京都大学防災研究所年報 第 47 号 C 平成 16 年 3 月
Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 47 C, 2004

Optimal Maintenance of Infrastructures under Natural Disaster Risk

Takeshi NAGAE*1 , Hirokazu TATANO and Norio OKADA*1 COE Researcher, DPRI, Kyoto University

Synopsis
This study proposes a framework for dynamic evaluation and management of infrastructures
facing natural disaster risk, especially, earthquakes. In this framework, the optimal maintenanceproblem is formulated as a stochastic impulse control problem, of which optimality condition isrepresented as a variational inequality problem (VIP). Our analyses reveal that the VIP reduces
to a standard-form linear complementarity problem (LCP). This enables us to develop an efficientalgorithm exploiting the recent advances in the theory of complementarity problem.
Keywords : non-steady-state Poisson jump; Dynamic Stochastic Control; Linear Complementarity Problem

1

problem, there are few studies which deal with natural
disaster risk regarding several important facts in sucha large-scale infrastructure management. Specifically,a) several natural disasters, in particular earthquakes,
have non-steady-state behavior; b) there are financial
constraints on maintenance cost whereby preventive
maintenance takes a certain amount of time. Despite of their importance, these features have been
neglected for either simplicity or computability in thepast literature.
The purpose of this article is to propose a prototype of quantitative method for preventive maintenance problem of a infrastructure facing natural disaster risk. The most significant aspects of our method areboth in a continuous time-space framework, and regarding above two features of the infrastructure management explicitly.
This paper is organized as follows. In Section
2, we formulate the optimal maintenance problem as
stochastic control problem in a continuous time-state
framework, where the optimality condition is represented as a Variational Inequality Problem (VIP). Section 3 sketches a numerical method for the problem

Introduction

Recent earthquake disasters have repeatedly
demonstrated the seismic vulnerability of grand-scale
infrastructures, especially road, bridge and so on.
While much attention has been paid on the preventive
maintenance of these infrastructures, there are few
studies which deal with natural disaster risk.
In the field of operations research, there areenormous numbers of studies which dealt with optimal maintenance problems of a deteriorating system(see Wang[6]). Stadje and Zuckerman [5] argues
an optimal problem with endgenous repair degree.
In the field of civil engineering, Rioja[4] discussed
a problem of maintenance versus new investment
in public infrastructures. Kobayashi and Ueda[3]formulated the optimal maintenance problem as astochastic impulse control problem. They further
analyse the problem by using the Markov decision
process technique.
Althogh these studies have developed quite
convenient methodology for the optimal maintenance1

exploiting the recent advances in the theory of complementarity problem. For this reason, we show the
optimality condition reduces to a Linear Complementarity Problem (LCP) via certain variable transformation techniques. This numerical method is applied in
Section 5. Section 6 concludes.
2

The Model

Consider one facility (e.g. one bridge in privateroad) managed by one administrator who intends tomaximize the revenue during a certain period [0, T ].
One can imagine the administrator holds a maintenance project as a PFI(Private Finance Initiative)
contract of which mature is T .
2.1

Formulation
We denote the state of the facility as an aggregate one-dimensional variable. This variable called thefunctional level, of which value at time t is denoted as
P(t). We assume that the functional level follows astochastic differential equation,dP(t) = x(t) dt − µ(t, P) dt+ σ(t, P) dW(t) − η(t, P) dq(t),P(0) = P0 ,

