITUTE OF ENGINEERING AND SCIENCE

OF BILKENT UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

MASTER OF SCIENCE

By

A'lu.stafa Karakul

August, 1996

fe·

Ts

İ8 1 .6

^ S S 6

I certify that I have read this thesis and that in my opinion it is fully adequate,

in scope and in quality, as a thesis for the degree of Master of Science.

oğrusöz (AdVisdr)^

I certify that I have read this thesis and that in my opinion it is fully adequate,

in scope and in quality, as a thesis for the degree of Master of Science.

Assoc. Prof. M. Cemal Dinçer

I certify that I have read this thesis and that in my opinion it is fully adequate,

in scope and in quality, as a thesis for the degree of Master of Science.

Assoc. Prof. Erkan Türe

Approved for the Institute of Engineering and Sciences:

Prof. Mehmet Bara}^

ABSTRACT

OPTIMAL OVERHAULS AND REPLACEMENTS OF

DETERIORATING AND STOCHASTICALLY FAILING

EC^UIPMENT

Mustafa Kaicikiil

M.S. ill Industrial Engineering·

Supervisor: Prof. Halim Dogrusdz

August, 1996

In tliis tlu'sis, the |)гоЬ1егп of optiruaJ overhauls ancl rcplacenu'iits of de(,e- ı■¡oı■a.l.¡ng and siocliastically failing e(|uipmenl, is studied and solved lyy a new conec'pt. Stratified Dynamic Programming, as we call it, witli forwai’d i4'cur- sions. In Uic Stratified Dynamic Progi-amming model, time' avei-a.ge rat(' of cost of i4'plac('ment cycle, wliicli takes into a.ccount tim e value of money, is ns('(| as the o|)timiza.tion criterion. VVe |)rovide the algoi'ithm to solve tin' mod('l d('V4'loped and make sensitivity a.na.lysis to establish an insight into tin' prohh'in situation of such character. It is shown tliat, dyiia.mic |)rogi'a.mmiiig with Гопуа.1ч1 recursions is the most suita.hle approach for solving ratlu'r com- ph'.x ı·epla.c('meut problems, beacuse the necessity of prefi.xing the пит1кч· of s(.ag('s is avoided, and in a sense optimized. The review of related litei'aturc' indicat('s tlia.t this study fills a. gap.

Key words: P,('|)lacemcnt, Overliaul, Maintenance, Stratified Dynamic Pro­ gram m ing, T im e Avera.ge Ra.te of dost.

ÖZET

YIPRANAN VE RASSAL OLARAK BOZULAN

EKİPMANLARIN EN İYİ REVİZYON VE YENİLEME

KARARLARININ TESBİTİ

Mustafa Karakul

Endüstri Mühendisliği Bölümü Yüksek Lisans

Tez Yöneticisi: Prof. Halim Doğrusöz

Ağustos, 1996

Bil yi|)i-a.ııa.ıı ve rassaJ ola.ra,k bozulan ekipmanların c'iı iyi i'evizyon v(' yc'iıik'me ka.ra.rlan |>robleıni araistınlch ve “KatmaııLı Dinanıik Progra.m- lama.” a/lını vcu'diğimiz bir modelle çözüldü. Modelde paranın zama.na. bağlı d('ğ('i'iıd dikkate alan “zaman ortalamalı maliyet ora,nının minimiza,syonu" c'iı- iyilenu'ölçütü olarak kullanıldı. Bu modelin çözümü için gerekli l)ir algoritm a gc'liştiridi v(' bilgisa.ya.!' progra.mı yazıldı. Böyleee verilen bir referans para.- ıiK'trc'k'ri kümesine uygun l:)ir örnek çözüm elde edildiği gibi parametrelerdeki d('ğismel('r('cluyarlılık analizi de ya.pılarak, bu tip probkmıleriıı ta b ia tın a nülüz ('dilnu'si sağlandı. Bu çalı.‘;ima, a,yrıca, bu tip karmadık yenileme problem- l('rinin çözümünde, en uygun yakla..'5 imın Dinamik Rrogramlaıua olduğunu da göstc'rmifştir. Algoritma.da “ileri yineleme” tekniğini kullanma, a.ça.ma sayısını onr('d('iı tc'spit gi'inği olmadığı için kaçınılmazdı, ve bir a n lam d a a.'^a,ma sa.yısmm optim izasyonuna da imkan vc'rmi.'^tir. Literatür tara.ma.si, bu ara.çtırma.nm biı- boşluğu doldurduğunu göstermiştir.

Analılar sözcükler: Yenileme, Revizyon, Bakım, Ka.tmanlı Dinamik Rro- g ıam lam a. Zaman Ortalamalı Maliyet Oranı.

ACKNOWLEDGEMENT

I a.m iii(ld)t('d to I’rol. Di. Jlalim Dogruscjz lor his invalnabK' guidance, ('iicom-a.gi'nu'ui and above a.11, lor the euUuiwiasm which he inspirc'd on nu' during this stiuhy.

I a.m also iiuh'bted to Assoc. Prol. iVI. Cema.l 1)11Ц'ег arid Assoc. Prof. Krkan 'I'iiie For showing keni intei'est to tlie snlrject m a tte r and a.cceptiiig to read and r('view this tlresis.

I would like to tha.nk to Prol. Dr. .Ja.y B. ( ¡hosh for lus invaluable commenl.s and lu'lp during tliis study.

I would also like to thank to my classmates Abdullah Dai^ci, Erdem (¡ündü//, Mnıa.l. Aksu, Bayram Yıldırım, Sıraceddiıı Orıeıı, Saınir l'lllıedhli and Sa.va.ş Da.yamk lor their fric'ndslıip and patience'.

k'inally, I would like to tha,nk my parents, brothers, sisl.e'rs anti Aynur for tİK'ir pati('nce, (4 icoura.gement a.nd the moral support they [irovided.

C o n te n ts

I I n t r o d u c t i o n

I. I Srf)|)(' oF File Study

1.2 'I'lic ( 'om|)arative Ijitc'raturo Itcvievv...

I

I .d Oi'ganizatioii oF the Thesis and a Sliort Sum m ary oF I'diidings . . 12

2 F o r m u l a t i o n o f t h e P r o b l e m 13

3 C o n s t r u c t i o n o f t h e M o d e l 21

'I'Ik' Dynamic Programming Model 2d

d.I.I l)eriva.tion oF the FuiictionaJ l 'k | u a t i o n s ... 25

d. 1.2 'I’Ik' Forward Algorithm 27

d.2 ( 'om|)utationa,l Reduction R e s u l t s ... dl

4 N u m e r i c a l A n a l y s i s 36

1.1 .\ Discussion on I m p l e m e n ta tio n ... d7

1.2 I'i.xample... dS

CON'ri'lN'l'S

I.;} Sí'iisitivity AnaJy.sis

3.1

Sensitivity of the Model to SaJva.ge Va.liie

3.2

Sensitivity of tlie Model to 0|)('i-ating (iosls

1.3.3

S('iisitivity of the Model to Maiiiteiia.iiee Costs

1.3.1

Sc'iisitivity of the .Moch'l to HcviJ fntei'est Rate

15

15

18

53

58

5 Conclusion

60

B IB L IO G R A P H Y

62

A Proofs of the P roperties in Chapter 3

66

B R esu lts of the First A lgorithm

69

L ist o f F ig u res

2.1 ()|)era,(,iiig (Jliai'cicteristicis ol’ tlic .System uiider Coiisidei-atioii . . 10

2.1 .Midiisia.ge Dynamic 10-ogramnnng Diagra.m... 25

2.2 'I'lu' i-elation between T A ( X '{ T ° ) and T A O C '{n J '°). .22

1.1 ,S'„ versus 7 ” ... K) 1.2 7 v('i'sus , 7i*, 7’*... 17 1.2 //,, versus 7’*... 10 l. l A//, versus /'’° . , ? r , 7 ’*. .50 1.5 C(i v('rsus ,7/.*,7'*. 51 l.f) A c versus / T , //?... •'"77 1.7 A 7 ;(l) versus /'’°.,7i.*,7'*... .55 1.8 A 7 ; /7 versus A’, 77.*,7'*... .55 1.0 (',■ versus 57

I.IO lvx|)('el.ed bil'etime versus 7r*, 7'*. b8

1.1 I n versus 77.*, 7 ’*... 50

L ist o f T ables

. I Surge 1 oi l.lie Results of Algoritlmi 2 for flic lUxampk' Rroblr'in 10

.2 Stag(' 2 of the Rxrsults of Algoritlini 2 for tlie 1Axa.ni|)l(' Riobhun II

.2 Sl.age 2 of th(^ Results of Algorithm 2 lor the lUxample Problem 42

. I Stage 4 of the Results of Algorithm 2 for the lUxample Ib'obh'm 12

.0 Sl.age ■') of the Results of Algorithm 2 for the Exam|)h' Probh'iu 12

.0 Stage () of the Results of /Mgorithm 2 for the Example Pi-obhuii 12

.7 Sta.ge 7 of the Results of Algorithm 2 for the lAxa.mple Pi'obh'm I I

.8 Effect of ,S'o on tlie Optima.1 P o li c y ... Ki

,0 Effect of 7 oil the ()|)timai Policy 17

fO I'fflV'ct of/i() on the Optim al P o l i c y ... If)

, I 1 Effect of A /i oil the Optim al P o l i c y ... bO

12 Effect of C() oil tlie O ptim al P o l i c y ... bl

U2 Effect of A c on the O ptim al Policy 52

11 Effect of (X){[) on the O ptim al Policy b f

lb Efh'ct of A O l i on the O ptim al P o lic y ... bb

U S ' ! ' O F T A F L E S XI

I.K) I'yircci of c,. on tlie 0 |)tim al P o l i c y ... 50

1. IT Pilcct of A on Uie Optima.! I\)licy (

1.18 Itil'cci of (V ()ii tlie Optimal P o l i c y ... .T!)

B.l Sta.ge 1 of the Results of Algoritlmi 1 for tlie Ifoxample Probh'in 72

C h a p ter 1

In tr o d u c tio n

1.1

S cop e o f th e S tu d y

111 liliM-a.t.iiri', I,lie (’oiiccpt oF re|;)laceinent is deiined extensively, l^ew oF tlu'sc'

arc' ,i>^iv('ii Ik'Io w.

