Asset Management System for Educational Facilities
Considering the Heterogeneity in Deterioration Process
Kengo OBAMA *, Kiyoyuki KAITO**, Kiyoshi KOBAYASHI***Kyoto University*
Osaka University**
Kyoto University***
ABSTRACT: In asset management of infrastructures, predicting deterioration of structures is one of an essential
technique to make a decision on the optimal maintenance policy. However, as for large-scaled infrastructures thatconsist of a huge number of structural components, in order to estimate their deterioration process with highaccuracy, the heterogeneity of individual components has to be considered because each component possesses
different material characteristics and designs and is in unique service under various environmental conditions.
This paper focuses on especially educational facilities from among infrastructures, and constructs its asset
management system considering the heterogeneity in deterioration process of individual components. This system
mainly consists of 3 functions as: 1. database, 2. deterioration prediction and 3. life-cycle cost evaluation. The
database stores the basic structural and component’s information and visual inspection data. Based on theseinformation and data, the deterioration prediction is statistically carried out. Specifically, the deterioration process
can be basically expressed by random proportional hazard model, and the heterogeneity can be modeled asprobability fluctuation in the hazard rate. Furthermore, the time-dependent hazard rate is formulated by the
Weibull hazard model. The heterogeneity of the hazard rates across the individual characteristics of components is
explained by the random proportional Weibull hazard model in which the hazard rates are subject to Gamma
distribution. In the 3rd function, through the comparisons of the life-cycle costs between the multiple repair/renew
strategies, the optimum one is decided. Here, as the deterioration process of individual components can beformulated by the Markov transition probabilities defined by the estimated hazard rates, the proposed life-cycle
cost evaluation method organically links to deterioration prediction results via Markov decision process. In
addition, an empirical study employing visual inspection data for an actual university facility is carried out toverify the validity and applicability of the system.

KEYWORDS: educational facilities, random proportional weibull hazard model, asset management system
1. INTRODUCTION

infrastructures. In fact, it has been pointed out that for
educational facilities that were built in the early stages

In the same way as civil infrastructures, Japaneseof the period of high economic growth, their repair andeducational facilities have been built continuously

reconstruction costs began to surface around 2005,

from the period of high economic growth. In generalsooner than for civil infrastructures. It can easily be

the expected lifetime of educational facilities is about

deduced that these costs weigh down the management

30 years, which is short in comparison to civilof educational facilities, and it is absolutely necessarydecision makings regarding the planning of repair andreconstruction strategies.
In the asset management of educational facilities,
lifecycle cost is an important evaluation index that
determines repair and reconstruction strategies. In
addition, deterioration prediction results are reflected

Repair and Reconstruction Cost
[Trillions Yen]

to develop asset management to support various

14
12Road Infrastructures
Educational Facilities
Others

10
8
6
4
2
0
1985

1990

1995

in the evaluation of lifecycle costs, and so theestablishment of deterioration prediction technique isalso an important issue. In general, deterioration

2000

2005

2010

2015

Years

Fig.1.1 Transitions in the repair and reconstruction
costs of infrastructures

prediction methods can be roughly classified as: 1.
physical deterioration prediction methods based on themechanical deterioration mechanisms of structuralcomponents and 2. statistical deterioration methods
based on past inspection data. However, foreducational facilities, repairs and reconstructions are
sometimes carried out based not only on physicaldeterioration but also on the users’ usability and visualfactors (aesthetics). Therefore, when attempting tocarry out deterioration predictions for educational
facilities, it is preferable to employ a statistical
Statistical deterioration prediction methods are
methods that take vast amounts of deterioration
information and model the regularities behinddeterioration processes. In recent years there has beenremarkableaccumulationofresearch

intodeterioration models using hazard functions. Hazard
models are distinctive because in characterizing thedeterioration process of each facility they respond to
the structural characteristics of the facility andenvironmental conditions to give individual hazard
rates.

However,asaa more exhaustive deterioration prediction, it isnecessary to develop a deterioration prediction methodthat takes into account the heterogeneity of thedeterioration process for individual components.
With an awareness of the above issues, in this study
the authors propose a random proportional hazard
model that expresses the heterogeneity of theindividual deterioration processes of facilities as ahazard rate probability distribution. Furthermore, theauthors propose life-cycle costs evaluation model withdeterioration prediction method.

