New Approach for Efficient Road Maintenance on Urban Expressway
Based on HEML (Hanshin Expressway Logic Model)
Yasuhito SAKAI*, Kiyoshi KOBAYASHI**, Haruhiko UETSUKA***
Hanshin Expressway Company Limited*
Graduate School of Management, Kyoto University**
CTI Engineering Co., Ltd.***

ABSTRACT: Hanshin Expressway is an urban expressway network in Osaka-Kobe area accommodating
more than 900,000 vehicles per day. Since it is one of the most important transportation infrastructures in thearea, the highest level of Rehabilitation and Maintenance (R&M) quality is always required and we have met
the need by developing many efficient maintenance methods. However, due to the lacks of the sufficientbudget for highway R&M, the highway administration bodies are requested to reduce the expenditures for
R&M activities.
Hanshin Expressway has accelerated to develop a new approach for efficient Road R&M after its
privatization. In this paper, the logic model for R&M (named HELM; Hanshin Expressway Logic Model),which was developed by the Hanshin Expressway Corporation, is presented to administrate the whole system
of R&M activities in an efficient way. This paper presents the risk management diagrams for the highway
patrol, by which the administrative bodies can efficiently design the optimal patrol frequency given thepredetermined risk levels based on HELM. The paper is concluded by summarizing the ongoing research
agenda to improve the HELM throughout the PDCA cycles of the real R&M activities.
KEYWORDS: Logic Model, Rehabilitation and Maintenance, Risk
to improve the efficiency of maintenance work.
1.

INTRODUCTION

Kaito et al. [2], for example, modeled the occurrenceof road impediments as a Poisson process, proposed

Maintenanceof

expressways

includesvariousservices: maintenance of bridges, tunnels, and otherstructures; roadway cleaning and inspection for
ensuring the safety of road users; maintenance of
information

systemsforprovidingtraffic

information continuously; and cleaning of parking
and other facilities for the amenity of road users [1].
The road managers are required to keep expressways
always in good condition while improving theefficiency of maintenance.
Management systems are being developed recently

a risk management model on impediment occurrence
to discuss desirable road patrol strategies that would
reduce the cost of patrol, and verified theeffectiveness of the model. Aoki et al. [3] modeled
the deterioration of lighting in expressway tunnels,
and proposed and verified the optimization of tunnel
lighting systems in consideration of life-cycle cost
and the risk of unlit lights. These studies focused on
the efficiency of individual maintenance activities
rather than the entire maintenance work.
The authors have developed a maintenance logic
1/11

model for managing roadway inspection and othermanagement targets (outcomes), risk management

routine maintenance that directly affects road users,

levels (outputs), and maintenance activities (inputs),

and proposed a method to perform evaluation andthe adequacy of risk management levels for theverification appropriately according to the PDCA

entire maintenance and repair work as well as(plan-do-check-act)

paper

service levels can be discussed comprehensively.

discusses methods to pursue the efficiency andThis study foresees future commissioning ofadequacy of maintenance risk management. Whilemaintenance and repair work on the basis of

the risk is defined in various ways, this paper defines

specified performance. The study therefore aims to

it as the product of the probability of damage and thedevelop a methodology that allows specification ofmagnitude of damage. The maintenance risk includes

performance based on a clear basis from theaccidents, major repairs, customer complaints, as

viewpoint of adequate risk management.

cycle

[4,

5].

This

well as management flaws caused by neglecting
inspection, repair, cleaning, and other maintenancework. This study organizes the objectives andmeasuresofmaintenance

risk

2.

LOGIC MODEL DEVELOPMENT

management

systematically as a logic model and discusses the
optimization of maintenance risk levels by using
outcome and output indicators. In other words, therisk of high-risk management items can be reducedby raising their maintenance levels, and the level ofoverly-low-risk maintenance items can be reduced to
cut their maintenance cost. As a result, cost reduction
can be achieved while suppressing the risk of theentire management items in a balanced manner.
The occurrence of road impediments, such as
potholes, fallen objects, and sand, varies with theroute. The probability of the road users to encounter
impediments can be reduced by increasing thefrequency of inspection, which means reducing thetime impediments are left unsolved. To reduce theprobability of encountering impediments in areasonably balanced manner, higher and lower
inspection frequencies should be applied to heavyand low-traffic routes, respectively. While defining
the risk as the product of the probability ofencountering impediments and the volume of traffic,
the risk of the entire route network can besuppressed adequately by controlling the risk of eachroute to achieve its target level. By using a logic
model to analyze the relationship between the risk