(1)
(2)

where µ, σ, η are given functions of (t, P), and the initial value P0 is a given constant. The dynamics (1)consists of four terms. The first term represents theenhancement of the functional level by the repair of
the facility. We denote the increment of the functional
level by x(t) called the maintenance flow. per unit
time. The second term, µ dt, states that the expected
depreciation. The third term σ dW(t) adds a stochastic element of the depreciation in which dW(t) is theincrement of a standardized Wiener-process W(t) andthe coefficient σ(·) denoted the instantaneous standard
deviation of the functional level perturbation. The lastterm, η(t, P) dq(t) represents the discontinuous downward jump caused by the disaster, say an earthquake.
While dq(t) is the increment of a non-stead-state Poisson process whose intensity at time t is denoted by agiven function λ(t), the coefficient η(·) represents thefragility of the functional level against the disaster.
Suppose that the revenues from the facility
consist of three elements. The first one is the instantaneous revenue from the facility, of which value attime t is represented as a given function π(t, P(t))*1 .
The second element is the maintenance costwhich assumed the following linear function of themaintenance flowC(t, x(t)) ≡ Ax(t),(3)

where A is a given constant. The last element is theterminal payoff earned only at the expire date of thecontract T . We denote the terminal payoff by a givenfunction F(T ) ≡ F(P(T )).
Suppose that there is a financial constraint on themaintenance cost. Specifically, we assume that the instantaneous maintenance cost has an upper limit C̄.
Therefore, the maintenance strategy at each momentx(t) should satisfy the following constraint.
0 ≤ x(t) ≤ k̄ ≡ C̄/A,∀t ∈ [0, T ].

(4)

The administrator decides the maintenance strategy {x(t)}Tt=0 in order to maximize his own revenue
from the facility during the contract period [0, T ]. This
profit maximization problem is formulated as the following stochastic control problem.
[P]max. J(0, P0 , x(·)), s.t. (4),{x(t)}Tt=0

where ρ is a given constant which represents the discount rate of the cash flow stream, and J(t, P) is defined as follows:
J(t, P, x(·)) ≡"Z TEe−ρ(s−t) {π(s, P(s)) − C(s, x(s))} dtt+e−r(T −t)#
F(P(T )) P(t) = P . (5)Eq.(5) represents the net present value of the cash flow
stream during [t, T ] under the maintenance strategy{x(s)}Ts=t .
2.2

Optimality Condition represented as a VIP
We define the value function of the problem [P],under a situation in which the functional level P(t) = P
is observed at time t, as follows:
V(t, P) ≡ max. J(t, P, x(·)), s.t. (4),{x(s)}Ts=t∀(t, P) ∈ [0, T ) × R+ ,(6)

where the terminal condition isV(T, P) ≡ F(P),∀P ∈ R+ .

(7)

Applying the DP (Dynamic Programming) principle,
we obtain the following HJB (Hamilton-JacobiBellman) equation
V(t, P) = max. π(t, P) − C(t, x(t))*1 We can suppose the administrator intends to maximize thesocial benefit from the facility. In that case, the instantaneous
revenue should be redefined as the ’instantaneous benefit’,correspondingly.

x(t)

−ρ dt

+e

E [V(t, P) + dV(t, P)|P(t) = P] ,s.t. (4). (8)That is, the value function V(t, P) describes the maximum expected present value of revenues given the current functional level is P. Using the Ito lemma, the
HJB equation (8) reduces tomax. π(t, P) − C(t, x(t)) + x(t)VP (t, P) + LV(t, P) = 0,
x(t)

(9)
where L is a differential operator defined as follows:(

LV(t, P) ≡

)
∂
∂
1 2
∂
− µ(t, P)+ σ (t, P) 2 − (ρ + λ(t)) V(t, P)
∂t
∂P 2
∂P
+ λ(t)V[t, P − η(t, P)]. (10)2Because there are no adjustment cost or costs associated with changing the level of maintenance, theproblem [P] has a ’bang-bang’ solution: the instantaneous maintenance level will be0 if VP (t, P) < A∗
x (t, P) = .
(11)k̄ if VP (t, P) ≥ A
Substituting the above optimal strategy into the HJB
equation (9), we obtain the following two differential
equation.
π(t, P) + LV(t, P) = 0,(x∗ (t, P) = 0)(12)π(t, P) − C̄ + L1 V(t, P) = 0,(x∗ (t, P) = k̄),

