Asset Management System for Educational Facilities Considering the Heterogeneity in Deterioration Process

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1 Asset Management System for Eucatona Factes Conserng the Heterogenety n Deteroraton Process Kengo OBAMA *, Kyoyu KAITO**, Kyosh KOBAYASHI*** Kyoto Unversty* Osaa Unversty** Kyoto Unversty*** ABSTRACT: In asset management of nfrastructures, prectng eteroraton of structures s one of an essenta technque to mae a ecson on the optma mantenance pocy. However, as for arge-scae nfrastructures that consst of a huge number of structura components, n orer to estmate ther eteroraton process wth hgh accuracy, the heterogenety of nvua components has to be consere because each component possesses fferent matera characterstcs an esgns an s n unque servce uner varous envronmenta contons. Ths paper focuses on especay eucatona factes from among nfrastructures, an constructs ts asset management system conserng the heterogenety n eteroraton process of nvua components. Ths system many conssts of 3 functons as:. atabase,. eteroraton precton an 3. fe-cyce cost evauaton. The atabase stores the basc structura an component s nformaton an vsua nspecton ata. Base on these nformaton an ata, the eteroraton precton s statstcay carre out. Specfcay, the eteroraton process can be bascay expresse by ranom proportona hazar moe, an the heterogenety can be moee as probabty fuctuaton n the hazar rate. Furthermore, the tme-epenent hazar rate s formuate by the Webu hazar moe. The heterogenety of the hazar rates across the nvua characterstcs of components s expane by the ranom proportona Webu hazar moe n whch the hazar rates are subect to Gamma strbuton. In the 3r functon, through the comparsons of the fe-cyce costs between the mutpe repar/renew strateges, the optmum one s ece. Here, as the eteroraton process of nvua components can be formuate by the Marov transton probabtes efne by the estmate hazar rates, the propose fe-cyce cost evauaton metho organcay ns to eteroraton precton resuts va Marov ecson process. In aton, an emprca stuy empoyng vsua nspecton ata for an actua unversty facty s carre out to verfy the vaty an appcabty of the system. KEYWORDS: eucatona factes, ranom proportona webu hazar moe, asset management system. ITRODUCTIO In the same way as cv nfrastructures, Japanese eucatona factes have been but contnuousy from the pero of hgh economc growth. In genera the expecte fetme of eucatona factes s about 3 years, whch s short n comparson to cv nfrastructures. In fact, t has been ponte out that for eucatona factes that were but n the eary stages of the pero of hgh economc growth, ther repar an reconstructon costs began to surface aroun 5, sooner than for cv nfrastructures. It can easy be euce that these costs wegh own the management of eucatona factes, an t s absoutey necessary

2 to eveop asset management to support varous ecson mangs regarng the pannng of repar an reconstructon strateges. In the asset management of eucatona factes, fecyce cost s an mportant evauaton nex that etermnes repar an reconstructon strateges. In aton, eteroraton precton resuts are refecte n the evauaton of fecyce costs, an so the estabshment of eteroraton precton technque s aso an mportant ssue. In genera, eteroraton precton methos can be roughy cassfe as:. physca eteroraton precton methos base on the mechanca eteroraton mechansms of structura components an. statstca eteroraton methos base on past nspecton ata. However, for eucatona factes, repars an reconstructons are sometmes carre out base not ony on physca eteroraton but aso on the users usabty an vsua factors (aesthetcs. Therefore, when attemptng to carry out eteroraton prectons for eucatona factes, t s preferabe to empoy a statstca eteroraton precton metho. Statstca eteroraton precton methos are methos that tae vast amounts of eteroraton nformaton an moe the reguartes behn eteroraton processes. In recent years there has been a remarabe accumuaton of research nto eteroraton moes usng hazar functons. Hazar moes are stnctve because n characterzng the eteroraton process of each facty they respon to the structura