" KVc'pla.eeirient dcxdsioii is an investinent decision, or a conipoiK'iit in an in- \'('stin('!it clc'cision making problem. It may lie viewed a.s making investmenl wIic'ii a pic'ce oF ec|iiipment is to be taken out oF a system so tlial a new ('c|nip- mc'iil. is iiistaJlc'd to take |)la.ce oF tlic' old takc'ii out.'’

Ilallni Dof'jrusorj

Ivepla.ec'nu'iit in a dyna.mic society is the displaccMnent oF capital goocls Irom (.lic'ii· (imction or service. It is Fnnctiona.l robbery.”

( l( ovfjr T( rhovfjli

IF all tlic' (v.|nipment re(]nired no maintena.nce, were as good a,s new to the' ('ink collapsed iinally a.ll at once in a liea.|) oF junk and if tlicw wc're not dis-

pla.cc'cl bc'Foi'e the end oF their pliysical endurance by improved sniistitntc's, tlien

the' pi'obic'm of when to replace would lie as simple as the |iroblem oF whem to

laix'ly met. If not all, the ma.jorit,y of the e(|uipineiits sud’er a debaisemenX of rimetion over their life. This debasement of function nuw In' eithei' (|iiantita.- tive or qualitative. 'I'hat is, tiiere may be either a decrea.se in the am ount of s('i-vie<' ı·('nder('d a.s the unit ages or a deterioration in tiie (|ua.lity of the service, or l)oth. .Moreover, in a dynamic technology, they're ,subj('ct to tlu' com|)('ti- tion of iiii|)roved substitutes, so th a t the quality of their sc'rvicx' ma.y decline r< l.alivc lo aimilablt alternatives even when it does not deteriorate absolutely, t'ons<'(|iK'nc.es of deteriora.tion appea.r in one oi· a combination of the following:

i) d('cline in the I'ate of revenue,

ii) increase in the rate of expenses,

iii) d('cline in tlie salvage value.

CIIAPTER J. INTRODUCriON

2

.Anotlu'!' cha.ra.cteristic th a t should be i.a.k('ii into accomit is tin' stochastic In'ha.vioi·. 'I'ypically, in a world governed by stocha.stic ev('iits, it is (piite natural th a t th(' <'(|uipments fail randondy, i.e. at ra.ndom tijiu's, and their failnia' rat(' (s('(' delinition L.1.1) increa.ses with a.ge a.nd use. As the ('(|uipment gc'ts ohh'r, costs a.ssocia.ted with lailures (repair costs, losses diu' to failures, ('tc. ) i iici'('a.s('s.

.\s tecimology develo|)es, new complex ma.chinery and e(|uipment are g ra d ­ ually ix'pla.cing tlie labor force in m anufactni’ing. Since the share of ma.chiiK'ry and ('(|nipment in tlu' total m anufacturing costs increases, they have to fu' iis<'d mor(' effectively and efficiently. Tliis situation points out tlu' im])orta.nc(' of maintc'nance |)lamiing clearly. When we talk about m aintenance we nu'an ı■<'|)a.irs, overhauls etc...

W Ik'ii dea.ling with the repla.cement decisions a.ssociated with l.lie e(|uip-

mc'iits d('scribed al)ove, it is necessa,ry to know something a.l)out tiu' failure' time's, ll is I'are'ly the ca,se tha,t the engine-X'r can predict when a failure will occur but it is usuaily possible to obtain information abe)ut tlie probability e)f the' life'time' e)f the' systems.

Aiiotlier ini|:K)i'l,a.iit. as|)cd, oF ı·eplcVceüıeIlt. problems is time vaJue ol' money. VV(' a.i'(' (Ve(|uent]y concerned with rephrcrmient decisions foi- fairly ('X|)('iisiv(' pi('ces of |)la.nt, oi- ev(‘ii the |)la.nt itself, wliere tlu' replacement intei-va.l is iiK'a.-

sm-('d in tc'i'ins of years, ra.tlier than weeks or montlis as is the case of i-eplace-

iiKMit of minor components. VVlien this is the ca.se, account must Ik' taken of th(' fa.ct th a t tlu' value of money changes with tinux Money s|)('nt in tlie future' is woi'tli less than money spent at present, therc'fore' money spent in tlu' fiitiii'e musl. Ix' discounted to its |)resent da.y value to enable comparisons of a.It('rnativ(' r('|)lac('ment decisions.

rilÁ P T E ll I. INl'IiODifCTION

:)

In this tlio'sis, tli(' system we a.r(' analyzing deteriorates with time, where d('t('rioi'a.tion is realize'd lyy the inci'ease in the o|)('.rating e.xpenses. l(. is a. i('pairabl(' system which fails stocliastica.lly with exponential lifetime dis1.ril)u- tion. I'ailurc's are ic'paired miniina.lly whenever a failure occurs, hbr l'epairabh' systc'ins, vninnud repair concept wa.s first introduced l)y HarlowMllimtei· [2]. lbid('i· minimal r('pa.ir, it is assumed tlia,t tlu' repair a.ction returns tlu' syst('in in(.o opc'rational sl.ate such a. way th a t tlu; system characteristics arc' I.Ik' sa.nu' as just b('foi’(' failmx' i.e. the system is a.s good a.s old. In other woixls. minimal repail· of lailures does not clumge the sy stem ’s failure rate. .A forma.l définition of minimal repair is provided by Nakaga.wa&Kowada [20].

Since operating ex])enses increase by time, overhauls should be scheduled in oi'd('r to revise' the system. Overhauls can not bring the system to an as good as a. iK'w condition but it pos.sesses a nearly new condition. By ea.ch ovc'rha.ul condition of tlie system departs from its new condition somewhat., both an iiici'ea.s(' in the after overhaul operating cost ra.te a.nd an increase' in the' failure' i-al.e'. Me)i-e'e)ver, e'e)sts e)f overhauls incre'a..se |)re)gressively, i.e. by the uuml)e're)f pre'vieeus e)ve'i-|iauls maele. So there exnnes a. tim e tha.t. no me)re e)ve'rha.id pays itse'lf aiiel the' system should be replae’.eel.

We' assiimeel tliat the system uneler ce)nsidera.tie)n ele)es ne)t have' tee'lmole)g- ica.l life', se) we e'e)iisieler only the economical life. Mere the t.erm l(cnolopical l.if( is use'el as the e)pera.tional tim e perioel be'ye)iid whie·!) the ee|uipment e'an not be' ma.ele e)pe'rable by any nmans. For tliis re.'a.son the time' horize)n we'

('ll. \ РТ1Ш I. INTIiO D U CTiO N

(•oii.sick'r i.s A plant, a. nuclear reactor, a. sliip etc.. ma.y l)e vievv('(l, in a.p|>ro.xima.tion, examples of sucli systems tliat vve a,re woi'king on.

Since tli(' syst('ins uiuler consiclera.tions are fairly expcmsive and long-la.sting, time value' of money is considered. And although in sncli long horizons llnu' av('ra.g(' valiK' of salvage will he microscopic and wdll have no alfect on the policy d('velo|)ed, vve include it for the sa.kc' of complete'iiess..

So oiii' ol)j('ctive is to determ ine optimal scliedule of overlnuds and tin' ( iiiK' of i('pla.cem('nt of the ('(|uipm('nt so a.s to miinmize the time average unit cost pe'i’ unit tinu'. 'I'ime a.verage unit cost concept tai\irig tim e value of money into a.ccount in geiK'ra.l, was first introduced hy Halim Dogrnsoz [19]. A preci.se d('linition of this concept will be given in (fhapter 2.

W<‘ assume th a t the system is replaced with an identical one. So the r<'- pla.c('ments form a regenerative |)rocess. 'I'lierefore in order to solve' the infinite hoi’izon prohh'tn it is sulficient to determ ine the optimal policy for tlie first ı■('pla.cem('nt cycle. VVe develop a. forwa.rd dyna.mic programming algorithm to d('t('i inin(' tli(' optim al ove'rhaul and replacement time policy, 'f'lu' tim e average' imil. ee)st pe'r unit time which is a. function of the numl)er e)f e)verhauls I.e) he' ma.eh' in the lirsf. i’epla.c,ement e\ycle is a mumoelular fuue'tion with a. glejhal m in­ imum. It ele'e-1 'ea.se's until a minimum a.nel l)e.'gius te> increase a.fterwarels. 'I'he' forvva.rel re'e-ursive algoi'ithm se)lvevs foi' highe'.r and higher nuiuhei' e)f ex)nsee-id.ive' ove-rlia.ids t.e) he' maele' hefoix' repla.e:ement a.nel wdien it rea.clie's a. minimum it stops. 0|)tima.l overlia.ul and replacement time's a.re eletermiueel ce)rre:'sponeliug to th a t miinmum.