adiffer for each facility. Therefore, in order to carry out

hazard

rate

is

given

deterministically, the deterioration process for facilities
that have the same structural characteristics andenvironmental conditions will be identical. Regarding
this point, even when structural characteristics andenvironmental conditions are the same, it is morenatural to consider that the deterioration process will

use of random proportional hazard model. Below, insection 2 the basic concepts of this study are
consolidated, in section 3 the random proportional
Weibull hazard model and its estimation methods are
explained, in section 4 as a empirical study ofapplication, a university facility is taken up and someanalysis carried out based on its visual inspection data,and in section 5 proposing method of life-cycle cost
evaluation.
2. BASIC CONCEPTS OF THIS RESEARCH
2.1 Current state of educational facilities
Educational facilities includes public, national andprivate schools (elementary, junior high and senior highschools), research institutions, day cares, kindergartens,
universities,

research

institutions,museums,art

galleries, libraries and community facilities, as well as

targeted for repair or reconstruction if they fail to open

social education facilities, lifelong learning facilities

and close and seem to damage usability for users.

and cultural and community facilities fall into thisFurthermore, if there is partial damage to the tiles ofcategory. In the same way as other social infrastructures,

exterior walls or partial deterioration of paint, repairs

as a part of economic policy in the postwar era, themay be carried out for aesthetic reasons. Therefore, in

educational facilities in Japan were constructed as partthe asset management of educational facilities, evenof repeated social infrastructure development. Afterwhen the authors speak of deterioration predictions a1970 the stock value of educational facilities rapidly

simple physical deterioration prediction targeting

increased, and in 1997 educational facilities accounted

structural safety is not sufficient, and rather a generalfor 12.2% of all social infrastructures (with a grossperformance prediction that includes structural safety

stock value of 70 trillion yen). Furthermore, among

considerations is necessary. At the present time, othereducational facilities, the average serviceable life ofthanschools and academic facilities is considered to be

performance evaluation does not exist, and a statistical

about 30 years, short in comparison to the averagedeterioration

serviceable life of civil infrastructures. In fact, in 2003,

performance prediction method) based upon visual

the gross total area of elementary and junior highinspection data would be effective.

visualinspection,
prediction

this

kind

methodofgeneral(statisticalschools was 160.9 million square meters, of which
41.5% was 20- to 29-year-old facilities and 29.0% was
30 to 39, the aging of which is beginning to be

2.2 Hazard model and the heterogeneity of thedeterioration processactualized. The results of repair and reconstruction cost

In traditional hazards analysis, it is assumed that theestimations based on this data are shown in Fig.1.1.

target facility is entirely built of the same material,

Simply because their average serviceable life is short,with the aim of modeling deterioration phenomena thatthe repair and reconstruction costs of educational

arrive randomly in accordance with certain hazard

facilities are becoming greater than those of civil

functions. In hazard analysis, the occurrence process of

infrastructures. In conclusion, the asset management ofrandom deterioration phenomena is modeled, and theeducational facilities is a problem that has tremendous

hazard function, a deterministic probability model, issocial urgency.

used. However, in large-scale facilities, such as the

As points to consider in the asset management ofeducational facilities targeted in the empirical study ofeducational facilities, when carrying out evaluations of

this paper, it is not necessarily possible to express thethe condition of the facility or components, one musthazard rate of each individual facility component withconsider not only structural safety, but also usability

the same hazard rate. Rather, it is more natural toand convenience for users, and furthermore aesthetic

consider that the hazard rate for each type of

aspect, all of which may be listed as importantcomponent will have a different respective hazard rate.

evaluationthey

For the management and operation of large-scalecharacteristically have a large number of components

facilities, the consideration of repair and reconstructionthat users will come into direct contact with andplans for these many components is a critical issue. In

components that users will view directly. For example,

this way, as a method that expresses the heterogeneity

in the case of doors and window frames, even thoughof a hazard rate that considers the differences in thethey do not in any way influence safety, they will becomponent types, we can consider 1) a method infactors.

In

other

words,

which differences in component properties are

In mixed hazard models, it is considered that

expressed as dummy variables and deterioration

heterogeneity parameters controlling the hazard

estimation is carried out, and 2) a method in which it isfunction are distributed with being subject to aassumed that the hazard rate will be subject to aprobability density function. In addition, a hazard

particular probability distribution for each component

function is defined by probabilistic convolutions of thegroup after which a deterioration estimation is carriedprobability distribution of the hazard function andout. The fist method has the advantage of being simple

heterogeneous parameters. In regards to a mixed

and easy to understand. On the other hand, it ishazard model, Kaito et al. have modeled the arrival

problematic because as the number of components

process of road obstacles and made a case study of theincreases the number of dummy variables (whichapplication to asset management. On the other hand,

express component properties) increases, and theeducational facilities are composed of a large number

estimation accuracy of the model decreases remarkably.

of component types, such as roofs, exterior walls,

In addition, an increase in explanatory variables isdoors and eaves. To put it another way, it can bedirectly connected to an increase in field observation

anticipated that there exist component groups thatand inspection items, increasing the burdens in practice.

require homogenous hazard rates, and that each

Furthermore, because the heterogeneity of thegroup’s hazard rate has an inherent probability

deterioration process may be controlled by factors thatfunction. In this study, it is considered that a mixed

are not possible to observe, it is essential that a morehazard model in which these probability error items

efficient deterioration prediction method be developed.

are assumed to have a gamma distribution.