characterstcs of the facty an envronmenta contons to gve nvua hazar rates. However, as a hazar rate s gven etermnstcay, the eteroraton process for factes that have the same structura characterstcs an envronmenta contons w be entca. Regarng ths pont, even when structura characterstcs an envronmenta contons are the same, t s more natura to conser that the eteroraton process w Repar an Reconstructon Cost [Trons Yen] Roa Infrastructures Eucatona Factes Others Years Fg.. Transtons n the repar an reconstructon costs of nfrastructures ffer for each facty. Therefore, n orer to carry out a more exhaustve eteroraton precton, t s necessary to eveop a eteroraton precton metho that taes nto account the heterogenety of the eteroraton process for nvua components. Wth an awareness of the above ssues, n ths stuy the authors propose a ranom proportona hazar moe that expresses the heterogenety of the nvua eteroraton processes of factes as a hazar rate probabty strbuton. Furthermore, the authors propose fe-cyce costs evauaton moe wth use of ranom proportona hazar moe. Beow, n secton the basc concepts of ths stuy are consoate, n secton 3 the ranom proportona Webu hazar moe an ts estmaton methos are expane, n secton 4 as a emprca stuy of appcaton, a unversty facty s taen up an some anayss carre out base on ts vsua nspecton ata, an n secton 5 proposng metho of fe-cyce cost evauaton.. BASIC COCEPTS OF THIS RESEARCH. Current state of eucatona factes Eucatona factes ncues pubc, natona an prvate schoos (eementary, unor hgh an senor hgh schoos, research nsttutons, ay cares, nergartens, unverstes, research nsttutons, museums, art

3 gaeres, brares an communty factes, as we as soca eucaton factes, feong earnng factes an cutura an communty factes fa nto ths category. In the same way as other soca nfrastructures, as a part of economc pocy n the postwar era, the eucatona factes n Japan were constructe as part of repeate soca nfrastructure eveopment. After 97 the stoc vaue of eucatona factes rapy ncrease, an n 997 eucatona factes accounte for.% of a soca nfrastructures (wth a gross stoc vaue of 7 tron yen. Furthermore, among eucatona factes, the average servceabe fe of schoos an acaemc factes s consere to be about 3 years, short n comparson to the average servceabe fe of cv nfrastructures. In fact, n 3, the gross tota area of eementary an unor hgh schoos was 6.9 mon square meters, of whch 4.5% was - to 9-year-o factes an 9.% was 3 to 39, the agng of whch s begnnng to be actuaze. The resuts of repar an reconstructon cost estmatons base on ths ata are shown n Fg... Smpy because ther average servceabe fe s short, the repar an reconstructon costs of eucatona factes are becomng greater than those of cv nfrastructures. In concuson, the asset management of eucatona factes s a probem that has tremenous soca urgency. As ponts to conser n the asset management of eucatona factes, when carryng out evauatons of the conton of the facty or components, one must conser not ony structura safety, but aso usabty an convenence for users, an furthermore aesthetc aspect, a of whch may be ste as mportant evauaton factors. In other wors, they characterstcay have a arge number of components that users w come nto rect contact wth an components that users w vew recty. For exampe, n the case of oors an wnow frames, even though they o not n any way nfuence safety, they w be targete for repar or reconstructon f they fa to open an cose an seem to amage usabty for users. Furthermore, f there s parta amage to the tes of exteror was or parta eteroraton of pant, repars may be carre out for aesthetc reasons. Therefore, n the asset management of eucatona factes, even when the authors spea of eteroraton prectons a smpe physca eteroraton precton targetng structura safety s not suffcent, an rather a genera performance precton that ncues structura safety conseratons s necessary. At the present tme, other than vsua nspecton, ths n of genera performance evauaton oes not exst, an a statstca eteroraton precton metho (statstca performance precton metho base upon vsua nspecton ata wou be effectve.. Hazar moe an the heterogenety of the