1.2

T h e C om p arative L iteratu re R ev ie w

Maiidx'nane’e nmelels lor eleteriorating and stocliastically falling e'ejuipme'nt have' attra.e i.e'el the' inte'rest e)f many scholars a.nel pi'a.ctitie)nei's lor the' past thi'e'e d('ca.de's. One e>f the' ma.jor rease)ns for this is lliat maintenane'e' nmele'ls can l)e

a.|)|)li('<l to a vai'iety of aro'a,s sucli as industry, military, licaltli and tlio ('iiviron- iiH'iit. As systc'ius b('com(' more'complicated and require new teclmologic's and nu'tliodologies, more sopbisticaXed maintena.nce models aud control |)olici('s ai’(' ii('('d('d to solve' the m aintenance problems.

Beldi'(' going into tlie detailed litera.ture review of the replacement de'ci- sion making |)roblems and the snbseepient chapters, it may be nsel'ul to give' de'(initie)ns e)f some eoncepts and term s conceive'el and useel in the stuely e)i mainte'iia.iiex' pe)lie’ies, by various autherrs.

'I’lie stuely e)i m aintenance policies involve's a broad range e)f elecisie)ii making pre)ble'ins sucli as Re|)lacement, He[)alr, Spa.re Pa.rts Su|)|)ly ariel lnvente)ry, a.nel Alle)e'a.tion e)f Sta.nelby Units, etc. .

He'liability the'oi'y is an im portant part of maintenance which is exine'eiMieel wil.li elete'rmining the pix)ba.bility th,a.t a. system, pe)ssibly ex)nsisting e)l nia.riy (■ompe)ne'iits, will rmictioii during the mission time. During tliis missie)ii time the're' may eeeeair seiine unele'sirable events due te> environmental anel internal ce)nditie)ns. 'I’he'se unelesira.ble events, so e'alleel ra.ilure's, exuise' elisruptieins in the' prexx'ss. railuix' is the ix^sult of a joint actie)ii of many unpix'elie'ta.l)le ra.iiele)m pre>ex'sse's geiing e)ii inside the operating system as well as in the environme'nt in whie'h the syste'in is operating.

Ke'plaex'menl. pe)licies iucxrrporate tire' stuelies al)out the' ste)cha.stic n ature e)f the' failure's e)f tlie syste'rn with optim ization metlioels to a.e-hieve a elesiix'el ameiunt e)f epiality. QuaJity is e.\pix'sse:'el in terms of e|uantita,tive nu'asurei's sueii as, syste'in ix'lia.bility, system a.va.ila.bility or exist of maintena.nex', e'tex Systrin rrliahilily and syslein av<ı/iJмJ)il:i.l·y are defined in teiins of the' lifetimee)f a syste'in. D e f i n i t i o n 1 .1 .1 . LifeHmc of a- system is the random time from the Ix'giiming of IIk' oj)('ration until the apix'arance of a failure and il· is the sourer' of tlx'

uue('rtainty in imdnteuance decision mfd\ing.

CIIAr'I'ICn I. IN'ni()DUCTlON

r,

D e f i n i t i o n 1.1 .2 . Let X he the random variable denoting the lifetime of the system. Then /^(A" > .Vo), which is the prohahility that .\ exceeds a vaJue.ro.

is lh(' system rcHabiUty (or the survival prohahiUty) for a mission time of.ro·

D e f i n i t i o n 1.1 .3 . System avaHahility is the probability tlia.t for a sp ed lied ¡x'fiod of time the system is available for operation.

( 'HAP Tl'lll I. I N T H O D U C T I O N 0

l·'ailurc ride (oi' Im.-jard rate) r(/-), a.iul cuinulalive failure rate /(*(/), play kc'v rol('.s in n)a.iiit.('na.iirc (Iccisioii making. I'a.ilurc rate of a.n o(|ui|)ineiit a.t l.ime / i.s propoi'l ioiial to the prol)ability tha.t tlic e(|uipment will (ail in tli(' ne.xt small int('rval of tiiiH' given th a t it is good at tlu' star!, of the interval.

D e f i n i t i o n 1 .1.4. Let /''(/.) be the distribution function of the lifetime X . and supi>ose the density function f{l.) exists. Then the failure rate r{t) is,

.IV)

·,·(/.) =

I - /-(/,)

( l . l )

IF I' ft) do('s not necessaTily possess a. density, i.e. it' it has discontinnitic's, d('(inition Foi· Failure rate still applies. However, witliout loss oF genei'alil.y, thi'ongliont tins study, it is assumed th a t A lias an ex|.)onential distribution. IF /■(/) given in (1.1) is increasing with /, then liFetime distribution /'’(/) d an ln(■r('a.sing I'ailurc' Kate (IFH.) distribution. ( lonversely, it is a Dc'cixuvsing

I'ailnrc' Kate (l)FK ) distril)ution il r[l) is decreasing with /.

D e f i n i t i o n 1 .1 .5 . The cumubd.ive failure rate ll{t) is deiined as ll(t) — fo '-(/)<lt.

Having introduced some concepts related to the reliability theory W(' may now d('scril)(' tlu' ma.iiitenance do'cision ma.kiiig prol)lems. K.('pla.cem('nt (h'cision making pi'oblems nsnally consider two type oF |:)olicies: age re|)la.c('in(vnt and block i('|)la,c('nu'nt policies.

D e f i n i t i o n 1 .1 .6 . Ago' replacement policy is a policy where the system is ro'/)/aco‘o/ upon failure or at a.ge T, whichever comes first. Usually. C] is assumed to be lh(' cost of replacement at lailure and cy is assumed to be the cost ol

CIlAI^Tim I. ¡NTIUXDUCTION

ı■('¡)l;ı(^('nıcııí■ ai age 7', where c\ > 0 2· ¡'or age replacement, the average lo/ig-

nin cost per unit time is given by;

C, Fi T) + C 2 f r n

/0

(1.2)

D efinition 1.1.7.

Block rcphiccincnt policy is a. policy where the system is

r(^pbice(l (¡1)011 fciihire ciiicl ai tiines The expected cost per unit

/.///K'

following a. block replacement policy at intervcil T over an infinite time

sj)xin is given by;

wIk'I'C' ni('l') is the expected iiuinber oi Cailures in [0,7') (reiievva.1 riinctiou) conx'spoiKliiig to tlu' uuclerl,yiiig lifetime distribution. For the derivation of the aJ)ov(' roriuu1a.s refer to Barlow a.nd Proschan [3]. Note th a t (1.2) and (1.-3) ignoix's tim e value of moirny whicli may not be a.|)propriat(' ii tinu' ('xtc'iids.

Ivijxrltcl vT/Ki/r concept has originated Irom the discussion about. ;;/rc/ niaiiili vMVcc. a.|)|)('a.ring in [12],[28], [29]. In tlu'se studio's, it is a.rguo'd th at, duo' to repairing tlie wrong |)art, or repairing the laulty part partially, or dam aging SOUK' a.dja.co'ut parts wltile repa.iring the laulty pa.rt , l.lie maini.o'nanco' action may not be a.s ])erfect. as it is assumed to be. In the study of B row life 1’rose! 1 an [M] a. IVa.mo'work For iinperjccJ: niainLcvJincc was esta.blish(id a.nd uset'ul ix'sidts wo're derived. According to their discussion, imperfect repair can be defined as Fo)llo)ws :

D e f i n i t i o n 1 .1 .8 . Under imperfect repair, a system is repaired at failure. Wil li probability ]) it is returned to the as good as new state i.e. perfect n'pair. and with probability {1 — p) it is returned to the functioning state as good as old i.e. minimal r('pair. Imperfect repair is the generalization of minimal rep;dr sinc(' an imperfect repair with p = 0 is a minimal repair.

So'vo'i'al |)0 '0 )plo' have resea.rched anol sui'vo'yeol tlie a.rea o)F eo|uipment maiu- lainability. h'ii-st wo)i’ks on this subject were publishoxl in miol 19()0’s. Tho'se exco'llo'iit vvo)rks vvei’e ro'poirted in the book by Ba.rlo)wfe.Pro)so’han [3] (l9()-b)

and Mi(' suivey pa.|)('r l),y McCall [27] (19()5). Latei·, several olliers followed, iiKlnding Ba.i-loW(CPr().sdia.ii [4], Jardine [25]. 'I’lic following· suivcw |)a.|)('rs, PicvrskallafeVoidker [32], .Slicrif&Sniilh [-33], Valdoiz-FloresfeFi'ldiuan [35] and, ClioC Parlar [14],covers a wide variety of litei-ature in inaintenanc('.

rilAP'riCIl I. INTIiODUCTlON

8

McCall [27] and Slierif&Smitli [.33] divided maintenanc(' models into two distinct cat('goi'i('s l)as('(l on knowledge of the systeins. The prinu' categoiT's th('v ns('d were the "pi'eventive” a.nd “prepa.redness” maint('nanc(' models with and without complete information. In tlu' preventive m aintenance models state' of th(' stocha.stically failing system is always known with cei'tainl.y. On the otln'i- side', pre'pai-e'elne'ss m aintenance moelels also eleials with stoe'ha.stie-ally failing ('(|nipme'nt hut the' state e>f the system is assnined to be unknown unless e'ithe'r iiispee tie)n e)r le'pla.e'ement is carried out.