2.1 Outline of logic model
According to the theory of new public management(NPM) [6], there is always a hypothetical logic that
explains what outcomes are intended by the activityof any policy or project. A logic model is a tool todefine and show systematically final outcomes,
intermediate outcomes required for achieving thefinal outcomes, and the action required for achieving
the outcomes. The final outcomes in the present caseinclude the improvement of customer satisfaction
and the reduction of risk that vehicles on the roadface. The logic model clarifies the intention of thepolicy or project to be evaluated by describing what
effects are generated and what final outcomes are
obtained by conducting the policy or project; thisinvolves description of multiple steps and procedures
as well as schematic representation of their mutual
relevance [7].
Logic models have been established as a basic rule
to support the NPM theory and have been used forcarrying out administrative and fiscal reforms. In
Japan, the Government Policy Evaluations Act
became effective in 2001 and governmental agencies
are developing basic plans for evaluating policies.
However, systematization of objectives and policies
2/11

using logic models has not been achieved yet. Some

2.2 HELM construction and indicator setting

manuals for developing logic models were proposedIt

in the West [8], and logic models have been usedmaintenance is conducted to ensure the safety ofwidely in particular in Anglo-Saxon nations.

road

Application of logic models to asset management

maintenance has been recognized on both individual

has been reported in Australia and other nations [9];and departmental bases but has not been grasped

to our knowledge, however, there has been nosystematically. As a result, there have some

reported logic model specifically designed for asset

problems: the levels of risk management may varymanagement. A logic model shows various eventswith the route and time for the same maintenance(elements) that may occur in the intermediate

operation, and conformity in risk management levels

processes between specific activities and final

may not be achieved between different maintenanceoutcomes, and connects the elements with a single

operations. With the aim of conducting the risk

line or multiple lines to clarify the procedure ofmanagement of the entire maintenance work

achieving the outcomes. To indicate the degree ofeffectively, the present study organizes the objectives

change or improvement in the policy or project,and measures of the entire maintenance work

outcomes are presented in multiple steps (forsystematically.

example, intermediate and final outcomes) as shown

management levels to be achieved and thein Table 1 [7, 8].

maintenanceisunderstoodusers.

conceptually

CausalThe

that

relationshipstudy

operations

to

shows
be

in

routineroutinetheconducted

risk
for

achieving the levels by using a logic model called
Hanshin Expressway Logic Model (HELM). In the
HELM, the inputs include routine maintenance
operations and their frequency, and the outcomes
include the safety of road users. The causal
relationships in the intermediate processes are
systematically described by using intermediateoutcome and output indicators, so as to allow
quantitative evaluation as much as possible. A policy
evaluation model was developed to evaluate theTable 1. Elements of logic model

relationship between inputs and outcomes. The
developed HELM is a large-scale model. To outline

The logic model has the following features:- The process between activities (input resources)

the configuration of the model, only part of themodel is shown in Figure 1 for the sake of brevity.

and final outcomes is shown by connecting theelements with a single line or multiple lines.
- Outcomes are presented in multiple steps.
In this way, the logic model allows us to show theprocess of deriving outcomes of a policy or projectplainly and objectively; otherwise the process tends
to be vague.

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Finaloutcomes

Intermediateoutcomes
Reduction ofdamagecaused byon-roadimpediments

Ensuring
safetyofvehicleson roadReduction ofdamagecaused bydrainagedefects

Reduction ofdamagecaused byroad-surface
defects

CustomerloyaltyandappreciationReduction ofdamagecaused byroad-surface
defects
Reduction ofdamagecaused bynighttime
visibility

Intermediate outcome indicators

Outputs

InputsNumber ofaccidents caused
by fallen objects

Number ofnear-accident occasions
caused by fallen objects

Finding of
fallenobjects

Road surface
cleaning(man power)

Number ofaccidents causedby bouncing stones

Number ofnear-accident occasions
caused by bouncing
stonesRemoval of
sediment
from roadsurface

Road surface
cleaning(machines)