(13)

where
L1 ≡ L + k̄

∂
.
∂P

(14)

Since one of the two strategies 0 or k̄ must be
optimal, either (12) or (13) holds as equality. Hence,
V(t) ≡ {V(t, P)|∀P ∈ R+ } is the solution to the following VIP (Variational Inequality Problem) held at timet.
[VIP(t)]()
min. −π(t, P)−LV(t, P), −π(t, P)+C̄−L1 V(t, P) = 0,
∀P ∈ R+ .
where the terminal condition is Eq.(7).
3

Numerical Methods

Since the optimal maintenance problem [VIP(t)]has no analytical solution, we argue a numericalmethod in this section. First we reformulate theproblem [VIP(t)] in a discrete framework. We thenshow the reformulated VIP reduces to a standard
form LCP (Linear Complementarity Problem) via
certain variable transformation techniques, whereby
we develop an efficient algorithm exploiting the recent
advances in the theory of complementarity problem.

3.1

Discretization
Let us consider a discrete grid in the timestate space with increments ∆t and ∆P, and leti, j≡ f (i∆t, j∆P) denote an arbitrary function at thegrid points, where the indices i ∈ {0, 1, · · · , I} andj ∈ {1, · · · , J} characterize the location of the pointwith respect to state and time, respectively. In thissetting, the [VIP(t)] is reformulated as follows:[VIPi ]

nmin. −πi − Li V i − Mi V i+1 ,o− πi + C̄1 − Li1 V i − Mi1 V i+1 = 0,(15)where V i ≡ {V i,1 , · · · , V i,J }, πi ≡ {πi,1 , · · · , πi,J }are given J-dimensional vectors, and 0, 1 are Jdimensional vectors with every elements 0, 1,
respectively. Li , Mi , Li1 , Mi1 are J × J matrix obtainedby approximating of the differential operators L, L1 ina certain finite difference scheme. (See Appendix A).
Correspondingly, the terminal condition (7) reduces to
V I = F,(16)

where F ≡ {F 1 , · · · , F J }.
Note that the [VIPi ] can be treated as an independent problem when the next period’s value function
V i+1 is known. Therefore, we can solve the [VIPi ] in a
successive manner as follows: First, the value function
at the expire date V I is known as the terminal condition (16); Then the unknown variable V I−1 at the I−1th
period can be obtained as the solution of the [VIPI−1 ],in which V I is a known variable. Repeating this procedure successively, we can obtain whole unknown variables {V i |i = 0, 1, · · · , I − 1}.
3.2

Reduction to A Standard Form LCP
The optimal maintenance problem [VIPi ] is noteasy to handle because of its non-standard form.
Consequently, we show that the [VIPi ] can reduces
to a standardized form LCP (Linear Complementarity Problem) via a certain function transformation,
whereby an efficient algorithm for calculating the
problem [P] as shown in the Section 3.3.
First, let us consider the following new unknown
variable.
Xi ≡ −πi − Li V i − Mi V i+1 ,i = 0, 1, · · · , I − 1. (17)

If Li is nonsingular, we can denote the V i as the following linear transformation of Xi .
 −1
V i ≡ − L i X i − gi ,
(18)
where

 −1 higi ≡ − Li
πi + Mi V i+1

is a known variable at the ith period.

(19)

Substituting the variable transformations (17),(18) into [VIPi ], we obtain the following standard
form LCP.