Pai'le)W(CPre)schan [4] and Pierska.lla&Ve)elker [32] classified the works in maintainal)ility in an e)tlier way. 'I'lieir classification se’heme was base'el e)ii the' meeele'l tv|)e's. Se) moelels useel in the maintenaiie’e prol)lems e’an be foiinel by I he' he'lp e)f the'se pa|)ers. 'I'he prime calxigories the-ey useel were' the “elise're'te'- tinu'” a.uel ”e'e)ntinue)us-time:'” maintenane-e moelels with anel without e-omple'te' iufe)rmatie)u.

\5elelez-Fle)i'e'S(C4'e'ldma.n [35] focused on the we)rk done on single-unit sys- te'ms sine-e' the' I97G surve'y. 'I'he'y classifie'd the single>unit mainte'nance' nu)ele'ls iiite) fe)Ui· e-ale'geerie's: inspe-'ctie)ii moelels, minimal repair moele'ls, shock me)elels and e)the'i· i'e'|)la.eement pre)blems.

In Che)«k·Pa.rlai' [II], altliough this рарел' surveyys the literature relaXe'el te> optimal mainte'iia.iice anel replacement irmdels for m ulti-unit syst.ems, tlie an- tlmrs suggeste'el a new e-la,ssiiica4ie)n scheme by ta.king into a,e'e’o u n t the previe)us a.p|)re)a.e-|ie's. Tlie prime e:al,eigorie:s the.'y prendele is the ‘‘pre'ventive” anel “pre'- pare'elne'ss” mainte'nane'.e models with a single'-unit e)r multi-unit syste'in. hor single'-unit syste'ms the'y provide several snb-cate'gorie's ba.,se'el e)ii moelels type's a,s ce)ntre)l tlie'e)ry ivmelels, slmck models, age replae'ement nmelels, minimal I'e'- pair nmele'Is, anel inspectioii/rnaintenance models. The intere'ste'd reader shoulel re'fe'r to [I I] foi· the detailed classification scheme of the multi-unit systems.

( 'ІІЛРТІШ I. INTRODUCTION

III i,ІИ' s(4|ii('l, SOMU'of I,ho most important studies a.bout maiiitoiianco mod-

('Is and tİK'ir comparison with this stud,y is provided.

KaiofcOsaki [2(і] reviewed some discrete and continuous lifetime d istrib u ­ tions. 'I'hey applied them to the a,ge replacement am.l block r<'|)la.cement prob- h'liis. TIk' i4'sults of tlieir study is provifled by tables whicli can be usc'd a.s a ı■('fer('nc(' guide'.

ІІауіч' [2d] workc'd on tlu; decision making |)roblem of wlu'ther to repair or rephui'. 'ГІИ' system he considered was deteriora.ting ov('r time. 'ГІіе sysi.em was to bc' i4'|)aired or replaced Iry a new oiu' when the de'terioration h'vel i4'a.ch('d a critical l('vel. According to repiaciMuent re|)airs are (|uite clieap Ind, its ('llV'ct is c-omparatively less effective', /\fter repairs ne'w failnres may e'ome in slmrter times. Ce)s1,ly re'|)lacements renews the systeem. 'I'lie' {,ra,ele-e)lf betwe'en re'pair a.nel re'|)la.ce'iuent is пюсіек'еі as a semi-Ma.rke)v elee’isie)ii pre)e’e'ss a.nel tlie long-rnn a.ve'rage e-e)st pevr unit tim e is minimizexl.

In a similar study, Yun&Bai [1] e'onsidereel an a.ge repbu-ement ре)1іе’у with i('paii· ce)st limits. ТІк'у assumeiel minimal repairs at failures. Uneler this пюеіе'і. syste'in was te) be' ı■e'|)la.e·eel eitlier if tlie; estima.te'el re'jrair e’e)st. e'.xe’eeels a. e a.l- ciilate'el e'eest limit value L e)r at age 7', whie'hever e)e'curs lirst. ba.ter tlie'v (onsiele're'el a. re'pair e'ost limit pe)lie;y for a system with impe'rfect re'pair [-'І!)]. 'ГІи'іг aim was to find an o|)timal e'ost limit L, for the elee'isie)ii e)f wliether Іч) re'pa.ir e)i· re'place at failures, e)ver an infinite time 1югі/д)П. They assumeel a ra.iiele)m repa-ir e'ost a.nel if the estima.ted repa.ir cost is Ік'уоікі the level of L, it is е'еч)ііотіе'а1 to repbu'e tlie systenn.

Na,kagawa.ik·K’e)wa,da [20] analyzeel syste'ms with minimal repairs at failure's. 'ГІи'у a|)|)lie'el the' re'sults obtalneid to a repbu'ement prolrlem. d'he'y assiimeel that the system under consideration was ininimally repaired upon Failures and was re'pbu'e'el e'ithe'r a,t age T e)r at the n"'' failure whichever e)e'e’urre'el lirst. 'ГІіе'у assıııiK'el lixe'ej герачг e'osts. In this stuely theey prendelexl the e'onelitie)ns iineler wliie-h t he' e)|)timal iiumbe'r of repairs is finite and ппіеціе.

('11ЛРТ1'Ж í. INTIIODVCTION

l lial а.Г1,('г ('adı rc'pair 1.1к' failure; rate; ¡ис1ч^аа('(1.'Г1к'3( coiiísid('i4'(J t.vvo policic's. OIK' for siiiglt'-uiiit H^^stc'iTis а.1 к 1 огк.' for mulii-uiiit, s\(si<'ins.

(dd'roux ('t. al. [Id] also сои8 1 с1 ("ге'( 1 а.м a.gc' rc'plaaíiiK'iü policy vvifli ininiinal i4'|)airs al. failure's but, l.lu'y usexi ra.iidom ro'pair costs. lf(;'|)la.c('uioiit at failure' is assuiiK'd to ta.k(' place if the random cost (■ of repa.ir is bigger than e')C|, where C| is the' coustaut cost of replacement at failure a.nd 6 is a known percentage. 'Г'||('У ha.d provided the' cost fuiiction over an infinite time liorizon and the' solution algorithm to find tlie optima] replacement policie's.

O ur study is e|iiite differemt from tlie above studie's in tlial. uoue of the

above' studie's considered overluuds a.nd time' value of moue'y which we' couside'r

as a prima.ry conce'rii. VVe' a.re minimizing lime nvcra/jc unit cos! jx r unil lime

which is a crile'ria none of theMii usexl. In our stud^g we tried to model more ge'iK'i'alize'd |)robl('ms. Although it is not geii('ra.l enough, tlie model we' used showe'el that more general problems can be solvexl by tlie a.pproa.ch we' use'd. In the' .4('e|ii('l. W(' revie'W some works of authors who considered overhauls.

First studie's on such problems sexmi to be the one's iiia.de by White' [d7], [•Ui]. Ill [.■?()], the' autlior used tlie next overha.ul epoch a.s the ele'cision variable' ill (he' dyiiamic |)rogra.nmiing formulation of tlie prolrlem of dete'rmining the' optimal ove'rhaul policy of a system which is subject to give' .service for n units of (,ini('. Ill Ills stud}e, he maximizexl the ex|)ect('d profit (uiidiscouiited) ove'r the' iK'xt n units of time. 'I'lieii lie let n to increase to infinity and under certain vc'iy g('iiera.l coiiditioiis determinexl the maximum |)rolit |)er unit time, lii [d7], he' exl.('iie|('d the a.bove' inode'l by incorpora.tiiig/ei//(m.s which force' immediate' ove'ihauls. lie coiisiderexl the problem of overhauling a furnace', sulije'ct to failure's. At spe'cilic tim e intervals (a. fixexl multiple of the basic time |)eriod) from the' time'of completion of a.n overliaul or from the time' the' last ins|)('ction was made', the' de'cision of wliether to overlnuil or not to overha.ul wa,s to be' ma.de'. 'I'lu' ra.te' of |)roduction dependexl on the a.ge since la.st overhaul, aiiel on production rate' in tlie last tim e period, in a j)roba.bilistic manner. 'I'liis could be' ze'ro wIk'ii a failure occurs. A decision to overluvul, prior to a failure, ke'pt

( 'll A I. INTllOD U C riO N

of inoiioy. An overluuil lor a Failure was luiicli more <'xpensive in U'rms oF l)o(.|i moiK'Y and lime. IF, on inspeclion, the decision was nol to overlia.nl, a Failmi' pi'ior to llie next ¡ns|)eciion wa.s Forcing overhaul llien. lie d('t<'i-miii('d an oplimal ovc'rhani policy in terms oF production ra.te and a.ge (since la.st ov('rliaul) oF tiu' (H|uipment.