Educational facilities are constructed from an
extremely large number of components, and estimating

3. RANDOM

a hazard model that makes use of dummy variables is

MODEL

PROPORTIONAL

HAZARDnot practical. In order to express the heterogeneity ofthe deterioration process, there are limits to the

3.1 Random proportional Weibull hazard modelrefinement of a Weibull hazard model by increasing

The random proportional Weibull hazard model is a

explanatory variables and so on. As long as innate

Weibull hazard model that considers the heterogeneity

facility information is expressed as explanatoryof the hazard rate between components. The details ofvariables, the estimate accuracy and efficiency will

hazard models in general may be found in theinevitable decrease. Therefore, in this study the authors

references.

have employed a mixed hazard model in which,

A certain component of a facility is considered to be

depending on the type of component, the heterogeneityclassified into N kinds of component types. A total ofof the hazard rate is expressed as a probability

Ni of the ith (i = 1,…,N) component type exist.

distribution to model the deterioration process of a

Furthermore, focus on the jth (j = 1,…,Ni) component

facility.

of type i. The time that has elapsed since the

Research into hazard analyses that consider thecomponent was reconstructed is represented by theheterogeneity of hazard rates is accumulating. In

random variable ζ i j . Suppose that the arrival rate ofparticular, there is a large accumulation of studies

deterioration events for each component conforms to aregarding mixed hazard models in which there exists a

Weibull deterioration hazard function

heterogeneous hazard rate for each individual sample.

λi (ζ i j ) = γmε i (ζ i j ) m−1(3.1)

In equation (3.1), γ is a parameter that expresses theadvantage that it can express the exponential family

arrival density, m is an acceleration parameter thatprobability distribution function that is defined on the

represents the tendency of the hazard rate to increaseinterval [0, ∞ ] . Here, suppose that the parametertheγ represents the average hazard arrival ratio between

heterogeneity parameter below), which represents thetypes, and the heterogeneity parameter ε i is aheterogeneity of the hazard rate of type i, has been

observation from a Gamma distribution with average 1

added

The

and variance φ −1 and is a probabilistic error term.

heterogeneity parameter takes a common value for

The Gamma function is defined on the interval [0, ∞ ] ,components of the same type. However, when

and with respect to an arbitrary explanatory variable

component type differs, it takes different values. In

and probabilistic error term, the right side of equation

actuality, the heterogeneity parameter takes on a(3.1) is assured to take the positive value. In general,

deterministic value, but is an impossible parameter forthe probability density function g (ε : α , β ) of the

an observer to observe. In addition, the lifespanGamma distribution G (α , β ) can be defined asover time and the parameter

to

the

Weibullhazardε i (called

function.

f i (ζ i j )probability density functionof type ig (ε : α , β ) =~component j, and the survival probability F (ζ i j ) arerespectively expressed as{

f i (ζ i j ) = γmε i (ζ i j ) m−1 exp − γε i (ζ i j ) m{~
F (ζ i j ) = exp − γε i (ζ i j ) m

}

}

1

β α Γ(α )

ε⎞
⎟⎟
β
⎝
⎠
⎛

ε α −1 exp⎜⎜ −(3.3)

The average of the Gamma distribution G (α , β ) is
(3.2a)(3.2b)

Now, the value of the heterogeneity parameter is oneof the observations from random variables that cannot

μ = αβ , and the variance σ 2 = αβ 2 . In addition,
Γ( )is a Gamma function. Furthermore, theprobability density function g (ε : φ ) of the standard
Gamma distribution that has an average 1 and varianceφ −1 is expressed as

be directly measured by the observer, but it is known
to be distributed in accordance with the probabilityg (ε : φ ) =φ φ φ −1
ε exp(− φε )
Γ(φ )

(3.4)

density function g (ε ) . That is the Weibull hazard
model (3.1) has an identical deterioration acceleration
parameter m for all types of components, but for eachcomponent the arrival ratio γmε i differs proportionally,
and the individuality of deterioration is expressed.
Regarding hypothesis testing of the homogeneity(below, proportionality) of the acceleration parameter,
we make another investigation. In this study, for eachtargeted component, a Weibull hazard model in which
the hazard arrival ratio is a observation from aprobability distribution is called a random proportional
Weibull hazard.
Here, suppose that the probability distribution of theheterogeneity parameter conforms to a Gamma
distribution. The Gamma distribution, as a special
form, includes the exponential distribution, and has the

3.2 Two Steps estimation method for the model
In a random proportional Weibull hazard model, a
total of 3+N unknown parameters exist, the arrival
density parameter γ, acceleration parameter m,heterogeneity parameter ε i (i = 1,…, N), which differs
for each component, and the distribution parameter