Algorithm 1 Fukushima Procedure
1: procedure F LCP(i, V i+1 )2:
Xi(1) ∈ R+J. initial feasible solution
3:n := 1
4:
repeat5:dn := Hi (Xi ). descent direction

Xi · Gi (Xi ) = 0, Xi ≥ 0, Gi (Xi ) ≥ 0,

[LCPi ]

where Gi (·) is a linear function of Xi defined as follows:
 −1
(20)
Gi (Xi ) ≡ +Li1 Li Xi − hi ,
and
h

i

≡ π − C̄1 − Li1 gi − Mi1 V i+1i(21)

is a known variable at the ith period.
Algorithm
The key of the above sections are twofold: the[VIPi ] reduces to standard form LCP; the [VIPi ] canbe solved in a successive manner. Thus, an efficientalgorithm the maintenance problem [VIPi ] can be denoted as follows: (a) Compute Xi as the solution of the[LCPi ], regarding V i+1 as a constant; (b) Obtain V i via
the variable transformation (18); (c) Repeat these two
steps by turns.
While there are a large variety of numerical algorithms for the standard form LCP such as [LCPi ],we adopt the merit function approach that appearedwith the recent advances in the theory of operations
research. Generally, this merit function approach isquite effective and does not require strict conditions
for convergence compared to the traditional methods,
for instance, diagonalization method, Lemke method,
projection method, and so on (See Ferris and Pang[1]).
This approach reduces the [LCPi ] to an convex
problem with one-dimensional objective function
Φi (Xi ) called the merit function. The merit function isa continuous RJ → R+ mapping whose value satisfies
that Φi (Xi ) = 0 when Xi is the solution of the [LCPi ]and Φi (Xi ) > 0 when Xi is not the solution. In thisstudy, we introduce the Fukushima merit function[2]defined as follows:

Find α such thatmin. Φi (Xi(n) + αd(n) )6:7:α∈[0,1]i(n+1)

i(n). line search

(n)

8:
X
:= X + αd. iteration
9:n := n + 1
10:until Xi(n) converges to the solution
11:
return Xi(n) .
12: end procedure

3.3

Φi (Xi ) ≡ −Gi (Xi ) · Hi (Xi ) −
where

1 i i
H (X ) · Hi (Xi ), (22)2hi
Hi (Xi ) ≡ Xi − Gi (Xi ) − Xi ,+(23)and [Z]+ is an operator of projection on the positiveJ
real spacenoR+ , of which kth element is defined askmax. Z , 0 .
An efficient algorithm for [LCPi ] using
Fukushima merit function (22) can be describedas the following procedure[2], which requires V i+1
previously obtained.
In summary, whole of the numerical method forthe optimal maintenance problem [P] is described asfollows:

Algorithm 2 Main Algorithm
1: V I = F. terminal condition
2: for i := I − 1 to i = 0 step −1 doCompute Xi by using the F
LCP(i, V i+1 ) procedure.
4:
Obtain V i by substituting Xi into Eq. (18)5: end for
3:

4

ExampleWe finally apply our method to a simple example and show several numerical results. The purposeof this section is to show an illustrative example ofour framework and to confirm the algorithm solves the
problem. For this reason, we adopt several restriction
on the model in this section. Remember these assumption do not affect the essence of our method.
First, let us assume that the dynamics of the functional level as the following geometric form:dP(t)= µ dt + σ dW(t) + η dq(t).
P(t)

(24)

We further suppose a steady-state Poisson process forthe catastrophic event whose intensity is denoted λ ≡λ(t)*2 .1
In the rest of this section, we show the value function and the optimal strategy under the following parameters:
T = 50,1,
λ=50µ = 0.02,

σ = 0.01,

ρ = 0.1

η = 0.5,C̄ = 0.5,

A = 1.0,(25)

and the linear instantaneous revenue
π(t, P) ≡ 0.1P − 0.6.
*2

(26)

Note the intensity is a reciprocal number of the recurrence
interval of the catastrophic events.