Hastings [22] worked on determ ining tlie optimal I'epair limits. In his study, Ik' ainu'd to sliow how dynamic |)rogra.mming metliods, a.nd in partic- nlai· .Markov I'eiK'wal programming ca.n Ik' nso'd to Formulate' a.nd solve' le'pair limit re'|)la.e-e'me'nt pi'obh'ius. He analy/x'd two prolrlems, tlie lirst eonsieh'i'ed e'e|iiipm('nt. e-ondition oF which is related to a.ge and the se'cond where' coiulition is re'lat.e'd to the luuuhci· of m ajor repairs (overhauls) the eepiipme'nt has ha.d. He' elisensseel t.he ca.se oF discounteid and undiscounted costs and Finite and in- linit.e' horizons. He developeM an improved genen-al Formulation Foi· Finite t.ime lioi'izon, stochastic, dynamic |)rogramming |:)rol)lems.

Davidse)!) [IG] ca.n'ied out a study to determ ine an optimal overhaul pe)lie-y For a.ir he'ale'rs in a Power Sta.tion oF Scotland h'de'ctrie'ity Board [I.'")]. In this sl nely, t lie' e'osts inciirrc'd using a.n ov('rha.ul policy, where' I.Ik' plant de't.ei'ioi’a.tes w'il li e)pe'i-a.tion, were overlnuil and iucrease'd luel costs. Only the Fuel cost in e'xee'ss eiF th a t ne'ce'ssa.ry lor operation iF the plant did not dete'riora.t(' W'e're inlliK'iie-e'el by an overlia.ul |)olicy. 'I'liis cost w'a.s termed as ’’('xce'ss Fue'l e'ost". anel the' e-|'it('i'ie)n te) !)(' minimizc'd was the sum oF overhaul and excc'ss Fne'l e eists ine-ui'i'e'el In'twec'ii conseentive a.imual major ma.intena.nc('overha.uls oF the' e'om|)le'(.e' boiler. He elevelope'd two m athem atical moeh'ls. 'The' iirst moeh'l was base'ei on the' assumption th a t tlie intervals betwee'n consecutive ovei'liauls were' e e)iista.nt and he use'd a ca.lculus Formula.tion a.nd a.pproa.ch to solve it, while the' se'e'ond moelel gave a. dynamic |)rogramming Formulation with Irackward re'e'ui'sions in vvhicli overha.nl interva.1 was not necessarily consta.nt. 'I'inie value' e)F monev was not considered.

ГиЛРТЕИ I. IN'rilODUCniON

12

1.3

O rganization o f th e T h esis and a Short

Sum m ary o f F in d in gs

VVc have' s(4'ii l.liaX dynaiiiic proyramininy is Uic most, siiita.ble approach Гог solving ili(' pi'ohicm at hand. Our a.pplica.tioii оГ dyiiamir |)i-ogramiidng is nontra.ditioiial in the sense tluit the niunl)er of stages is not pr('[ix('d, evc'ii it is a (h'eision varialrle. Гп the tra.ditiona.l applications nund)er of stages is known, so l)aekwa.rd and Гогуушч! recursions can be used inierchaugably. Hut in oiii· ca.s(' I'orwai'd I'eeursion is nearly a must. The algoritiun and the code we d('V('lop('d using tins approach invariantly converged to sensible' re'sults. 'I'lu' (h'taih'd a.cc.onnt of these results will be in (d iapter 4.

'Г1к' oiga.ni'/,a.tion ol the thesis is as (ollovvs: In the lu'.xt cha.ptc'r rorinula.tion of th(' r('plac('nu'nt |)roblems foi· deteriorating and stochast/ically railing ecpii])- iiH'iits will !)(' giv('ii. 4'he tliii'd cha.|)ter is devoted to tlu' construction of tiu' Dvna.mic Pi'ogramming Model used Гог solution. NuiuericaJ a.nalysis and the i('suits obtained are |)rovided in C hapter 4. The tliesis ends with the conclusion and sngg('st('d a.ve'.nnes lor luture resea.rch.

C h a p ter 2

F o rm u la tion o f th e P r o b le m

\Vhil(' rormiiUi.I.ing tlie ix'pIaceiTKMit prohlciiis ol' deterioi'al.iiig and .st.ocliastically failing ('(|nipinent one sliould take into a.ccouat of some very iinporta.nt a.s|)eets of tlic'S(' systi'ins. A.s in our ease, if tecimologically iinprovc'd substitute's are' not available', at lea.st if it is assumed like th at, e<|uipment should be re'pla,e-eel at. its e'e-e)iie)inie· life by a.n ielentie'a.l e)ue. So a.ria.lysis of tlie first, re'pla.e-ement. e'ye'le' solve's the iniiuite hori'/oii pre)bleiin. I'br tlie lbrnuila.tiou e>f the re'pla.e-eme'ut e>f ('e|iiipme'nt in a. elviiauiie'. euvireniment, i.e. teehnoiogie:al inipi-ejveinenl.s are' tei be' consiele'i'e'el. the be)ok I)y 'lerborgh [34], whicli ehi'serve's a spe'e'ial ine'ii1.ie)ii. c;i.n be' slieivvn as a goeiel relereue;e. Pra.cticall,y, evei-y Ltngine'ering L'k'euKiiny book a.ehlre'sse's this problem to se)me extent in elem entary te'rms [17].

In ge'iie'ral, me)st e'e|uipments are use'd be-'yond tlieir ee-e)ne)mie· life'. 'I'his mise-e)nee'ptie)n results from the scarcity e>f e-apital гс'ецпге^е! for tlie pur|)e)se. It se'e'ins th a t te) i-e'plae'e an ec|uipmeut still in running e-.onelitie)n is eonsidei-eel a waste'. In fact, while ti-ying to .sa,ve capital ex|)e'nelitures, more' wa.ste' incurs in eipe'i-ating e'xpe'iise's elue' to eleteriora,tion of the e:'e|ui|)ment e-einsiele're'el. Using an e'e|uipme'nt as le)ng a,s possible sa.ves e:a.pital e.xpeneliture's, but the're' exist, t.wei a.elve'i'se' I'lfe'cts vveirking a.ga.inst this. I'drstly, as tire ee|ui|)menl, gets erlele'r a. fime l ieina.l elegi-aela.t,ie)u e;an be erbserved eitlrer by a dee-rea.se iu tlie a.vailability e)i- by a eh'e-re'ase iu the epiality of tire .servie-e it givers, anel .see-onelly, inen-ease' ill ma.inl.e'uaiie-e' and ope'ratiug expen.se's. Dergrusoz et. ah [18] analyze'el the'

('IIAPTER 2. FOlWlULATION OF T llF PROBLEM

(l('n·('a.s(' ill the ava.ila.l)ility oi gai-bagc collection trucks while moch'ling tlu'ir ı■<'|)la.(■('nu'llt (h'cision.

Maiiitc'iiaiicc' ('X|)(viises consists oF tlie costs of the niajor ii'paii's, so called ovi'i'lianls, which revises the condition of the e(|ui|)meiit beldre it has failed, and th(' ri'paiis done upon tlie failure of tlie ('(|uipineut. A coinmoii assumption in ('(piipinnit repla.cement problems is tlia.t the failure ra.te a.nd other pro|)erties of an ilrmi can be related to a single para,meter referred to a.s a.ge. d'he a,g('- r('la.t('d model is nsnally valid |)rovided tha.t an equipment consists of a fairly large' number of components, a.nd that r('|)air work is carried out to ov('rconi(' particnlai· malfunctions a.nd not to raise the item to a. sta.ndard of ri'lia.bility signilira,ntly Irigher tlian the avera,ge of its contemporaries.

K.x|)('ii('nce suggc'sts th at the a.ge-related model is r('a.sonal)ly valid ('ven wIk'I'i' (piit(' extensive repairs a.re sometimes undertaken. However, there ('xists a class of problems where the condition of a.n item is better related to the inim- b('r of ovc'iiiaiils whicli it lias lia,d. In a. typical overhaul probh'in an ('(|uipm('nt is insp('ct('d from time to time and the cost of overliaul is ('stimatc'd; inspc'c- tion ma.v occni· eil.lier liecause the ('(|uipment fails, or rea,cfi('s a. c('rta.in levc'l of di'gra.dai iom or bi'comes scheduled for inspection under some system of rules. rii(' distribution of overhaul costs, the distribution of any minor repair costs, which ma.y occur Irefore the next inspection , mean tim e to next overliaul, and operating costs between each overhaul arc related to the num ber of overhauls

which l,li(' ('(juipment Inis already had.

In case of ı·('|)aira.ble systems, wliicli deteriorate by age and fail sf.ochasti- cally, rc'pairs can b(' minima.l or imperfect, d'lie a.ction suitalile for the' system nndi'r consideration should be applied while formulating tlie problem. As ('X- pla.iiK'd in (dia.|)ter 1, minimal repair does not clnuige the failure ra.tc' of the systi'iii vvhih' impe'rfect I'epair ma,y do so.