φ of the heterogeneity parameter. In the case of an
ordinary Weibull hazard model, it is enough toestimate the parameters γ and m from deterioration
data record. However, in the random proportional
Weibull hazard model, besides these two parameters, itis necessary to pursue the probability distribution
parameter φ of the heterogeneity parameter and theheterogeneity parameter ε i (i = 1,…, N) for eachvalue. To express this, it should be noted that thecomponent type.
Now, let us suppose that the deterioration history

likelihood function

Li (ξ i : θ)is defined as an

database of the facility is available. The database

expected value related to the probability function ε i ofcontains information relating to the time of

the conditional likelihood l i (ξ i : γ , m, ε i ) . Here, if adeterioration (repair) of all components from the time

variable transformation xi = ε i (φ + γτ i ) is carried out

the targeted components began service. Express thedeterioration record of components as Ξ = (ξ 1 ,L, ξ N ) ,Li (ξ i : θ) ={where ξ i = {(δ i1 , ζ i1 ),L , (δ iN , ζ iN )} (i = 1,…, N). Inis +φ −1

i
∞⎛
xi ⎞
⎟
⋅ ∫ ⎜⎜⎟
0 φ + γτi ⎠⎝iaddition, δ i is a dummy variable that takes the valuej=

1 if type i component j (j = 1,…, N) has deteriorated,
and takes the value 0 if it has not deteriorated, and ζ i j
is the period of service of type i component j, that is,when δ i j = 0 , ζ i j means the length of time from theprevious repair or reconstruction to the present. On the

{

N

i =1

= ∑ [φ ln φ − ( si + φ ) ln(φ + γτ i ) + ln Γ( si + φ )
N

No

{

j

δij

expressed as θ1 = γ , θ 2 = m}]

θ = (θ1 , θ 2 , θ 3 )However, each element of
(3.5)

}

⋅ f i (ζ i j : γ , m, ε i ){− ln Γ(φ ) + ∑ δ i j ln γ + ln m + (m − 1) ln ζ i j
j =1

j =1

(3.8)

i =1

expressed as
i

}

δij
Γ( s i + φ ) N iφφ⋅ ∏ γm(ζ i j ) m −1
si +φ
Γ(φ )
(φ + γτ i )j =1

ln L(Ξ, θ) = ∑ ln Li (ξ i : θ)l i (ξ i : γ , m, ε i ) for the observed data ξ i for type i is

}

(3.7)

expressed asgiven. At this time, the conditional likelihood

{

dx i

φ + γτ ifunction for the observed data Ξ = (ξ 1 ,L, ξ N ) can be

Here, suppose that the heterogeneity parameter ε i isiexp{− x i }is obtained. Therefore, the logarithmic likelihood

other hand, when δ i j = 1 , ζ i j indicates lifespan.

N(1−δ )~l i (ξ i : γ , m, ε i ) = ∏ F (ζ i j : γ , m, ε i )}δij
φ φ Niγm(ζ i j ) m −1∏
Γ(φ ) j =1

isand θ 3 = φ . The

maximum likelihood estimator of the parameter θHowever, in the above equation, it is explicitlythat maximizes logarithmic likelihood function (3.8)

indicated that the lifespan probability density function

can be given as θˆ = (θˆ1 ,θˆ2 ,θˆ3 ) , which simultaneously~
f (ζ i j : γ , m,εi ) and the survival function F (ζ i j : γ , m, ε i )satisfies

are described as functions of parameters γ, m and ε i .

∂ ln L(θˆ , Ξ)=0∂θ iHere, if the heterogeneity ε i is distributed according
to the standard Gamma distribution g (ε i : φ ) , thelikelihood function for the observation data ξ i is
Li (ξ i : θ) = ∫

∞
0

Ni

{

}

Furthermore, the estimator Σˆ (θˆ ) of the asymptotic
covariance matrix can be expressed as(1−δ )~∏ F (ζ i j : γ , m, ε i )
j =1

{

j

i

} g (ε : φ )dεφ=∏ {γm(ζ ) }Γ (φ )⋅ f i (ζ i : γ , m, ε i )
j

φ

Ni

j =1

∞

⋅ ∫0 εsi +φ −1iˆ ⎤⎡ 2ˆ (θˆ ) = ⎢ ∂ ln L(θ, Ξ) ⎥Σ⎢⎣ ∂θ∂θ′ ⎥⎦

δij

i

i

j
j m −1 δ i

(3.6)

i

exp{− (φ + γτ i )ε i }dε iNiNiabove equation, with respect to all type i components,
the heterogeneity parameter ε i takes a common−1(3.10)

However, the inverse matrix of the right side of theabove formula is the inverse matrix of a 3 x 3 Fisher
information

matrixthat

consistsofelements∂ ln L(θˆ , Ξ) / ∂θ i ∂θ j .