4.1

Value Function
Fig.1 displays the value function at each pointin the time-state space. While the two horizontal

1
1
the three case of the catastrophic intensity, λ = 20
, 100
and ∞1 (= 0). The last case represents a situation of
no disaster risk exists. Fig. 3 shows that as the catas20

value function
25
20
15
10
5
0
-5

15

suspend
10

-10

=1/

=1/20
20
15

state10

5
0

0

10

5

15

20

25

30

35

40

45

5

50

=1/

full maintenancetime
0

Fig. 1 Value Function

5

10

15

20

25

30

35

40

45

50

0

Fig. 3 Optimal Maintenance Strategy

15

full maintenancesuspendtrophic intensity increases (i.e. the reccurence interval
decreases), the threshold shifts upward (i.e. the administrator intends to ’fully maintain’ frequently).

value function

10

5

5

Conclusion

0

-5

-10

P*(0)
0

2

4

6

state

8

10

12

14

Fig. 2 Value Function at t = 0

axes represents time and the state variable (i.e. thefunctional level), the vertical axes represents the valuefunction. This figure shows that the value function isincreasing respect to the functional level at arbitrary
moment.
Fig.2 shows a slice of Fig.1 at the initial time
t = 0. In this figure, the horizontal axis represents
the functional level and the vertical one denotes thevalue function, respectively. This figure shows that thevalue function consists of two functions connected at
the level P∗ (0), at where the optimal strategy switchesfrom ’full maintenance’ to ’suspend’. In other words,
if the functional level is below P∗ (0), the administrator
maintain the facility as much as possible. Contrarily,
the administrator does nothing but wait for a momentin the case of the functional level exceeds P∗ (0).
4.2

Maintenance-Suspend Threshold
The switching threshold at every moment P∗ (t)is displayed in Fig.3, of which horizontal axis represents time and the vertical axis is the functional level.
In this figure, three thresholds are shown respectively

This study proposed a prototype framework forquantitative analysis on an optimal maintenance problem of a certain infrastructure faces catastrophic risk.
In this framework, we have shown that the finite difference method is still convenient to deal with nonsteady-state Poisson processes. Further we have revealed not only a ’bang-bang’ strategy is optimal under a linear maintenance cost, but also the optimality
condition can be formulated as a variational inequality problem (VIP). We finally have shown the VIP reduces a linear complementarity problem by using certain variable transformation techniques. This make us
enable to develop an efficient algorithm exploiting therecent advances in the complementarity problem theory.

Aknowledgement
This study was suppoted by the Japanese Ministry of Education, Culture, Sports, Science and Technology 21st Century Center of Excellence (COE) Program, awarded to the Disaster Prevention Research Institute, Kyoto University (Grant No. 14219301). The
fist author would like to thank Dr. Akamatsu (Tohoku
University) for his technical comments. The authors
also thank the secretaries and students of Integrated
Management for Disaster Risk research division for
their kindly help which have kept the authors awayfrom wearying chores.

∂2 V(t, P)
V i+1, j+1 − 2V i+1, j + V i+1, j−1
≈
β
∂P2
(∆P)2
References+ (1 − β)[1] Ferris, M. C. and Pang, J.-S. eds.: Complementarity and Variational Problems: State of the Art,
Society for Industrial and Applied Mathematics,
1997.
[2] Fukushima, M.: Equivalent differentiable optimization problem and descent methods for asymmetric variational inequality problems, Mathematical Programming, Vol. 53, pp. 99–110, 1992.
[3] Kobayashi, K. and Ueda, T.: Perspectives andresearch agendas of infrastructure management,
Journal of Infrastructure Planning and Management, Vol. 61, No. 744, pp. 15–27, 2003.
[4] Rioja, F. K.: Filling potholes: Macroeconomic
effects of maintenance versus new investmentsin public infrastructure, Journal of Public Economics, Vol. 87, pp. 2281–2304, 2003.
[5] Stadje, W. and Zuckerman, D.: A generalized
maintenance model for stochastically deteriorating equipment, European Journal of Operational
Research, Vol. 89, pp. 285–301, 1996.
[6] Wang, H.: A survey of maintenance policies ofdeteriorating systems, European Journal of Operational Research, Vol. 139, pp. 469–489, 2002.
Appendix A

By applying the above finite difference scheme,
LV(t, P) is rewritten as follows:
LV(ti , P) ≈ Li V i + Mi V i+1 + λi Di V i ,(27)

where Li is a trigonal matrix defined as follows:
 i,1
b
ai,2

 0

i
L ≡  ...