As till' ('(|uipni('nt gets older, operating expenses consisting of labor, ('lu'rgy, mainti'iia.ncc'and repair costs, etc. increase indicating the (h'te'rioratioii by a.g('. Ovi'iiianls, although can not bring the systc'm to an as good as a lu'w condition, improve' tlu' condition of the system consid('ral)l,y. So, if the fre(|uency of tlu'

ov('rlia.iils increases. ()|)eratiiig costs incurred will dccrca.se hut on the otlx'i' sid(' ov('rliaul costs will grow. 'I'lic problem is to determine the optimal numhc'r and sclu'dule of tlu' overhauls.

( 7/. \ r r i ' M 2. l·X)¡lMI i L A T l O N O F T H E P H O B L E M 1 r,

In case of unr('pa.iral)le s^'stems, or under the a,ssumption th a t ¡-('pairs and ox’ei hanls bring tlie syst.em to the as new condition, i.e. repairs and ovei'liauls arc' synoipymous with ¡'('placement, tlie pi-ol)leni of when to ¡-epla.cr' is ( ¡•ii.ical in tin' s('us(' th a t railme re|)la,cenients (i.e. ¡■epla.centent only on a faihu-e of tin- e(|uipn¡('nt) cost ¡¡K)(-e tlian preventive ¡-('plac/'nients (i.e. ¡-('pla.ci'nu'nt whih' th(' e(|ui|)n¡('n(. is in a.u o|)era,ting state). 'This is so beca.use oF th(' lost p¡·oducti()¡¡ (s<'¡■vi(·('), and the daniag(' incurr('d u|)on the failure oF the ('(|ui|)n¡('nt, ('tc.. In such ca.s('s, th(' Failure rate oF the (:'(|uipmeut must be i¡¡c¡·('a.sing. H('cau.se if th(' Faibn-(' ¡-ate und('(· consideration is constant., as is the case wl(('n Failnres oc- cm- according to the nega.tive binomial (Γ¡st¡·ibntion, replacen(('nt b('F()¡·(' Faihn-(' do('s not ('ih'ct l.l((' |)i-()ba.bility tliat the ('(|ni|)ni('nt will Fail in the ¡¡('.xt instant, gi\-('n th a t it. is good now. Conse(|nently, it. is never a.dvisa.l)le to ¡-('placi' t.h(' ('(|ui|)ni('nt l)('Fo¡■(' Failure. OIrviously, wlu'n the lailui'e rate is (l('c¡·('a.sing, such as wh('u Failm-('s occur according to the hyper ('xponential (list¡·il)¡¡ti()n, p(-('- v('ntiv(' ¡-('placenient is not applicabh'. 'I'he p¡·obl(.'m her(' is to do a. |)¡·('V('nt¡v(' ¡■('pla.c('n¡('¡¡l. jn st İK'Fore a. lailnre. Bnt stochasticity ol tlu' Failm-('s sliould Ix' la.k('n ca.¡·(' oF. So¡ne discrete and cont.innous distributions For the lilctinx' oF t li(' ('(|¡¡ip¡¡¡('¡¡t.s a.¡■(' aiialy/x'd by Kaio a.nd ()sa.ki [20]. Possibh' dist¡■ibutions a.¡■(' VVeibull. N('ga.t.iv(' Binoniial, (¡anuna., I'lxponentiai a.nd .No¡■ma.l l)ist¡■ibutio¡¡.

()bj('ct iv('( ¡'iti'ria. suita.l)le tor the ¡•('|)lacenient ch'cision p¡·obl('¡ns a.¡■(' profit, ¡■('V('nu(', cost, down t.inie and reliability. While optindxing vvit.h ¡espc'ct t.o tli('S(' (·¡■ite¡·ia., if th(' syst('m unde¡' consideration is very ('X|)ensive and t.h(' {•('placement interval is measured in terms of years rather than wo'eks or months, account, ¡nnst !)(' taken oF the fact tliat tlie vaJue oF monc'y change's with t.inn'.

b('t., (V !)(' the ('(F('ctive interc'st ra.t(' per inten'st period; E ¡¡¡¡¡nb('¡'()F int.('¡■('st. p('¡■iods; .4, uniF(){-(n s('i'ies aniount (ocem's at t.lie end oF ('ach int('¡■('st. p('¡■io(l); l'\ Fut.¡¡¡■(' wo¡·th; P, |)r('sent wo¡■th. 'l'h('n, under the assumption oF discretr (■oiirpoiin.diiifj, vvhicli niea.ns th a t the int('¡'('st is com|)ounded at th(' ('¡id ol ('a.ch

Cll.\ rrl·:!} 2. FOIIMULATION OF TUF PHOBLEh'l

l()

niiit('-l('iigili iK'riod, sucli a..s a. inontli or a, yea.i·, and discrtlc m.s7/ //o«« spaced at tli(' ('lid ol'c'ciual time intervals, we lia.ve tlie iollovving rela.ti()iislri|)s In'twc'eii ,l,/''a,iid F: F = l ’( \ + rv )' and F = I _{\+«y1 = '1 r , v ( i + . v ) ' 1 and F = .4f ( l + < v ) ' - l 1 [ ( 1+ 0· ) ' - ! J 0( 1+, , · ) ' J = F t'V a.nd F = A (1 ) ^ — 1 [ ( l + c v ) ' - l J C'v

ircontinnons coin|)oiinding, wliicli assuiiR's th a t casli flows oeciir a.t discreie intervals (e.g. once per year), is considered than the abov(' relationships turn out to !)(' : /'' = and /■’ = F(-ivl. F F and F = .4 ( / > ■ < _ I ' l l _ I and F = e-''-l > - l )

ll('r(', note tlia.t cv is tlu' (vilective annual interest rate, coniponiKh'd coiitin- iioiisly.

l·'or the derivation of the above formulas and for nioi'o dc'tails o i k' can ix'ler to any l‘higineering Economy book [f7] a,nd the |)a|)cr by Dogrnsdz and Karalrakal

In most of the ap|)roaehes to the replacement decision making problem, basically two measures of effectiveness are used :

1) hresc'iit worth (discounted value) of all cash flow expenditiues up to an infinite' horizon with repetitive r('|)lacement of tire ('(|uipm('iit with its own kind.

2) l·'/(|uival('nt uniform cash flow I'ate (e.g. annuaJ vvortfi) of cash flow ('.x- p('iiditiir('s.

( '¡¡A rri^l l 2. [' 'O H M IIL A' I'I ON O F T I I F F R O B L E M

(■(|uival('iil. in l.lie sense UiaX iliey give the sa.me result. But iii some ease's, noiu'

of tli('S(' UK'asures a.re appropriate!. One t3q)iea.l aiiel simple situation vvhe're'

tlie'N' e-an not l)e' used is the case where the productie)n rate' is ne)t establisheel e)pe'ratie)nally; i.e. the fuuctional Ibnn e)f the pre)elue-.tiou ral.e' e-an iie)t he' e|e- te'rmiiie'el. iVloi'e' spe'ciile'ail.y, in me)st practical situa,tie)us, pi'oelue'tie)n rale is the' eliirere'ne’e hetwe'en the cash inilow (revenue) rate ,r(e/,/), anel e'asli outdenv (e'xpe'iise') i'ate',7;/.(e/, /). In nuuiy eavse's, expense rate e’.an be eleteriiune'el, but re've'niie rate' e-an ue)t l)e. In tlial. case, if revenue rate is assumed te> be inele'- (M'liele'nt e)f the' ele'e'ision varia.blt!s (aJterna.tivms), the'u minimization e)f pre'se'iit \ahie' e)(' e-e)sts, e)i- annual vve)rth of e:e)sts are useel. TIris has bc'en a e-omme)n ci ite'rion in the litera.ture, anel is very well known. But it abanelonme'iit time 7' is a de'e isieni variable, this a|)pre)ach may not work. Dogrusdz a.nel Kai-abakal

!)] gave' sue'h an e.xa.mple' witJi tlie rollowing situation:

1) .Manulae-turiug rate e/(e/,/) I’or tlie proelucts (.se'rvices) pi'e)eluee'el by the system ca.ii be' eletermine^'el, but tlie sale's |7i-ie'e is unknown e)r ne)t me'a.n- ingiul,

2

)

.Ml |)re)elne'tion is assumeel to be solel.

In this e-a.se', neither the miuimizatie)n e)f the present vvoi-th of e-osts ne)|- tlie minimizatie)!! e)f the annua.l worth of e-osts is appropriate, sine-e' the pre'sent valiK' of the revenue's is ne)t inelepende^nt e>f the elecisioii anel time, d'e) ove're-nme this pi-e)ble'm tlie'y eletermiued the prie-.e' the revenues we)ulel pay all e-ash e)ut.la.ys.