eteroraton process In tratona hazars anayss, t s assume that the target facty s entrey but of the same matera, wth the am of moeng eteroraton phenomena that arrve ranomy n accorance wth certan hazar functons. In hazar anayss, the occurrence process of ranom eteroraton phenomena s moee, an the hazar functon, a etermnstc probabty moe, s use. However, n arge-scae factes, such as the eucatona factes targete n the emprca stuy of ths paper, t s not necessary possbe to express the hazar rate of each nvua facty component wth the same hazar rate. Rather, t s more natura to conser that the hazar rate for each type of component w have a fferent respectve hazar rate. For the management an operaton of arge-scae factes, the conseraton of repar an reconstructon pans for these many components s a crtca ssue. In ths way, as a metho that expresses the heterogenety of a hazar rate that consers the fferences n the component types, we can conser a metho n

4 whch fferences n component propertes are expresse as ummy varabes an eteroraton estmaton s carre out, an a metho n whch t s assume that the hazar rate w be subect to a partcuar probabty strbuton for each component group after whch a eteroraton estmaton s carre out. The fst metho has the avantage of beng smpe an easy to unerstan. On the other han, t s probematc because as the number of components ncreases the number of ummy varabes (whch express component propertes ncreases, an the estmaton accuracy of the moe ecreases remaraby. In aton, an ncrease n expanatory varabes s recty connecte to an ncrease n fe observaton an nspecton tems, ncreasng the burens n practce. Furthermore, because the heterogenety of the eteroraton process may be controe by factors that are not possbe to observe, t s essenta that a more effcent eteroraton precton metho be eveope. Eucatona factes are constructe from an extremey arge number of components, an estmatng a hazar moe that maes use of ummy varabes s not practca. In orer to express the heterogenety of the eteroraton process, there are mts to the refnement of a Webu hazar moe by ncreasng expanatory varabes an so on. As ong as nnate facty nformaton s expresse as expanatory varabes, the estmate accuracy an effcency w nevtabe ecrease. Therefore, n ths stuy the authors have empoye a mxe hazar moe n whch, epenng on the type of component, the heterogenety of the hazar rate s expresse as a probabty strbuton to moe the eteroraton process of a facty. Research nto hazar anayses that conser the heterogenety of hazar rates s accumuatng. In partcuar, there s a arge accumuaton of stues regarng mxe hazar moes n whch there exsts a heterogeneous hazar rate for each nvua sampe. In mxe hazar moes, t s consere that heterogenety parameters controng the hazar functon are strbute wth beng subect to a probabty ensty functon. In aton, a hazar functon s efne by probabstc convoutons of the probabty strbuton of the hazar functon an heterogeneous parameters. In regars to a mxe hazar moe, Kato et a. have moee the arrva process of roa obstaces an mae a case stuy of the appcaton to asset management. On the other han, eucatona factes are compose of a arge number of component types, such as roofs, exteror was, oors an eaves. To put t another way, t can be antcpate that there exst component groups that requre homogenous hazar rates, an that each group s hazar rate has an nherent probabty functon. In ths stuy, t s consere that a mxe hazar moe n whch these probabty error tems are assume to have a gamma strbuton. 3. RADOM PROPORTIOAL HAZARD MODEL 3. Ranom proportona Webu hazar moe The ranom proportona Webu hazar moe s a Webu hazar moe that consers the heterogenety of the hazar rate between components. The etas of hazar moes n genera may be foun n the references. A certan component of a facty s consere to be cassfe nto ns of component types. A tota of of the th (,, component type exst. Furthermore, focus on the th (,, component of type. The tme that has eapse snce the component was reconstructe s represente by the ranom varabeζ. Suppose that the arrva rate of eteroraton events for each component conforms to a Webu eteroraton hazar functon m λ γm (3.