 0

 0
0

ci,1
bi,2
ai,3
..
.

0
ci,2
bi,3
..
.

···
···
···
..
.

0
0
0
..
.

0
0
0
..
.

0
0
0

0
0
0

···
···
···

bi,J−2ai,J−10ci,J−2bi,J−1
ai,J


0 

0 
0 
..  ,. 

0 

cJ−1 

bi,Jand Mi is a trigonal matrix, of which diagonal elements bi, j is replaced by di, j . For instance, when we
adopt the Crank-Nicholson scheme (i.e., β = 1/2),
i, j
i, j
/∆P)2
these elements are ai, j ≡ µ /∆P+(σ, bi, j ≡ − ∆t1 −
4
i, j
i, j
/∆P)2
(σi, j /∆P)2− (r + λi ), ci, j ≡ −µ /∆P+(σ, and di, j ≡24(σi, j /∆P)2
1, respectively. In this approximation, Di∆t −2

is a lower triangular matrix, of which ( j, k) element isdefined as follows:η(ti ,P j )
1 if k ≤ ∆P < k + 1
i, j,kδ ≡(28)
0 otherwise

The Finite Difference Method

We approximate the partial derivatives in the[VIP(t)] as follows

Correspondingly, L1 V(t, P) is approximated asfollows:∂V(t, P) V
−V
≈
,
∂t
∆t
∂V(t, P)V i+1, j+1 − V i+1, j−1
V i, j+1 − V i, j≈β
+ (1 − β)
∂P
2∆P
2∆P
i+1, jV i, j+1 − 2V i, j + V i, j−1
,
(∆P)2

i, j

L1 V(ti , P) ≈ Li1 V i + Mi1 V i+1 + λi Di V i ,(29)where Li1 , Mi1 are tridiagonal matrices as same as
Li , Mi but their elements ai, j , ci, j are replaced by
i, j
i, j
i, j
i, j
/∆P)2
/∆P)2
ei, j ≡ (−k̄+µ )/∆P+(σand f i, j ≡ (k̄−µ )/∆P+(σ,44under the Crank-Nicholson scheme.

自然災害リスク下での施設の最適補修戦略
長江 剛志*1 ・多々納裕一・岡田憲夫*1 京都大学防災研究所 COE 研究員

要旨
本研究の目的は，災害リスクに直面する社会基盤施設の最適補修問題に対する定量的分析手法のプロトタイプの提案である．具体的には，まず，最適補修問題を確率的インパルス制御問題として定式化し，その最適性条件が変分不等式問題(VIP: Variational Inequality Approach) として記述されることを示す．次に，この VIP が，適当な関数変換によって，標準形の線形相補性問題 (LCP: Linear Complementarity Problem) に帰着することを明らかにする．この分析結果に基づき，最近の相補性問題理論を活用した効率的解法を開発する．
キーワード： 予防的保全，非定常 Poisson 過程，確率的インパルス制御問題

COE-33

災害リスク下での社会基盤施設の最適補修戦略
長江剛志，多々納裕一，岡田憲夫(Z T[P] max. E

1 はじめに{x(t)∈}T0

道路や橋梁などの社会基盤施設（以下，施設）の多く
は，一般に，これらの施設の新規建設には多くの費用を必要とし，その供用期間は数十年以上にも及ぶ．そのため，これらの社会基盤施設に対しては，長期的な補修を

−ρt

e

)
{π(t, P(t)) − C(x(t))} dt P(0) = P̄ .