Number ofaccidents caused
by water on roadsurface

Number ofnear-accident occasions
caused by water onroad surface

Removal of
sediment
fromdrainagebasin

Number ofaccidents causedby frozen roadsurface

Number ofnear-accident occasions
caused by frozen roadsurface

Number ofaccidents causedby potholes andrutting

Number ofnear-accident occasions
caused by potholes andrutting

Number ofaccidents caused
by objects fallenfrom slab bottom

Number ofnear-accident occasions
caused by objects fallenfrom slab bottom

Number ofaccidents causedby poor nighttime
visibility

Number ofnear-accident occasions
caused by poornighttime visibility

Eliminationof frozen
road surface

Finding of
potholesand rutting
Finding of
slab-bottom
damageFinding andrepair ofunlit lights

Drainagecleaning
Measuresagainst snow
and ice(distributionofsnow-melting
agent)
Routineon-roadinspection
Routineslab-bottom
inspection
Inspection
and repair oflighting
systems

Figure 1. Tree diagram (partial) of Hanshin Expressway Logic Model (HELM)
2.3 PDCA cycle in maintenanceThrough the PDCA cycle, the maintenance work

Figure 2 shows the flow of the PDCA cycle, which

can be improved continuously, and accountability toincludesthe road users and the public can be strengthened.

HELM

construction,

calculationofindicators, and the review of routine maintenance
operations.

The

PDCA

cycle

of

Hanshin

Expressway Company Ltd. starts from constructing
an HELM. During this stage, it is important to set
priorities to intermediate outcome indicators andselect inputs that would maximize the performance,with the aim of achieving final outcomes effectively.
In the PDCA cycle, the achievement and change of
output and outcome indicators are evaluated byregularly

measuringtheintermediate

HELM

indicators on a yearly or monthly basis. If any
indicator

departs

fromthecorresponding

benchmark value, the causal relationship between
the outcome, output, and input indicators should be

Figure 2. PDCA cycle for maintenancereevaluated and, if necessary, the relationshipshould be revised and inputs selected again.
4/11

3.

LOGIC MODEL DEVELOPMENT

each route, and the sum of all the risks is regarded
as the risk of the entire route network, which is

3.1 Consideration of risk in maintenancegiven by

The risk in this study is defined as the product ofthe probability of damage and the magnitude ofR=∑ (P × C )
i

i

(1)

i

damage. The risk in maintenance includes accidents,
major repairs, customer complaints, as well as

Here, R is the risk of a certain management item, Pi

management flaws caused by neglecting inspection,is the probability of impediment occurrence for the

repair, cleaning, and other maintenance work.

management item in route section i (i=1, 2,.., n),Risk levels should be optimized in maintenance risk

and Ci is the effect of the impediment occurrence inmanagement. Figure 3 shows schematically theroute section i.

distribution of the risks of management items,
before optimization, based on the probability and3.2 Method of management level setting

magnitude of damage. The acceptable zone in the

A clear basis is required for setting maintenancefigure indicates the risk management levels thatlevels. Let us concentrate on the final outcome

Hanshin Expressway considers desirable. When a"Ensuring safety of vehicles on road" shown incurrent risk is higher than the acceptable range of

Figure 1, which is part of the HELM. To achieverisk management (i.e., within the risk reduction

the final outcome, the intermediate outcomes sucharea), the maintenance level should be raised toas the occurrence of accidents and management

reduce the risk. When the risk is sufficiently lower

flaws should be reduced or eliminated. It isthan the acceptable range, the maintenance level

important to set management levels by using thecan be lowered to reduce the maintenance cost. As alogic model in relation to the required reduction in

result, we can reduce the risk of the entire

the occurrence of impediments (outputs).

management items while reducing the cost.

For each management item, the relationship

Intermediate
Low
Very low

Probability of damage

Highbetween the occurrence of impediments (outputs)

Acceptable
zone

Zone
requiringrisk
reductionand the occurrence of accidents and management
flaws (intermediate outcomes) should be surveyed
and analyzed for each route, or for each routesection more in detail. It is not always possible toobtain deterministic analytical results, such as"accidents or management flaws occur when thenumber of impediments reaches a certain level."