5 In equaton (3., γ s a parameter that expresses the arrva ensty, m s an acceeraton parameter that represents the tenency of the hazar rate to ncrease over tme an the parameter (cae the heterogenety parameter beow, whch represents the heterogenety of the hazar rate of type, has been ae to the Webu hazar functon. The heterogenety parameter taes a common vaue for components of the same type. However, when component type ffers, t taes fferent vaues. In actuaty, the heterogenety parameter taes on a etermnstc vaue, but s an mpossbe parameter for an observer to observe. In aton, the fespan probabty ensty functon f of type ~ component, an the survva probabty F( ζ are respectvey expresse as m { γ } m f γm exp m { } (3.a ~ F exp γ (3.b ow, the vaue of the heterogenety parameter s one of the observatons from ranom varabes that cannot be recty measure by the observer, but t s nown to be strbute n accorance wth the probabty ensty functon g (. That s the Webu hazar moe (3. has an entca eteroraton acceeraton parameter m for a types of components, but for each component the arrva rato γm ffers proportonay, an the nvuaty of eteroraton s expresse. Regarng hypothess testng of the homogenety (beow, proportonaty of the acceeraton parameter, we mae another nvestgaton. In ths stuy, for each targete component, a Webu hazar moe n whch the hazar arrva rato s a observaton from a probabty strbuton s cae a ranom proportona Webu hazar. Here, suppose that the probabty strbuton of the heterogenety parameter conforms to a Gamma strbuton. The Gamma strbuton, as a speca form, ncues the exponenta strbuton, an has the avantage that t can express the exponenta famy probabty strbuton functon that s efne on the nterva [, ]. Here, suppose that the parameter γ represents the average hazar arrva rato between types, an the heterogenety parameter s a observaton from a Gamma strbuton wth average an varance an s a probabstc error term. The Gamma functon s efne on the nterva [, ], an wth respect to an arbtrary expanatory varabe an probabstc error term, the rght se of equaton (3. s assure to tae the postve vaue. In genera, the probabty ensty functon g ( : α, β of the Gamma strbuton G ( α, β can be efne as α g ( : α, β exp (3.3 α β α β The average of the Gamma strbuton G( α, β s μ αβ, an the varance σ αβ. In aton, Γ ( s a Gamma functon. Furthermore, the probabty ensty functon g ( : of the stanar Gamma strbuton that has an average an varance s expresse as g ( : exp( ( Two Steps estmaton metho for the moe In a ranom proportona Webu hazar moe, a tota of 3+ unnown parameters exst, the arrva ensty parameter γ, acceeraton parameter m, heterogenety parameter (,,, whch ffers for each component, an the strbuton parameter of the heterogenety parameter. In the case of an ornary Webu hazar moe, t s enough to estmate the parameters γ an m from eteroraton ata recor. However, n the ranom proportona Webu hazar moe, beses these two parameters, t s necessary to pursue the probabty strbuton parameter of the heterogenety parameter an the heterogenety parameter (,, for each

6 component type. ow, et us suppose that the eteroraton hstory atabase of the facty s avaabe. The atabase contans nformaton reatng to the tme of eteroraton (repar of a components from the tme the targete components began servce. Express the eteroraton recor of components as Ξ ( ξ, L, ξ, where ξ {(, ζ, L,(, ζ }(,,. In aton, s a ummy varabe that taes the vaue f type component (,, has eterorate, an taes the vaue f t has not eterorate, an ζ s the pero of servce of type component, that s, when, ζ means the ength of tme from the prevous repar or reconstructon to the present. On the other han, when, ζ ncates fespan. Here, suppose that the heterogenety parameter s gven. At ths tme, the contona ehoo ( ξ : γ, m, for the observe ata ξ for type s expresse as ( ξ : γ, m, ~ { F( ζ : γ, m, } { f : γ, m, } ( (3.5 However, n the above equaton, t s expcty ncate that the fespan probabty ensty functon f : γ, m, an the survva functon ~ F( ζ : γ, m, are escrbe as functons of parameters γ, m an. Here, f the heterogenety s strbute accorng to the stanar Gamma strbuton g ( :, the