'I’lie'y e-a.lh'el this prie-e'

c.

lM-e)m the ee|uiva.lene-e between the growtli e)f re'venue's

anel the' gre)wth of e.\pe;Mise:'s:

/ ' cq{(L /,)e:-'’·' elt + ,S'(e/, = / ‘ in{d, / ) c - ''' elt + l{d)

Jo Jo

i,li('V (k'iiiK' iJir lim e average unii cost per unit prod/ticlion/serviee a.s:

i;/'ni.(e/,/,)e::'Mlt + /(e/)

-./() (If

CHAPTER 2. FORMULATION OF THE PROBLEM

18

'I'licy call V·’ as time average unit cost of tlie products (servisos), since |.|i(' prc'sciil. value of cqidj.) over tlie lil’e 7 ’ is e(|ua,l t.o I.Ik' prescuil, value of |.|i(' o|K'ra(.iug ex|)eiises, plus investment outla,y, niiiuis the salvage value. 'I'lie optimization critc'ria tlun· pi'ovide is : imn,i;re

'I'liis nu'a.sur(' ea.ii be applied to the determination of the ('conomie lilV'

of ('(|iiipment with decreasing productivity but increasing operating ('.K[)ens('s

with ag('. 'rraditionally, witli the assnmption of rephvc('ment repeatedly with identical ('(piipment in tlu; future, tlu' econonne life is obta.iiied as the solution

to th(' following problem [84, 10]:

шгп'г

Jo m{í)e-^■^^

dt + / -

S( T) t -a'r

— (

III tliis a|)proach thero' exists a tacit assumption that operating rate of cost,

m{t),

is monotone increasing vvitli age but |)roductivity remains tlu- same. If

ther(' is also a deterioration in tlie productivity the above model is not valid.

Dogriisdz et. al. [18] experienced this situation, see also [0].

In this study, we consider the replacement d('cision of a deteriorating and stochastically failing ecpiipiiu'iit maintained with repairs and ov('rhaiils. W(' assume' th a t there is no technologically im|)rov('d alternative's aiiel the'refore' the' syste'iii is re'placeel by an ielentical one. Se> repbu'ement time's Ibrm a re'ge'ii- e'lative' pre)e'ess. 'I'lie first optiina.l replae’.ement tim e and the' e)verha.nl pe)lie'v de'te'rmine'd for the lirst re'pla.ce;ment cycle is repeated in the' subseepie'iit e ye-Je's.

(t>nelitie)ii e)f the' e(|ni|)ment is re'bite'd te> the numbe'r e)f e)verha.uls. Ope'r- ating e-eest rate' fnnetion, lifetime elistributie)ii, o\mrhaul costs, all elepend e)ii the' iiiimbe'r , of overhauls ma.de. 'I'lie o|)erating exist rate function of the' e'epiipment which is overhauled n times, ha.s the following functieinal form: ()('„{l) — c„e·''"*'·“ '"* where is the tim e of the lA’'’ eiverliaul, c„ is the rate eif e)|)e'rating exist when /. = /„, anel is the rate eif increase of the rate' of oper­ ating exist, sex' figure' 2.1. Note th a t this function is elefinexl for /„ < / < /,,+ |. We assume' liiu'ar relatieinsliips betwex'ii c„ anel n, anel betwex'ii //,, aiiel n.

( 'I I л РТ1':И 2. F()llM U L\TiO N OF THF FIIOBI.EM

1!)

//„ > 0 Гог a.uy n > 0, and = //.q + iiAfi. Cor n >

rate of opei ating

l·'¡,í>;uı■(' 2.1: Opc'rafiiig (dia rad. or is tics ol the System imd('r ( 'onsidt'ra.tioii

LilV'tinK' of tlic system uiKler consideration is a.ssumed to Ix' ('xponential with pa.rameter

A,,.. A„

>

0

Гог any n > 0, and

A„

= vvli(M'e L„ is tlu' ('xpc'c'tc'd lile tim e o r tlie system wliicli lias ha.d n overliauls. W(' a.gain assnnu'

liiK'ar relationslii|) between L„. tuid n, i.e. = do — u A L for n > I.

We assume th a t upon failures a minimal repair is done instantaneously with a (ix('d cost of (

Cost of iF'' ovt'rltaul, (' O(ii), is a.lso assumed to be fixed. 'I’liere is again a liiK'ai' r('la.tionsliip betvvet'u (' (){u) and v/., i.e. ( F) {v) = C(){\ ) - \- n A O ll for // ^ I.

Л с . Л/ 0 A d and A O H are supposed to Ь(> known by tlu' decision makt'r. Tliis can b(' known either by the experiences of others witli tlx' sysl.t'in at. hand or I'oi· sonu'tinu' tlie system may be run and the statistics ma,y Ix' gatlxvi-ed.

T'lx' ı·(^pla,c('ment action, minimal repair and overliauls supposed to take no time', i.('. tlx'.y are performed instantaix'ously.

( 7//\ r r ic ii 2. l·X)lm(rL·ATI()N Ol·' '¡'III·: PROBLEM

2 0

Our ol)j('ct.ivo is to ininiiiiizoi tlie time a.vera.go ra.to' of cost (pc'i· unit l.inio). to (l('(,('i'iniii(' the optima] replacement time a,iKÍ the overliaul policy (optimal ov('i-lia,ul times). /Mtliough due to discounting the effect of a, decreasing salvage \aJiK' is lu'gligible for such long-lasting systems, W(' include it for the sake of coinph'teiK'ss. SaJvag(' value lias tlu' following fuiictionaJ form whicli rc'lh'cts lh(' d('( liiK' of salvage value' vvitli age /.:

,S'(/) = S ()< '''h vvlu-re ,S'i) a.nd 7 a.re su p p o s e d to Ix ' k n o w n b y the d e c isio n m a k ('r.

VVh<'i'<'. : Salva ge ' va in e a.t l.ime (age') /, = ().

,\iiel, 7 : 'I'he rate e>f elex'line in S'(/:).

In the' Ibllowing e-hapter, we eleve'lo|) a elynamic pre)grammiiig moele'l with Ibrwarel ı·e'e·nrsious fe)i' the se)lntion.

C h a p ter 3

C o n str u c tio n o f th e M o d el

TI

k

' I

k

'

s

I, a.pproa.di k) ha.ucllo tlie problem imcler coiisideration s(4'm,s I,о 1к'

dy- iiainic proijrainimny airproach. \Nc

use the word "ар|)гоа.е1Г’ b('ea.us(' dyiia.mie

piogra.niming is not. a. pariicular algoritlim in the sense that Dantzig’s sim-

plc'x algorilJim is a vvcdl-delined set of rule's for solving a. Гпк'аг programming

proble'in. Dvnamie progra.mmiug is an approacli to solving e('i tain kinds of op-

timi/a.tion |)roblems, some оГ which can also !)(' solved by other procedure's. It

was lii'st introelne'ed by Richard Ihi'llman. 'I'lie iirst syste'ma.l ic treatment e)i the'

sidejeel. was given in [b] in И)Ь7. Other l)oe)ks by Bellman and liis collol)e)i'a.te's

have' snbse'e|ne'iitly appearc'd on applie'd elynamic programming [7] and e)ii the'

applie-atie)n е)Г elynamic |)rogramming to e'ontrol the'ory [6, 8].

Dvnamie· pre)gra.mming is a way оГ k)oking at a |)rol)lem which may e-e)ii- (,ain a lai'ge number of interrelateid de?cisie)u variables se) that the pre)ble'in is reL'ai ele'el as il’ it. e'enisiste'el of a see|nence of pre)l)lems, eacli e)i vvhicli i-ee|uire'd the' optimizatie)n е)Г е)п1у е)пе or a lew variablevs. leleally, wliat is sought is te> snb- stitaite' se)lving n single-'-varia.l)le proble'ins for ,se)lving one v/.-vai'iable' |)roble'm. VVhe'iie've'r this is pe)ssible, as in oni· |)re)blem, it nsnally re'e|uire's vei-y much le'ss e e)inpntatie)nal eiforl.. Se)lving n e)ne'-vai'iable pi'oble'ins i'e'e|uire's a e-ompn- tatienial e'lfoi't |)rope)i-tie)nal to the nnml)er n. On the e)ther hand. se)lving e)iie' large'!· pre)ble'in with n varial)lc's nsnally ree|nire's a ce)mpntatie)nal e'lfort whie-h is ve'iy re)nghly pre)|)e)rtie>nal te> a"\ whe'i-e a. > 1 is some constant. In enir study.

( 'll л Г 'П Ш :l. C O N S T R U C n O N O F T H E M O D E L ·)·)

vv(' hav(' l.lic cliancr l.o t.ranslomi Ihe inau,y-variai)le |)го1 )1 еш to a. i-('a.soiiabl(' iiiiinbcr of two-variable problems. VVe first solve how long slioiild tiu' first overhaul interval be. Tlien we solve the problem of (let('rminiiig tlu' tim e of tİK' second overha.nl given tlie first overha.nl, etc.. VV(' contimK' in this manner s(4 |uentiall,y until we r('a.cli the optimal number of overhauls, a.nd repla.c-emeut (.imes.

'I'Ik' principle or point of view th at (uiabh's us l-o ca.rry out the tra.nsforma.- tion we have' just discusso'd is known as llic principlt of opiimalily, which was liist ('iiuncia.tefl by Bellman [5].

Л .si'cond e.xtremely importa.nt a.dva.nta.ge of dynaniic progra.mmiug ov('r almost all otİK'r computa.tiona.l methods is th at dynamic programming guar- ant('('s a.bsolutc' (global) ma.ximaor minima, avoiding the trap of rela.tive (local) optim a. Hence we need not concern ourselves wdth the annoying prol)h'm of local ma.xima. and minima..