Themaximum

likelihood

2

However, si = ∑ j =1 δ i j and τ i = ∑ j =1 (ζ i j ) m . In the(3.9)

estimator of the parameter is obtained by solving thethree dimensional nonlinear simultaneous equation(3.9). In this study, the maximum likelihood estimator∂ ln L0i (ξ i , ε i : θˆ )
=0
∂ε i
The

maximum

likelihood

estimator(3.12)
oftheheterogeneity parameter obtained in this way is an
estimator that was obtained with the given parameterθˆ = (γˆ, mˆ , φˆ) . In order to clearly describe this, thesolution of equation (3.12) is expressed as εˆ (θˆ ) .
i

From equations (3.11) and (3.12), ifεˆi (θˆ )isspecifically estimated, the following equation is
obtained:εˆi (θˆ ) =si + φˆ − 1
φˆ + γˆτˆi(3.13)

The above two-step maximum likelihood estimator
estimation flow is shown in Fig.3.1.
4. AN EMPIRICAL STUDY
4.1 Overview of the case of application
A random proportional Weibull hazard model
Fig.3.1 Estimate flow of the maximum likelihood
estimator
was obtained using the Newton-Raphson Method. If
the maximum likelihood estimator θ̂ is obtained, using
a covariance matrix estimator Σˆ (θˆ ) , t-test statistic canbe also estimated.
Next, with the parameter vector’s maximum
likelihood estimator θ̂ as a given, the maximum
likelihood estimator of the heterogeneity parameter
ε i (i = 1,…, N) is obtained. Here, the partial likelihood

function is defined asφˆφ N
L (ξ i , ε i : θˆ ) ={γˆmˆ (ζ i j ) mˆ −1 }δ ε is +φˆ−1∏ˆ(3.11)
Γ(φ ) j =1
ˆ

0
i

i

{

The condition state of this facility is accumulated
through visual inspections. The inspection period isthree years. The condition state of the facility isevaluated as either possible to use (○) or not possible
to use (×). This facility group has all been classified
into 33 regions and is located in each one. The oldest
facility was built 73 years ago. This time, the data usedin estimations was the most recent visual inspection
data, which was collected in 2006. Below, the authors
will use the technical term deterioration, but as
mentioned before, the visual inspection data used inji⋅ exp − (φˆ + γˆτˆi )ε i

estimation is attempted for a certain university facility.

i}

Here, τˆi = ∑ j =1 (ζ i j ) mˆ . At this time, the maximum

this study as deterioration is defined not only as
physical damage to components, but is also as loss ofpleasantness and convenience of the facility that isNilikelihood estimator of the heterogeneity parameter
ε i (i = 1,…, N) can be obtained as εˆio that satisfies

judged to need repair.
For the specific estimate target of the random
proportional Weibull hazard model, exterior wallcomponents, for which the greatest abundance of datawas obtained, were focused on. Exterior walls can be

Li (ξ i : θ)classified into five types: tile, multi-layer finish painted,=

Γ( s i + φ ) Nδφφ⋅ ∏ {γm(ζ i j ) m −1 }
s +φ
Γ(φ )
(φ + γτ i )
j =1

j

i

i

(4.2)

i

thin finish painted, metal, and concrete blocks. In
addition, because there are a large number of exterior

At this time, the likelihood proportionality statistic towalls, an extremely small number of components exist

test the assumption testing model (4.1) is expressed by

for which, due to an initial failure, the time period from~
LRi = 2 ln Li (ξ i : θ ) − ln Li (ξ i : θˆ )

] [

{[

the start of service until the deterioration time point[~
ln Li (ξ i : θ )

]}

(4.3)

] expresses partial likelihood

was remarkable short. For this reason, in this estimate,Here,exterior walls for which repair was carried out withinwhen there is not the restraint of the null

one year from start of service are deemed to be initial

hypothesis H 0i and ln Li (ξ i : θˆ )failure samples, and such samples were excluded inlikelihood under the restraint of the null hypothesis[] expresses partial

advance. After the above preliminary preparation, the~H 0i . In addition, if θ does not have a restraint, it

sum total by type of exterior walls samples that could

expresses the maximum likelihood estimator. Since thebe used in the estimate were: 77 tile samples, 35

number of parameters that can be restrained by themulti-layered finish painted samples, 52 thin finish

null hypothesis H 0i is 1, the likelihood ratio test

painted samples, 20 metal samples, and 26 concrete

statistic will have a degree of freedom of one. It

blocks samples. Therefore, the total number of exterior

follows that if the test statistic LRi does not enter thewall samples was 210.

rejection region LRi ≥ χ (2100−α ) (1) , null hypothesis isnotrejected

4.2 Proportionality assumption testing

Here χ

2(100 −α )In random proportional Weibull hazard models, aof freedom 1.

byα%significant

level.