ehoo functon for the observaton ata ξ s L ( ξ : θ { ~ F( ζ : γ, m, } { f : γ, m, } m { γm( ζ } s + exp ( g( : { ( + γτ } (3.6 However, s an m τ (ζ. In the above equaton, wth respect to a type components, the heterogenety parameter taes a common vaue. To express ths, t shou be note that the ehoo functon L ( ξ : θ s efne as an expecte vaue reate to the probabty functon of the contona ehoo ( ξ : γ, m,. Here, f a varabe transformaton x + γτ s carre out L ( ξ : θ ( + γτ m { γm( ζ } x + γτ s + s + s + ( exp { x } m { γm( ζ } x + γτ (3.7 s obtane. Therefore, the ogarthmc ehoo functon for the observe ata Ξ ( ξ, L, ξ can be expresse as n L( Ξ, θ [ n + n ( s o n L ( ξ + n( + γτ + n s : θ + { nγ + n m + ( m nζ }] (3.8 However, each eement of θ ( θ, θ, θ3 s expresse as θ γ, θ m an θ 3. The maxmum ehoo estmator of the parameter θ that maxmzes ogarthmc ehoo functon (3.8 can be gven as θ ˆ ( ˆ θ, ˆ, ˆ θ θ3, whch smutaneousy satsfes n L(ˆ, θ Ξ θ (3.9 Furthermore, the estmator Σ ˆ (ˆ θ of the asymptotc covarance matrx can be expresse as ˆ n L(ˆ, θ Ξ Σ(ˆ θ (3. θ θ However, the nverse matrx of the rght se of the above formua s the nverse matrx of a 3 x 3 Fsher nformaton matrx that conssts of eements n L(ˆ, θ Ξ / θ θ. The maxmum ehoo estmator of the parameter s obtane by sovng the three mensona nonnear smutaneous equaton (3.9. In ths stuy, the maxmum ehoo estmator

7 n L ( ξ, : θˆ (3. The maxmum ehoo estmator of the heterogenety parameter obtane n ths way s an estmator that was obtane wth the gven parameter θ ˆ ( ˆ, γ mˆ, ˆ. In orer to ceary escrbe ths, the souton of equaton (3. s expresse as ˆ (ˆ θ. From equatons (3. an (3., f ˆ (ˆ θ s specfcay estmate, the foowng equaton s obtane: ˆ s + ˆ ˆ + ˆˆ γτ (ˆ θ (3.3 The above two-step maxmum ehoo estmator estmaton fow s shown n Fg A EMPIRICAL STUDY Fg.3. Estmate fow of the maxmum ehoo estmator was obtane usng the ewton-raphson Metho. If the maxmum ehoo estmator θˆ s obtane, usng a covarance matrx estmator Σ ˆ (ˆ θ, t-test statstc can be aso estmate. ext, wth the parameter vector s maxmum ehoo estmator θˆ as a gven, the maxmum ehoo estmator of the heterogenety parameter (,, s obtane. Here, the parta ehoo functon s efne as L ( ξ, : θˆ ˆ ˆ ˆ exp mˆ { ˆˆ γm( ζ } { ( ˆ + ˆˆ γτ } s + ˆ (3. Here, mˆ ˆ τ. At ths tme, the maxmum ehoo estmator of the heterogenety parameter (,, can be obtane as o ˆ that satsfes 4. Overvew of the case of appcaton A ranom proportona Webu hazar moe estmaton s attempte for a certan unversty facty. The conton state of ths facty s accumuate through vsua nspectons. The nspecton pero s three years. The conton state of the facty s evauate as ether possbe to use ( or not possbe to use (. Ths facty group has a been cassfe nto 33 regons an s ocate n each one. The oest facty was but 73 years ago. Ths tme, the ata use n estmatons was the most recent vsua nspecton ata, whch was coecte n 6. Beow, the authors w use the technca term eteroraton, but as mentone before, the vsua nspecton ata use n ths stuy as eteroraton s efne not ony as physca amage to components, but s aso as oss of peasantness an convenence of the facty that s uge to nee repar. For the specfc estmate target of the ranom proportona Webu hazar moe, exteror wa components, for whch the greatest abunance of ata

8 was obtane, were focuse on. Exteror was can be cassfe nto fve types: te, mut-ayer fnsh pante, thn fnsh pante, meta, an concrete bocs. In aton, because there are a arge number of exteror was, an extremey sma number of components exst for whch, ue to an nta faure, the tme pero from the start of servce unt the eteroraton tme pont was remarabe short. For ths reason, n ths estmate, exteror was for whch repar was carre out wthn one year from start of servce are eeme to be nta faure sampes, an such sampes were excue n avance. After the above premnary preparaton, the sum tota by type of exteror was sampes that cou be use n the estmate were: 77 te sampes, 35 mut-ayere fnsh pante sampes, 5 thn fnsh pante sampes, meta sampes, an 6 concrete bocs sampes. Therefore, the tota number of exteror wa sampes was. 4. Proportonaty assumpton testng In ranom proportona Webu hazar moes, a