Zone
requiring
cost
reduction

Routes on which no management flaws occurred ina given year can be identified by using accumulated
statistic data, and the average risk of such routes

Small

Large

SevereFatal

Magnitude of damage
Figure 3. Concept of risk optimization

can be defined as a management level. In this way,
it is possible to conduct management work based on
the target "causing no management flaws in the
future." The risk management level should be

Because the volume of traffic (magnitude of

determined in consideration of the risk occurrence

damage) depends on the route, risk is calculated for
5/11

properties of the management item. The procedure

to delineate an exact risk level Rlevel below whichof setting a risk management level is summarized inmanagement flaws will not occur. Therefore, a zone

Figure 4.

of risk management is defined by setting margins
on both safe and unsafe sides of a certain risk level.
The routes that fall within the area A of Figure 5 areconsidered to be overly managed, and their
management levels may be lowered in view of cost
reduction. On the other hand, the routes that fallwithin the area B are overly risky and their
management levels should be raised.
There naturally is a trade-off between risk and cost.
To reduce risk and cost simultaneously, inputs
should be reviewed for each route.

Figure 4. Flow of management level setting
3.3 Risk optimization method
There

are

two

trade-off

objectives

in

risk

management: (1) controlling the risk within a

Figure 5. Conceptual diagram of risk curve

desirable range, and (2) reducing the cost of
management. These objectives should be pursued
4.

under the following conditions.

Rlevel − Ri ≤ Rm arg in

(2)

∑

(3)

Cost i′ ≤i∑

RISK OPTIMIZATION IN DIFFERENTMANAGEMENT MEASURES(FEASIBILITY STUDY)

4.1 Routine on-road inspection and normalized

Cost ioccurrence of potholesiAmong the policy evaluation models that constitute∑ R′ ≤ ∑ R
i

i

i

(4)

i

the HELM (see Figure 1), we focus on a model
related to potholes (pavement damage requiring

Here, Rlevel is the designated risk management level,urgent repair) found through routine inspection.

Ri is risk for route i, Rmargin is the margin of theFirst, risk management levels for potholes are set bymanagement level, Costi is maintenance cost for

using the logic model.

route i, Costi' is maintenance cost after risk

In the HELM, the frequency of routine on-road

optimization for route i, and Ri' is optimized risk for

inspection is adopted as an input for controlling theroute i.

risk associated with the occurrence of potholes. The

When determining a risk management level fromfrequency of inspection varies with the route andthe occurrence of management flaws, it is difficult

section, and the frequency in fiscal 2005 was
6/11

generally two or three times a week. Figure 6 shows

volume represents the magnitude of effect, the risk

the number of pothole occurrences found by routineof normalized occurrence of potholes is given by

on-road inspection in fiscal 2002, 2004, and 2005.
The figure indicates that the risk of potholeoccurrence varies widely with the route and section.
The normalized occurrence of potholes, obtained by
dividing the number of pothole occurrences by totalroute length (in km) and inspection frequency, isadopted as an output indicator. Figures 7 and 8
show the data of normalized occurrence of potholes
and the flow of setting management levels,
respectively.

Figure 8. Flow of management level setting (in thecase of potholes)

R pi = Ppi × C i

Figure 6. Number of pothole occurrences(5)Here, Rpi is the risk of normalized occurrence of
potholes for route i, Ppi is the normalized
occurrence of potholes for route i, and Ci is theaverage spot traffic volume in 24 hours. The
normalized occurrence of potholes is given by
dividing the number of pothole occurrences foundthrough routine on-road inspection in a year by totalroute length and inspection frequency:

Pi = xi f i li

(6)Here, xi is the number of pothole occurrences in ayear for route i, fi is the frequency of routine
Figure 7. Normalized occurrence of potholes

on-road inspection in a year, and li is total route
length.

Because traffic volume varies with the route and

The combinations of normalized occurrence ofsection, the effect of potholes (accidents andpotholes and spot traffic volume that provide amanagement flaws) also varies. Considering that

constant risk of the normalized occurrence ofthe normalized occurrence of potholes represents

potholes are expressed as a hyperbola (see Figure 5).

the probability of occurrence and that traffic

The area on the upper-right side of the hyperbola is
7/11

higher in the risk of normalized occurrence of

0.03

potholes than the management level given by the0.025

hyperbola. Once a management level corresponding

Before reorganization
After reorganization
Urban trunk line
C-1

Urban trunk line
C-2

0.02

to a certain risk level is set for the occurrence ofIncrease inspection frequency
0.015

Urban trunk line
A-1

potholes, the entire risk can be leveled out by
strengthening inspection in appropriate routes and

0.01

Reduce inspection
frequency

0.005

sections.
Risk optimization was attempted with regard to

0
0

20000

pothole management flaws. The final objective wasgiven as "causing no management flaws in the