(1) expresses χ distribution with degree2

proportionality assumption is made in which all typesof components have an identical acceleration

4.3. Hazard model estimation

parameter m̂ . Therefore, the differences in the

In the random proportional Weibull hazard model

deterioration process between exterior wall types (tile,

estimated in this study, since the five kinds of types

multi-layered finish painted, thin finish painted, metal

estimated were tile, multi-layer finish painted, thinand concrete block) can be considered to befinish painted, metal, and concrete blocks, N = 5. It

aggregated in the heterogeneity parameter. Based onfollows that there are a total of 8 unknown parameters

actual data, the authors propose an assumption testingthat need to be estimated for the exterior wall: themethod to determine whether or not theproportionality

arrival density parameter γ, the acceleration parameterassumption is effective before making a prediction. Form, the heterogeneity parameters ε i (i = 1,…,5), which

type i (i = 1,…, N) an assumption testing model to test

differ for each component, and the heterogeneity

the proportionality assumption is formulated byparameter’s distribution parameter φ. Following the⎧⎪ H 0i : m = mˆ and γˆ, φˆ⎨ i⎪⎩ H 1 : m ≠ mˆ and γˆ, φˆ

process in section 3.2., the estimated parameter of the(4.1)

Here, once again write the likelihood function, basedon the database, of component i as

random proportional Weibull deterioration hazard
model are listed in Table.1. However, β fulfills
γ = exp( β ) . In addition, ε i (i = 1,…, 5) isheterogeneity parameter values that represent tile,
multi-layer finish painted, thin finish painted, metal

Table 4.1 Estimation results for an exterior wall using a random proportional Weibull hazard model

β

m

φ

ε1

ε2

ε3

ε4

ε5

Maximum likelihood estimator-15.46

4.12

4.16

0.35

0.60

1.82

1.27

0.62

(t-value)(-11.18)(11.18)(1.17)

Log likelihood-323.57

Table.4.2 Likelihood ratio test statistic
LR1i

Tile (i = 1)

4.45

Multi-layer finish painted (i = 2)

1.72

Thin finish painted (i = 3)

4.60

Metal (i = 4)

4.60

Concrete blocks (i = 5)

0.40

Table.4.3 Expected lifespanExpected
Types
Lifespan(Years)Tile

49.9

Multi-layer finish painted

43.8

Thin finish painted

33.5

Metal

36.5

Concrete blocks

43.5

and concrete blocks, respectively. The maximum
likelihood estimator of the acceleration parameter inTable.1 is m = 4.12. Generally, if m = 1.00 thedeterioration probability can be considered to betime-independent, but it can be said that the target
component is clearly the time-dependent type forwhich the deterioration probability will increase withtime. In addition, in the random proportional Weibull
hazard model, a proportionality assumption is madesuch that each type of component has an identicalacceleration parameter m̂ . Under this assumption, theheterogeneity of the Weibull hazard function of eachcomponent is aggregated in the heterogeneity
parameter value ε i (i = 1,…, 5). It follows that there isthe characteristic that depending on the magnituderelation of ε i , the deterioration velocity of each

Tile
Multi-layer finish painted
Thin finish painted
Metal
Concrete Blocks

1.0
Survival Probability

Types

0.8
0.6
0.4
0.2
0.0
0

10

20

30

40

50

60

70

80

Elapsed time [year]

Fig.4.1 Survival Probability of Exterior Walls

component

canbecompared.

From

Table.1,

ε 11 < ε 12 < ε 15 < ε 14 < ε 13 , so it is obvious that in theexterior wall the deterioration progress of thin finish
painted is the fastest and that of tiles is the slowest.
Furthermore, likelihood ratio test statistic LRli(i =1,…,5), which is meant to test the hypothesis testingmodel for the proportionality hypothesis, is indicatedin Table.2. Here, whenα = 99, χ (2100−α ) (1) = 6.6 , so it
is understood that the null hypothesis H 0i in which all
types found in the walls have an identical acceleration
parameter m̂ is not rejected.
The survival function and expected lifespan createdfor each type based on the random proportional
Weibull hazard model are indicated in Fig.4.1 andTable.3, respectively. In the figure, the 5 survival
probability curves indicate the average survival
function for each type of exterior wall. The period ofservice in which the survival probability of exteriorwall component is 50% is 50.0 years for tile, 44.2

years for multi-layer finish painted, 33.7 years for thinlifetime is over or not can be obtained by only periodic

finish painted, 37.0 years for metal, and 43.7 years forinspection. Now, it is assumed that periodic inspection