proportonaty assumpton s mae n whch a types of components have an entca acceeraton parameter mˆ. Therefore, the fferences n the eteroraton process between exteror wa types (te, mut-ayere fnsh pante, thn fnsh pante, meta an concrete boc can be consere to be aggregate n the heterogenety parameter. Base on actua ata, the authors propose an assumpton testng metho to etermne whether or not theproportonaty assumpton s effectve before mang a precton. For type (,, an assumpton testng moe to test the proportonaty assumpton s formuate by H H : m mˆ : m mˆ an ˆ, γ ˆ (4. an ˆ, γ ˆ Here, once agan wrte the ehoo functon, base on the atabase, of component as L ( ξ : θ ( + γτ s + s + m { γm( ζ } (4. At ths tme, the ehoo proportonaty statstc to test the assumpton testng moe (4. s expresse by LR n L ( ξ : ~ θ n L ( ξ : θˆ (4.3 { [ ] [ ]} ~ Here, n[ ( : θ ] L ξ expresses parta ehoo when there s not the restrant of the nu hypothess H an n[ L ( ξ : θˆ ] expresses parta ehoo uner the restrant of the nu hypothess H. In aton, f ~ θ oes not have a restrant, t expresses the maxmum ehoo estmator. Snce the number of parameters that can be restrane by the nu hypothess H s, the ehoo rato test statstc w have a egree of freeom of one. It foows that f the test statstc LR oes not enter the reecton regon LR χ ( α (, nu hypothess s not reecte by α% sgnfcant eve. Here χ ( α ( expresses of freeom. χ strbuton wth egree 4.3. Hazar moe estmaton In the ranom proportona Webu hazar moe estmate n ths stuy, snce the fve ns of types estmate were te, mut-ayer fnsh pante, thn fnsh pante, meta, an concrete bocs, 5. It foows that there are a tota of 8 unnown parameters that nee to be estmate for the exteror wa: the arrva ensty parameter γ, the acceeraton parameter m, the heterogenety parameters (,,5, whch ffer for each component, an the heterogenety parameter s strbuton parameter. Foowng the process n secton 3.., the estmate parameter of the ranom proportona Webu eteroraton hazar moe are ste n Tabe.. However, β fufs γ exp(β. In aton, (,, 5 s heterogenety parameter vaues that represent te, mut-ayer fnsh pante, thn fnsh pante, meta

9 Tabe 4. Estmaton resuts for an exteror wa usng a ranom proportona Webu hazar moe β m Maxmum ehoo estmator (t-vaue (-.8 (.8 (.7 Log ehoo Tabe.4. Lehoo rato test statstc Types LR Te ( 4.45 Mut-ayer fnsh pante (.7 Thn fnsh pante ( Meta ( Concrete bocs ( 5.4 Tabe.4.3 Expecte fespan Types Expecte Lfespan (Years Te 49.9 Mut-ayer fnsh pante 43.8 Thn fnsh pante 33.5 Meta 36.5 Concrete bocs 43.5 an concrete bocs, respectvey. The maxmum ehoo estmator of the acceeraton parameter n Tabe. s m 4.. Generay, f m. the eteroraton probabty can be consere to be tme-nepenent, but t can be sa that the target component s ceary the tme-epenent type for whch the eteroraton probabty w ncrease wth tme. In aton, n the ranom proportona Webu hazar moe, a proportonaty assumpton s mae such that each type of component has an entca acceeraton parameter mˆ. Uner ths assumpton, the heterogenety of the Webu hazar functon of each component s aggregate n the heterogenety parameter vaue (,, 5. It foows that there s the characterstc that epenng on the magntue reaton of, the eteroraton veocty of each Survva Probabty Te Mut-ayer fnsh pante Thn fnsh pante Meta Concrete Bocs Eapse tme [year] Fg.4. Survva Probabty of Exteror Was component can be compare. From Tabe., < < 5 < 4 < 3, so t s obvous that n the exteror wa the eteroraton progress of thn fnsh pante s the fastest an that of tes s the sowest. Furthermore, ehoo rato test statstc LR (,,5, whch s meant to test the hypothess testng moe for the proportonaty hypothess, s ncate n Tabe.. Here, whenα 99, χ ( α ( 6. 6, so t s unerstoo that the nu hypothess H n whch a types foun n the was have an entca acceeraton parameter mˆ s not reecte. The survva functon an expecte fespan create for each type base on the ranom proportona Webu hazar moe are ncate n Fg.4. an Tabe.3, respectvey. In the fgure, the 5 survva probabty curves ncate the average survva functon for each type of exteror wa. The pero of servce n whch the survva probabty of exteror wa component s 5% s 5. years for te, 44.