40000

60000

80000

100000120000140000Spot traffic volume (vehicles per day)Figure 9. Risk-based reorganization of management

future," and the management level (intermediateoutcome) of Hanshin Expressway was set as "theaverage risk of routes that have been maintainedwith no management flaws (32 routes in fiscal 2002,
2004, and 2005)." The management level for the

The calculation results in Figure 10 show how therisk of normalized occurrence of potholes andinspection cost in the entire route network arechanged by using the above approach of risk
optimization. In the present case, both the overall

number of pothole occurrences is given byrisk and cost were reduced by reorganizing

R p ,level =∑Rinspection and smoothing out the risk of different

p 0i

i

n

(7)

routes.

Here, Rp0i is the risk of normalized occurrence ofPlanedstate

Planed stateincreased to reduce the risk toward the target level.

Presentstate

In routes with high risk, inspection frequency is

Planed state

flaws.

Planedstate

Present
state

potholes in each route that had no management

In routes with low risk, inspection frequency is
reduced to cut maintenance cost. It is assumed thatthe number of pothole occurrences in a year isunchanged before and after the optimization, andrisk is estimated by changing only inspection

Risk(normalized occurrence of
potholes spot traffic volume
total road length)

Total cost(entire route
network)

Figure 10. Estimated reductions in risk and cost byreorganization of routine on-road inspection

frequency. As a result, as shown in Figure 9, thenormalized occurrence of potholes in each route fellwithin the range of risk management.

4.2 Routineslab-bottom

normalized

occurrence

inspectionofandslab-bottom

damageWe now focus on the optimization of the risk
related to slab-bottom damage found by routineslab-bottom

inspection.

Hanshin

Expressway

conducts slab-bottom inspection routinely with theaim of avoiding damage to third parties.
The procedure of risk optimization is the same as
8/11

that for the risk of normalized occurrence of1

potholes. The frequency of routine slab-bottom

Number ofmanagement flaw
occurrences2

inspection depends on the route and section, and theactual annual frequencies in fiscal 2005 were six

1

inspections on land and one inspection above the122sea. The output indicator is the normalized
occurrence of slab-bottom damage, which is
obtained by dividing the number of damage
occurrences (requiring urgent repair) found throughroutineslab-bottom

inspectionbyinspection

Urban
Radial Radial Radial Radial Radial
trunkline A route A route B route C route D route E

Figure 12.
occurrences

Numberofslab-bottom

damagefrequency and the total length (in km) of elevatedsections.
An analysis of slab-bottom damage revealed thatthe number of damage occurrences increased with

The risk of slab-bottom damage is defined as

Rui = Pui × C ui

(8)time elapsed after the completion of upper andHere, Rui is the risk of slab-bottom damage, Pui islower structures. Damage occurred most frequently

the normalized occurrence of slab-bottom damage,in the upper and lower structures completed in theand Cui is the total length of elevated sections. The

late 1960s (see Figure 11).

normalized occurrence of slab-bottom damage is
obtained by dividing the number of damage
occurrences (requiring urgent repair) found throughroutine slab-bottom inspection in a year byinspection frequency and the total length of sections
that may affect third parties. This indicator isexpressed as

Pui = xui f ui lui
Figure 11. Number of slab-bottom damage
occurrences on road sections completed in different
years
Because one route may consist of sections
constructed in different years, the analysis was doneon a section basis. Figure 12 shows the number ofslab-bottom damage occurrences and the number of(9)Here, xui is the number of slab-bottom damage
occurrences in a year, fui is the frequency of routineslab-bottom inspection in a year, and lui is the total
length of sections that may affect third parties. The
management level for the normalized occurrence ofslab-bottom damage is given bymanagement flaws in each section.

Ru ,level =

∑R
i

n

u 0i

(10)

Here, Ru0i is the risk of normalized occurrence ofslab-bottom damage in each route that had no
management flaws.
The relationship between inspection frequency andthe risk of slab-bottom damage was analyzed to see
9/11

if the risk is within the range of management. In the

input indicators that constitute the HELM. Some ofpresent case, as shown in Figure 14, both the risk ofthe management items are insufficient in dataslab-bottom damage in the entire route network andaccumulation, and the logic model developed in thisoverall cost were reduced by risk optimization.

study should be considered as a prototype. It isnecessary, in the future, to monitor these indicators
continuously and evaluate their causal relationship.