In all othou' optimization tecimiqiK's, certain kinds of coustra.iuts can cause signilicant problems. I'br example, the imposition of integrality on the va.rial)les (as l.li(' numl)er of oyou'luuds performed should l)e integer) of a problem a.r(' more' ti4 )ublesome in otluu· methods. Howeyer, in dynamic progra.mming tlu' r(4 piii4 'ment th a t some or all of the variabh's be integers greately simplilio's th(' com|)ul.ational process. In our model, we considered discrete' time points for th(' ova'iliaul a.nd rei)la.cement decisions, which is quite' realistie· siiie'e' in any ce)mpauy such huge' inye'stment de'e-isious (ove'rliauls, re'placement etc.) are

discusse'el e)idy e)uce е)г twie'e a year.

A furtİK'r e'liaracteristic e)f dymunic pre)gra.mming is the “embeelding” e-haı-

acte-ristie· e)f the' functional e'e|ua.tions. W hat this means is th at. se)lutie)ii e)l the'

prol)le'ni of replacing after n overhauls autom atically eletermine^s the opUiual se)lutie)iis to the pre)blems of replae'e after n —1 overhauls, ii — 2 overhauls , e'te·.

Our usage e)f elynamie· pre)gra.mming is elillerc'nt from tra.elitie)na.l usage in one' i-e'spe'ed. tliat l.he number of sta.ge's, n, te> be' e:overe'd is not preelete'rnune'el, sine-e' re'place'iuent tim e is a de'cision variable (an unknenvn). Tins prohibits tlie

(

7

/.\ r'l'K Il:}. ('ONSTIUJCTION OF THE MODEL

2:5

iisa.g(' oC ha.rkvvard rc'cursion, and tlKU-eiore forces forvva.i'd ix'cursioii.

In ( he iK'xt. sect.ion we will derive l,he d,yiiainic progra.nmiing mod('l a.nd (Jk'ii d('velo|) (1k' foi'wa.rd algoi'itlmi lor solut.ion. In (.he last si'el.lon we will state' sonu' (•oni|)ntation r('duction results a.nd U|)da.te our algoi'HJim wil.li respc'cl. (.o IIk'sc' r('snlts.

3.1

T h e D y n a m ic P rogram m in g M od el

In Uh' model, our infinite planning liorizon is divided into three' months time iinil.s, i.('. to (piarters. However, it may be divided even to one month , I mold hs, () months or one yea.r time units etc.. The algorithm we de've'lop h an­ dle's all. VVe' assume tliat tlie failure.is occur at the' end of tlu'se time units a.nd

I

he' inslantaiK'ous re'|)airs take' place at, tlie end of the time' unit. Ove'rhauls are' alse) suppose'el (e) take place at the e'nd of the time units and are' elone instan- (,ane'e)usly. VVe' a.naJy/,e the proldem as twee stratilieel interre'la.teel sul)|)i'e)ble e)l

'ins

1 ) 'I'lie pi'e)l)lem of ele'termining tlie optimal eiverliaul peilicy until i'epla.e'e'- iiie'iit.

2 ) 'The' pi-e)l)le'iii of eletermining the optima.1 la'placemeid· time.

Ill (,Ik' rest e)f this subsextion, we give a brief discussion eni the general nature of I he' eleri va.tieHi e)f tlu' elynaniic programming model by e.xa.mplifying on the lii'st subixible'iii anel in the' next subseiction we elerive the functional e'e|ua(.iems useel in iJiis meiele'l. 'riie last subsextion is fully eleveite'el to the forwai'el algorithm te> seilve' l.he' elynainic i)re)gramming model we have elevelope'el.

In ge'iie'i-aJ, the' structure' of a elynainie· preigrammiug moele'l is ma.ele up eil ( he' lolleiwiiig :

a) a. se'e|ue'iice' e>l sta.te vai’ia.bles,

( I I A P'VKll :{. ( K ) N S 'r i W C 'r i O N O F T H E M O D E L 21

( · ) i x ' r i i r i x ' i i r e r e l a . t i o n s .

In oiir iikkU'I, iniinber of overhauls ina.dc before replacenu'nl, is takcui a.s the

sta.gi' vai’iable. If we consider a. problem with

n

stages, tiu're must Ix' spc'ciiied

(•('ri.ain sc't of |)ara.meters describing the state of tlie systium In this model,

tiiiK' of tlu' ovc'rlia.ul is cliosen as the sta.te variable.

x„, iUc

nunilx'r of tinu'

units passed since the prcwdous overliaul ((//. — 1)'’'^ overhaul) is om- decision

variable'.

Xn

is the decision which determines tlie state of the system at stage

I ’ while' tlx' state e>f the system at stage ’/d is /;„. Tlx:' stage transfe)rma.tie)m

wliie li eh'termine's / „ , _ i wlien the current state is /„, and the' a.ctie)ii take'ii is

can be' wi'itte'ii as :

I n — \ — ^ II I ^ II· )

( k)st a.sse)ciateel witli tlie stage transformation at ea.e*h stage' a.s a fnii(‘tie)n e)f th(' state' variable of the cui'rent and the previous sta.ge and the' ele'e’ision maelc' at t he' enin'e'nt sta.ge is;

= r„(;r„,/.„-I,/,,) ^vnd =

Pn(xn.,Ln)

vvhe're

ILi.,.)

is the

slair

I

raiisil/on JuncUoii.

Se>,

X

n

— ^

n

(

'^'ii

"> di.) ·

In(lii )

tlu' ininimum time average rate of cost ( per unit time) up to sta.ge'

// when the' state variable (time of the /d'^' overhaul) is /„. It is (‘ailed the

stall

fii.nrtwiL

/■,(/„) is simply the minimum cost of the first

n

stage's when the system

state' is /„.

,/(.)(()) =

Ixiumm.

n

/„(/,,) = min 5]'·./(·<■,a,/)

th( forward rccurrcvcr r( lalion

L X n ) = m i n { r „ , ( * ^ ^ / . i /„,) + n —1 •I'll -1 11—1 [I'U-i) = m i n y ] .,=0 v )

( 'll A PTI'lll S. CONS'l'llU(!TION OF THE MODEL

•>r>

v v ( ' r a i l i v v v ' i ' i t e ) a , s :

( : U )

• ' l l · I

l''igui'c Miill.istago' ]).yiiamic Programming Diagram.

3.1.1

D eriv a tio n o f th e F u n ction al E q u ation s

In oi'di'r to solve ilu' firsi. problem, t.lu' problem oF deb'rminiiig the optimal o\('rliaul |)olicy until replacement, we derive the Following Functional ecpial.ion:

= 0 ) = IT = a very big number. In Fact, it is inlinity becau.sc' oF tinu' avi'raging to ‘0 ’ time units.

r . r . , 1 _ (

.1

IliLl) ~

- ( 1+ I p/n ( b ii •i/i. )

Jn-\{l-n

) (^

~

j

\v|i('i'(' n = 1,2,... and /„, = v/.,// + I, ...oc·. Kunctional Forms oF the sta.t(> Fune- ( ion, state transition Function and the stag(' costs are as Follows.

,Stag(\ n : Nu i uIk'i· oF o v i'i’h a u ls ma.de. S(a.t(\ /„ : 'I'iine oF the n"' overhaul.

Di'cision. :r„ : num ber oF time units passed since the pi'evious ov('i-ha.ul, i.('. (// — I )■"' ovn-haul.

С І І Л Г 'І 'Е І І :і. C O N S T R U C T I O N O F T I I F M O D F L 2G

ІІ(Ч'(' поіч' tliat:

, is I.Ik' (■<>iilinu<)UH (юпіроіиі<Іли.д prcs( III worih faclor o\' ѵоиі'ипюнн coii- slaiil (uiiiioriu) ca.sh flow ra.te.

. is tli(' (^)iitmuous coinjioiiiidinij cajiihil rf covcry fuclor (tinu' a.vera.giiig a.cior).

'Vom now on, ior tlio sake of simplicity, wo will intcrclia.iiga.l)ly ns(' (/-’| / l , n , /) and ( /11/^, n , /.) For tlie above lactors ros|)cctivoly.

State Transition Function

/«-I =

CÁ-i-nCn) = l-n - ■•'.I

,

whore n

= 1,2,... and

t = n, n +

1,....

Stage Costs

b(d ОІЦ· sysl.('m b(' in stage ii’. Sup|.)ose th at we а.іч' at st.a.te I„ wlu're n < /„ , since time' of the n"' ovei'liaul can not be Ic'ss than 'n'. And deiiiK';

la^^'ıı■, Ui.)) = Pi’es(nit worth oF the cost incurred during the tim e in- t('i-va.l (/„ - /„]. ТІИП, /■„(·'·,,, In - ■'·„.) = / c„,_| с""-'<"~'" + '"'>с” "" ds + CO{n)( ~"'" In +

E

Ab,(.s)r.Vc-"·'· . s - = / , „ . - f I

Ay,(.s) is th(' e.xpected number oF Failuins within the tim e units. Since the liFe- tiuK' distribution oF the system we are considering is e.\:])onential(A„), wh('r(' 'll' d('iK)t('s the number of pi'evious overha.uls made to the .system. Failures Form a poisson pi'occ'ss and Ab,(.s) = A„, tluvn

.4 = 1.,, + I • s = l

V 1 _ ^ J '!·(' , (.ho iiit,('grals <\.vc very oa.sy t.o mai.ia.go :

'П„-.Гп