concrete blocks, and as mentioned before, theis carried out at the time t 0 , t 0 + d , t 0 + 2d , K . Then,

deterioration progress of thin finish painted is thethe discrete-time axis t kd , which is expressed by

fastest and that of tiles is the slowest. When thin finish

initial time t 0 and time interval d , is defined as

painted has been in use for 25 years, its survival

t kd = t 0 + kd (k = 0,1,2, K)(5.1)

probability is about 81.3%, when in use for 50 years its
survival probability is about 0.03%. From this, we cansee that for thin finish painted, as the period of service
becomes longer, the deterioration probability becomeslarger at an accelerating pace. In addition, when tile
has been in use for 25 years, its survival probability is
about 96.5%, and when in use for 50 years its survival
probability is about 50.3%, and in the same way as
multi-layer finish painted, metal, and concrete blocks,
as the period of service becomes longer, a tendency forthe deterioration probability to become larger at anaccelerating pace can be confirmed.
From the above, it can be understood that even forthe same exterior wall component, for each type there
is wide variation in the heterogeneity parameter. Morespecifically, to estimate a hazard model for alarge-scale facility such as an educational facility,
which is made up of various kinds of components, it
can be said that a random proportional Weibull hazard
function that uses a mixed distribution is effective. In
addition, the deterioration survival probability can be
estimated for individual components, which could not
be considered if the hazard rate were simply treated
deterministically. Therefore, it is possible to expect thisto contribute to the refinement of asset management.

Next, consider the issue of managing components,
which are composed of N samples, at the same time.
To discuss it easily, suppose that all sample types are
the same. Furthermore, letξ = (d , u ) be therepair/renewal strategy for educational facilities withthe use of repair/renewal interval d and the maximum
length of service time u . Under the strategy ξ ,condition of components is inspected at the time

t 0 , t1d ,K, t kd ,K . In addition, all components whichhave been used for the time ud are repaired or
renewed. Thus, at time t kd , the number of components
classified by service time is expressed as the condition
parameter vector n ξ (t kd ) = (n0ξ (t kd ),K, nmξ −1 (t kd )) .

The condition parameter nlξ (t kd )(l = 0, K , m − 1)stands for the number of components which have been
in service for ld at the time t kd . Then the relative
frequency of components classified by service time
can
ξ

be
ξ

expressedas

π l (t ) = nl (t ) / N (l = 0, K , m − 1) . Moreover,let π lξ (t kd ) = (π 0ξ (t kd ),K, π mξ −1 (t kd )) be relative
d
k

d
k

frequency vector. Obviously the following equation is
obtained:u −1∑ π ξ (t ) = 1
l =0

l

d
k

(5.2)

Here, focus a certain component of which time in
5. LIFE-CYCLE COSTS EVALUATION

service is ld at the time t kd . In addition, pld is theprobability which this component doesn’t reach the

5.1. Modeling of repair/renewal process

stop of service until the next inspection time. Then, theA certain component of a facility is considered to be

expectation of relative frequency, which is expressed

begun service at the time t 0 and to be stopped service

as π lξ+1 (t kd+1 ) , that the component whose time of use

at the time τ = t 0 + ζ . The lifetime of the component
is expressed as ζ . But the information whether theis ld doesn’t reach the stop of service at the inspection
time t kd and then the component is to be time of use

(l + 1)d at the next inspection time t kd+1 , is defined

5.2. Formulation of transition probabilityasFormulate the transition probability matrix P ξπ ξ (t kd+1 ) = pld π ξ (t kd )
l +1

(5.3)

l

On the other hand, the component reached stop ofservice at the periodic inspection time t

d
k +1

is

renewed a new component immediately. Therefore

with the use of random proportional Weibull hazard
model. Probability pld that the component is time ofuse

ld at the inspection time t kd and then it is to

be available at the next inspection time t kd+1 , isdefined asservice time of this component is reset to zero at theinspection time t kd+1 . And the component whose timeof use is (m − 1)dPr{ζ ≥ (l + 1)d }Pr{ζ ≥ ld }pld =is to be renewed a newcomponent at the next inspection time. Hence, at theperiodic inspection time t kd+1 , the expectation ofrelative frequency, which is expressed as π 0ξ (t kd+1 ) ,of the components being time of use zero is defined as(5.8)

Moreover, the following equation is obtained with theuse of survival probability F (ζ )

pld =

~
F ((l + 1)d )~F (ld )