5.

SUMMARY

A logic model was constructed for the maintenancework of Hanshin Expressway Company Ltd., and amethodology to set risk maintenance levels (output
indicator)
Figure 13. Risk of slab-bottom damage in different
road sections

for

risk

optimizationinroutinemaintenance was proposed. The study findings are
summarized below.
− To manage the maintenance of expressways
efficiently,

maintenance

operations

wereorganized systematically and a logic model
specific to Hanshin Expressway (HELM) wasconstructed.
− A methodology of the PDCA cycle wasdeveloped to carry out maintenance work
efficiently and effectively by quantitatively
measuring the outcome and output indicators of
Figure 14. Estimated reductions in risk and cost byreorganization of routine slab-bottom inspection

the HELM.
− A method to set risk management levels basedon outputs such as the finding of potholes and

4.3 PDCA cycle and continuous improvement of
HELM

slab-bottom damage was presented. Instead ofdelineating an exact risk level, margins were set

Among management items that constitute theto define a target range of management in

HELM, we focused on the output and outcome

consideration of various uncertainties.

indicators related to the risk of pothole occurrence− To

controltherisk

towardthetargetand the risk of slab-bottom damage. We thenmanagement level, the frequency of inputs

discussed methods to determine the corresponding(routine

management levels. Other than these two items,

inspection) was reviewed and reorganized. As athere are many management items in the HELM. At

result, it was shown that both risk and cost could

the present stage, we have developed a primary

be reduced.

on-roadandroutineslab-bottom

policy evaluation model pertaining to the causal− As an example of setting management levels for

relationship between many outcome, output, androutine inspection, we focused on management
10/11

flaws related to potholes and showed anProceedings of the Japan Society of Civil

approach of setting a risk management target.

Engineers,− We presented an objective of suppressing, as

No.

805/VI-69,

pp.

105-116,

December 2005.

much as possible, management flaws that may[4] Sakai, Y., Nishibayashi, M., Arakawa, T.,

affect third parties, with regard to setting risk

Kojima, D., and Kobayashi, K.: Study on

management levels for routine slab-bottom

HELM logic model intended for effective

inspection.

maintenance of expressways (in Japanese),

As discussed above, risk and management levels

62nd Annual Conference of Japan Society ofshould be evaluated in the PDCA cycle byCivil Engineers (submitted), 2007.

constructing an HELM, reorganizing maintenance
operations,

and

the

Feasibility study on risk optimization ofoutcomes. The PDCA cycle allows continuous

expressway maintenance based on HELM logic

improvement

model (in Japanese), 27th Japan Road Congressofcontinuously
maintenancemeasuring[5] Sakai, Y., Uetsuka, H., and Kobayashi, K.:

efficiency.

The

present study has developed an HELM as aprototype model for the causal relationship between
intermediate outcomes, output indicators, and input(submitted), 2007.
[6] Osumi, S.: New public management (inJapanese), Nippon Hyoronsha, 1999.

indicators. We need to continue measuring various[7] Agriculture, Forestry and Fisheries Promotion

indicators to improve the logic model and theFoundation; and Agriculture, Forestry andmeasuring methods. Because it is expected that

Fisheries Policy Research Institute: Guide for

performance-specified maintenance services will be

logic model development (in Japanese), August

ordered more frequently in the future, the issues of2003.

setting, measuring, and monitoring adequate risk
management levels will become all the more
important.

[8] W.K. Kellogg Foundation: W.K. Kellogg
Foundation Evaluation Handbook, 1998.
[9] Australia NSW Government Asset Management
Committee: Total Asset Management Manual,
1992.

REFERENCES[1] Hanshin Expressway Company Ltd.: Standard
specifications on civil works IV - Relevant
standards - Standard specifications on civil
works maintenance (in Japanese), September
2002.
[2] Kaito, K., Kobayashi, K., Kato, T., and Ikuta,
N.: Frequency of patrol on roads and risk ofimpedimentoccurrence(inJapanese),

Proceedings of the Japan Society of Civil
Engineers (F), Vol. 63, No. 1, pp. 16-34, 2007.
[3] Aoki, K., Yamamoto K., and Kobayashi, K.:

Optimum policy on inspection and replacementof tunnel lighting systems (in Japanese),
11/11