10 years for mut-ayer fnsh pante, 33.7 years for thn fnsh pante, 37. years for meta, an 43.7 years for concrete bocs, an as mentone before, the eteroraton progress of thn fnsh pante s the fastest an that of tes s the sowest. When thn fnsh pante has been n use for 5 years, ts survva probabty s about 8.3%, when n use for 5 years ts survva probabty s about.3%. From ths, we can see that for thn fnsh pante, as the pero of servce becomes onger, the eteroraton probabty becomes arger at an acceeratng pace. In aton, when te has been n use for 5 years, ts survva probabty s about 96.5%, an when n use for 5 years ts survva probabty s about 5.3%, an n the same way as mut-ayer fnsh pante, meta, an concrete bocs, as the pero of servce becomes onger, a tenency for the eteroraton probabty to become arger at an acceeratng pace can be confrme. From the above, t can be unerstoo that even for the same exteror wa component, for each type there s we varaton n the heterogenety parameter. More specfcay, to estmate a hazar moe for a arge-scae facty such as an eucatona facty, whch s mae up of varous ns of components, t can be sa that a ranom proportona Webu hazar functon that uses a mxe strbuton s effectve. In aton, the eteroraton survva probabty can be estmate for nvua components, whch cou not be consere f the hazar rate were smpy treate etermnstcay. Therefore, t s possbe to expect ths to contrbute to the refnement of asset management. 5. LIFE-CYCLE COSTS EVALUATIO 5.. Moeng of repar/renewa process A certan component of a facty s consere to be begun servce at the tme t an to be stoppe servce at the tme τ t + ζ. The fetme of the component s expresse as ζ. But the nformaton whether the fetme s over or not can be obtane by ony peroc nspecton. ow, t s assume that peroc nspecton s carre out at the tme t, t +, t +, K. Then, the screte-tme axs t, whch s expresse by nta tme t an tme nterva, s efne as t t + (,,, (5. K ext, conser the ssue of managng components, whch are compose of sampes, at the same tme. To scuss t easy, suppose that a sampe types are the same. Furthermore, et ξ (, u be the repar/renewa strategy for eucatona factes wth the use of repar/renewa nterva an the maxmum ength of servce tme u. Uner the strategy ξ, conton of components s nspecte at the tme t, t, K, t,k. In aton, a components whch have been use for the tme u are repare or renewe. Thus, at tme t, the number of components cassfe by servce tme s expresse as the conton ξ ξ ξ parameter vector n ( t ( n ( t, K, nm ( t. ξ The conton parameter n ( t (, K, m stans for the number of components whch have been n servce for at the tme t. Then the reatve frequency of components cassfe by servce tme can be expresse as ξ ξ π ( t n ( t / (, K, m. Moreover, ξ ξ ξ et π ( t ( π ( t, K, π m ( t be reatve frequency vector. Obvousy the foowng equaton s obtane: u ξ π ( t (5. Here, focus a certan component of whch tme n servce s at the tme t. In aton, p s the probabty whch ths component oesn t reach the stop of servce unt the next nspecton tme. Then, the expectaton of reatve frequency, whch s expresse ξ as π + ( t +, that the component whose tme of use s oesn t reach the stop of servce at the nspecton tme t an then the component s to be tme of use

11 ( + at the next nspecton tme as t +, s efne ξ ( ξ π + t p π ( t (5.3 + On the other han, the component reache stop of servce at the peroc nspecton tme t+ s renewe a new component mmeatey. Therefore servce tme of ths component s reset to zero at the nspecton tme t+. An the component whose tme of use s ( m s to be renewe a new component at the next nspecton tme. Hence, at the peroc nspecton tme t +, the expectaton of ξ reatve frequency, whch s expresse as π ( t +, of the components beng tme of use zero s efne as π u ξ ( t+ + u ξ ξ ( p π ( t π ( t (5.4 Here, n terms of the repar/renewa nterva, u xu transton probabty matrx s efne as p p L p L P ξ M M M (5.5 p u L pu L Supposng that reatve frequency at the begnnng ξ tme s π ( t, then the expectaton of reatve ξ frequency, whch s expresse as π ( t +, at the arbtrary peroc nspecton t, s efne as ξ ξ ξ π ( t π ( t ( P (5.6 ξ where (P stans for the matrx that transton probabty matrx P ξ to the power of. In aton, havng repeate repar/renewa process for a ong tme, t reaches ong-term steay state. An, et ξ ξ ξ π ( π, K, π m be steay probabty vector whch cassfes components by the tme of use. Then, ξ steay probabty π s efne as π π ξ ξ ξ P ( Formuaton of transton probabty Formuate the transton probabty matrx P ξ wth the use of ranom proportona Webu hazar moe. Probabty p that the component s tme of use at the nspecton tme t an then t s to be avaabe at the next nspecton tme t +, s efne as p Pr{ ζ ( + } Pr{ ζ } (5.8 Moreover, the foowng equaton s obtane wth the use of survva probabty F (ζ ~ F(( + p ~ (5.9 F( p s an eement of Marov transton probabty matrx. Therefore, entfyng the form of the hazar functon λ (ζ, t can be erve concretey. In the case of ranom proportona hazar functon, probabty p s efne as m m m p ( exp[ γ {( + } ] ( Lfe-cyce cost evauaton For evauatng a fe-cyce cost, t s assume that repar/renewa process of eucatona factes system s steay state. Here, the number of component of type s expresse as (,,, repar cost per unt area of type s c, area of the th(,, component of type s s. Uner the peroc nspecton rue an repar/renewa strategy ξ (, u, an average cost C (, u whch s fe-cyce cost per unt of nterva s efne as C(, u s c π ( + L (5. where L s the nspecton cost for whoe eucatona factes system.