(5.9)

u −2

pld is an element of Markov transition probability

l =0

matrix. Therefore, identifying the form of the hazard

π 0ξ (t kd+1 ) = ∑ (1 − pld )π lξ (t kd ) + π uξ−1 (t kd ) (5.4)
Here, in terms of the repair/renewal interval d ,function λ (ζ ) , it can be derived concretely. In theu x u transition probability matrix is defined ascase of random proportional hazard function,⎛ 1 − p0d⎜
⎜ 1 − p1dPξ = ⎜ M⎜⎜1 − pud−2
⎜
⎝ 1

d
0

p
0

L
L

M
0

L

0

L

0 ⎞
⎟
0 ⎟
⎟
M ⎟
pud−2 ⎟⎟
0 ⎠probability pld is defined aspld (i ) = exp[−γε i {(l + 1) m − l m }d m ](5.10)(5.5)
5.3. Life-cycle cost evaluation
For evaluating a life-cycle cost, it is assumed that

Supposing that relative frequency at the beginning

repair/renewal process of educational facilities systemtime is π 0ξ (t 0d ) , then the expectation of relativeis steady state. Here, the number of component of type

frequency, which is expressed as π ξ (t kd+1 ) , at thei is expressed as Ni (i=1,…,N), repair cost per unit area

arbitrary periodic inspection t kd , is defined asof type i is ci, area of the jth(j=1,…,Ni) component ofπ ξ (t kd ) = π ξ (t 0 )( P ξ ) kξwhere ( P )

k

(5.6)

stands for the matrix that transition

probability matrix Pξto the power of k . In

addition, having repeated repair/renewal process for along time, it reaches long-term steady state. And, letξξξ

π = (π 0 ,K, π m−1 ) be steady probability vector
which classifies components by the time of use. Then,
steady probability π ξ is defined asξξ

π =π Pξtype i is sij. Under the periodic inspection rule andrepair/renewal strategy ξ = (d , u ) , an average costC = (d , u ) which is life-cycle cost per unit ofinterval is defined as∑ ∑C (d , u ) =

Ni
j =1

sij ciπ 0 (i ) + L

d ∑i =1 N i
N(5.11)

where L is the inspection cost for whole educational
facilities system.

(5.7)

N

i =1

CONCLUSIONS
Aoki, K., Yamamoto, K. and Kobayashi, K.:
In this study, a deterioration prediction for

Estimating

HazardModelsfor

Deterioration

component groups that make up educational facilities

Forecasting, Journal of Construction Management andwas implemented. In so doing, the authors focused onEngineering, JSCE, No.791/VI-67, pp.111-124, 2005

each component being made up of a small number of a(in Japanese).

variety of types, and pointed out that a hazard modelthat could express the heterogeneity of the hazard rate

Cabinet Office, Director-General for Policy Planning,

between types would be needed. In this way, toSocial Capital of Japan, Printing Bureau, Ministry ofoperationally express the heterogeneity of the hazard

Finance, 2002 (in Japanese).

rate, a Weibull hazard model was used as a base model,
and a random proportional Weibull hazard model was

Kaito, K., Kobayashi, K., Kato, T. and Ikuta, N., Road

formulated in which the proportional heterogeneity ofPatrol Frequency and Hazards Generations Risks,

the hazard rate was expressed as a gamma function.

Journal of JSCE F, Vol.63, No.1, pp.16-34, 2007 (inFurthermore, using an application case that targeted an

Japanse).

actual university facility, the effectiveness of the
proposed hazard model was positively verified. In

Kaito, K., Yamamoto, K., Obama, K., Okada, K. andaddition, in the application of the random proportional

Kobayashi, K., Random Proportional Weibull Hazard

Weibull hazard model proposed in this study to asset

Model: An Application to Traffic Control Systems,

management, there remain a number of issues. First, aJournal of JSCE F (submitted, in Japanese).

model that uses the hazard model proposed in thisstudy to estimate a multi-step deterioration process

Lancaster, T., The Econometric Analysis of Transitionneeds to be constructed. In general, the cost of repairs

Data, Cambridge University Press, 1990.

to infrastructures depends on the condition state ofdeterioration progress. For that reason, a variety ofGourieroux, C., Econometrics of Qualitative

alternative repair strategies exist. In view of this, when

Dependent Variables, Cambridge University Press,

developing repair strategies to minimize repair costs,

2000.

mixed hazard models that describe multi-step
deterioration processes need to be extended. Second,

Mishalani, R. and Madanat, S., Computation ofan application that can be used for asset management

Infrastructure Transition Probabilities using Stochastic

needs to be developed.

duration Model, ASCE Journal of Infrastructure

A part of this research has been carried out on the

Systems, Vol.8, No.4, 2002.

Program of Promotion of Environmental Improvement
to Enhance Young Researcher’s Independence, theTsuda, Y., Kaito, K., Aoki, K. and Kobayashi, K.,

Special Coordination Funds for Promoting Science

Estimating Markovian Transition Probabilities for

and Technology, Japan Ministry of Education, Culture,

Bridge Deterioration Forecasting, Journal of Structural

Sports, Science and Technology.

Eng.

/Earthquake

pp.241s-256s, 2006.
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