0と定式化される．

3 最適性条件問題 [P] の最適値関数を以下のように定義する．
(ZT)

e−ρ(s−t) {π(s, P(s)) − C(x(s))} ds P(t) = P

明示的に考慮した性能設計が必要不可欠である．こうしV(t, P) ≡ max. E

た社会基盤施設の補修戦略および性能設計を行うには，

この式に DP 原理を適用して整理すれば，時刻 t におい

以下の 3 つの特性を考慮する必要がある：a）地震などて成立するべき以下の HJB(Hamilton-Jacobi-Bellman)の予測不可能な自然災害によって構造物が破壊され，施方程式を得る．

設からのサービス水準が著しく劣化するリスク（災害リスク）が存在する；b）施設の自然劣化による減耗が非定常的な動学的不確実性をもつ；c）各年度の補修費用に上{x(s)∈}Tttmax. π(t, P) + LV(t, P) + x {−A + VP (t, P)} = 0, ∀P ∈ R+ . (2)0≤x≤k̄

ここで，L は偏微分作用素で，以下の式で定義される．

た枠組の下で，最適補修問題に対する定量的分析手法の

1
LV(t, P) ≡ Vt −µVP + σ2 VPP +(ρ−λ)V+λ(t)V(t, P−η(t, P))
2

提案を目的とする．

この作用素は，施設の性能水準 P の確率微分方程式 (1)限が存在する．本研究では，これらの要因を全て考慮しから一意に決まる．HJB 方程式 (2) は，補修速度 x に

2 定式化

関して線形であるため，最適補修戦略は x = 0 もしく

本研究では，社会基盤施設として高速道路における橋は x = K/A の Bang-Bang 制御となる．これより，HJB

梁や高架といった単一の構造物を対象とする．そして，方程式 (2) は以下の変分不等式問題 (VIP:Variational In-

有限の管理期間 t ∈ [0, T ) に施設から得られるサービス

equality Problem)：n[VIP(t)] min. π(t, P) + LV(t, P),

水準に影響する機能水準 e.g. 路面性状）を，一次元変数

P(t) で集約的に表現できるとし，以下の確率過程で表現
する．

dP(t) = x(t) dt − µ(t, P) dt + σ(t, P) dW(t)− η(t, P) dq(t),P(0) = P̄.

oπ(t, P) + k̄VP + LV(t, P) − C̄ = 0,

∀P ∈ R+ .

として書き直せる．終端条件は，V(T, P) = 0 とする．

(1)

この式の右辺第 1 項は補修 x(t) による機能水準の向上

4 最適補修問題の解法を，第 2, 3 項は自然劣化による減耗のドリフトおよびボ

最適補修問題 [P] は，各時刻 t ごとに成立する変分不ラティリティを，第 4 項は自然災害による不連続的な劣等式問題 [VIP(t)] に分解できることが判った．筆者らは，化を表す．ここで，q(t) は強度 λ(t) の非定常 Poisson 過

このような VIP が，適切な関数変換によって数理計画

程であり，η は t, P についての既知関数とする．

分野で良く知られる標準形の相補性問題に帰着すること

状 態 (t, P) に お い て 補 修 量 x が 選 択 さ れ て い る とを明らかにしている 1) ．この分析結果を用いれば，問題

き，この施設から単位時間当たりに発生する純便益を，[VIP(t)] に対しても，相補性問題の数値解法に関する最F(t, P) ≡ π(t, P) − C(x) とする．この式の右辺は，それぞ新の研究成果を活用した効率的計算法が開発できる．それ，利用者（租）便益および補修費用を表す．ここで，の詳細なアルゴリズムおよび計算結果については，発表

補修費用は補修速度に対して線形 (i.e. C(x) = Ax とし，会で報告する予定である．

任意の時刻において上限 C̄ が設けられているとする (i.e.

C(x) ≡ Ax < C̄)．
上述の枠組の下で，施設管理者は，供用期間 [0, T ) 中に獲得する便益の期待現在正味価値を最大化するように補修戦略 {x(t)|t ∈ [0, T )} を決定する．この行動は，

参考文献[1] 赤松隆, 長江剛志：不確実性下での社会基盤投資・運用問題に対する変分不等式アプローチ, 土木学会論文集, 2004, 投
稿中.