12 COCLUSIOS In ths stuy, a eteroraton precton for component groups that mae up eucatona factes was mpemente. In so ong, the authors focuse on each component beng mae up of a sma number of a varety of types, an ponte out that a hazar moe that cou express the heterogenety of the hazar rate between types wou be neee. In ths way, to operatonay express the heterogenety of the hazar rate, a Webu hazar moe was use as a base moe, an a ranom proportona Webu hazar moe was formuate n whch the proportona heterogenety of the hazar rate was expresse as a gamma functon. Furthermore, usng an appcaton case that targete an actua unversty facty, the effectveness of the propose hazar moe was postvey verfe. In aton, n the appcaton of the ranom proportona Webu hazar moe propose n ths stuy to asset management, there reman a number of ssues. Frst, a moe that uses the hazar moe propose n ths stuy to estmate a mut-step eteroraton process nees to be constructe. In genera, the cost of repars to nfrastructures epens on the conton state of eteroraton progress. For that reason, a varety of aternatve repar strateges exst. In vew of ths, when eveopng repar strateges to mnmze repar costs, mxe hazar moes that escrbe mut-step eteroraton processes nee to be extene. Secon, an appcaton that can be use for asset management nees to be eveope. A part of ths research has been carre out on the Program of Promoton of Envronmenta Improvement to Enhance Young Researcher s Inepenence, the Speca Coornaton Funs for Promotng Scence an Technoogy, Japan Mnstry of Eucaton, Cuture, Sports, Scence an Technoogy. REFERECES Ao, K., Yamamoto, K. an Kobayash, K.: Estmatng Hazar Moes for Deteroraton Forecastng, Journa of Constructon Management an Engneerng, JSCE, o.79/vi-67, pp.-4, 5 (n Japanese. Cabnet Offce, Drector-Genera for Pocy Pannng, Soca Capta of Japan, Prntng Bureau, Mnstry of Fnance, (n Japanese. Kato, K., Kobayash, K., Kato, T. an Iuta,., Roa Patro Frequency an Hazars Generatons Rss, Journa of JSCE F, Vo.63, o., pp.6-34, 7 (n Japanse. Kato, K., Yamamoto, K., Obama, K., Oaa, K. an Kobayash, K., Ranom Proportona Webu Hazar Moe: An Appcaton to Traffc Contro Systems, Journa of JSCE F (submtte, n Japanese. Lancaster, T., The Econometrc Anayss of Transton Data, Cambrge Unversty Press, 99. Goureroux, C., Econometrcs of Quatatve Depenent Varabes, Cambrge Unversty Press,. Mshaan, R. an Maanat, S., Computaton of Infrastructure Transton Probabtes usng Stochastc uraton Moe, ASCE Journa of Infrastructure Systems, Vo.8, o.4,. Tsua, Y., Kato, K., Ao, K. an Kobayash, K., Estmatng Marovan Transton Probabtes for Brge Deteroraton Forecastng, Journa of Structura Eng. /Earthquae Eng., JSCE, Vo.3, o., pp.4s-56s, 6.