Optimization of Maintenance Strategies for the Management of the National Bridge Stock in France

Optimization of Maintenance Strategies for the Managementof the National Bridge Stock in France

André Dominique Orcesi1 and Christian Francis Cremona2

Abstract: Maintenance and repair activities are crucial to prevent the deterioration of highway bridges and ensure users’ safety. In 2006,the French Ministry of Transportation initiated a survey to improve the maintenance strategies of its bridge stock. This study was justifiedby the introduction of new financial regulations in France. Each Ministry is now enforced to assess its performance through indicators andtargets. On behalf of the Ministry of Transportation, the Roads Directorate needs to determine the preservation policy for roads andbridges based on annual quality indicators such that target values are reached for the horizon 2020. During the last decades, inspection andmaintenance strategies applied on the most deteriorated engineering structures enabled to maintain an acceptable level of service andsafety. However, these management strategies can obviously be largely improved. Facing these stakes, the methodology presented in thispaper is elaborated to optimize the allocation of maintenance funds. The objective is to determine the global financial resources neededto satisfy the constraints on the quality indicators over the next 15 years. The proposed model is based on Markov chains fitted tocondition data collection. By including prediction models in the cost analysis, different maintenance strategies are evaluated. Applied tothe French national bridge stock, the procedure shows that a slight increase of the annual budget is required to fulfill the asset performancerequirements fixed by the Roads Directorate.

DOI: 10.1061/�ASCE�BE.1943-5592.0000125

CE Database subject headings: Bridge maintenance; Preservation; Markov process; France.

Author keywords: Bridge preservation; Maintenance policies; Bridge management; Markov models; Maintenance costs.

Introduction

In France, most bridges of the national inventory were built dur-ing the sixties and seventies �Fig. 1�. Many of them do not pro-vide an adequate level of service anymore and need essentialmaintenance actions �Orcesi and Cremona 2010�. In 2006, theFrench Ministry of Transportation initiated a study to improve themaintenance policy of its bridge stock. This initiative was justi-fied by the introduction of a new Law of Finance in France. Everyyear, each Ministry has to provide a set of performance objectivesfor the next year and an annual performance report for the pastyear. This enables to compare every year the performance resultswith the target objectives. Within this framework, the Roads Di-rectorate has to evaluate the preservation strategies for roads andbridges based on target values to be reached at year 2020.

The Roads Directorate applies several guidelines for inspec-tion, maintenance, and rehabilitation of highway structures �IT-SEOA 1979�. Until 1976, the aging of the national highway

1Res. Engr., Section Durabilité des Ouvrages d’Art, Division Fonc-tionnement et Durabilité des Ouvrages d’Art, Laboratoire Central desPonts et Chaussées, 58 boulevard Lefebvre, F-75732 Paris Cedex 15,France. E-mail: [email protected]

2Res. Engr., Directorate for Research and Technology, Ministryfor Sustainable Development, Tour Voltaire, F-92055 La Défense Cedex,France �corresponding author�. E-mail: [email protected]

Note. This manuscript was submitted on March 22, 2009; approvedon March 23, 2010; published online on March 25, 2010. Discussionperiod open until June 1, 2011; separate discussions must be submittedfor individual papers. This paper is part of the Journal of Bridge Engi-neering, Vol. 16, No. 1, January 1, 2011. ©ASCE, ISSN 1084-0702/2011/
1-44–52/$25.00.
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bridges was not a major issue and the priority was to build newinfrastructures. The budget for bridge maintenance was low andincluded in that for roads maintenance. Between 1976 and 1978,it was multiplied by three after the collapse of the Wilson archBridge in Tours, France �caused by scour development�. Thereaf-ter, this budget remained stable during several years, thenstrongly increased between 1996 and 2006, thanks to the intro-duction of a new scoring system that enables to better specifyfinancial needs. Indeed, the Roads Directorate initiated in 1994 anew scoring system, named IQOA �quality assessment of engi-neering structures� to assess the structural condition of the na-tional bridge stock. This scoring system was developed to providea global assessment of the national bridge stock by assessingbridges every 3 years �i.e., by applying each year IQOA inspec-tions on a third part of the asset�. It is noted that this 3 yearsinspection process is part of a more general inspection frameworkthat also includes annual routine inspections and the 6 years de-tailed inspection program. During the IQOA inspection process,the structures are rated according to five IQOA levels �Table 1�.The Roads Directorate draws up priorities, giving first priority tostructures rated 3U, then 3, etc. Nevertheless, maintenance ac-tions on structures scored 2E may be prioritized before structuresscored 3 to avoid a rapid evolution of the structural degradationand expensive rehabilitation costs. For instance, repainting, re-placing waterproofing systems or protecting against externalchemical aggression �e.g., deicing salts, chlorides� may be valu-able maintenance activities performed on 2E bridges. The RoadsDirectorate is conscious that this preventive maintenance strategycan be largely improved.

This paper presents the analysis made by the writers to evalu-ate different maintenance scenarios for the Roads Directorate in
France. The objective is not to define the maintenance activities
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for each bridge, but to provide the Ministry of Transportation witha general approach combining the results of the IQOA scoringprogram with the financial resources for the management of na-tional bridges. Several prospective scenarios are defined, based onthe minimization of the annual maintenance cost. Constraints areintroduced for each prospective scenario and the quality indica-tors, as introduced in the Law of Finance, are evaluated.

Overview of the IQOA Scoring System

Consistency and Condition of the Asset

The study is carried out on the national bridge inventory �around9,000 bridges�. During an IQOA inspection, several componentsare inspected: equipments �pavements, footways, cornices, retain-ing devices, expansion joints, etc.�, protecting components �wa-terproofing layers, anticorrosion coating, etc.� and structuralcomponents �deck, supports, bearings, foundations, etc.�. Byusing catalogs of defects, the inspectors are able to provide ascore for each component and structural part. The final score isthen the worst score of all components. The IQOA score is notconsidered in France as a condition index in the sense that all thedefects have the same weight in the final IQOA value for a bridge.For a specific bridge, the IQOA score covers various problemsfrom structural ones to nonstructural ones. The main objective ofthe IQOA program is to provide a snapshot of individual bridges’condition, and then, a snapshot of the overall bridge stock quality�by aggregating all IQOA scores�. The difference of defects be-tween IQOA scores 2 and 2E and IQOA scores 3 and 3U issubstantial. Scores 2 and 2E represent serviceability defects whilethe two others represent structural deficiencies. Fig. 2 details thesurface area distribution of the IQOA scores for the national

Table 1. IQOA Scores

Score Apparent condition

1 Good overall state

2 Equipment failures or minor structural damage.

Nonurgent maintenance required.

2E Equipment failures or minor structural damage.

Urgent maintenance required.

3 Structural deterioration.

Nonurgent maintenance required.

3U Serious structural deterioration.

Urgent maintenance required.

0% 10% 20% 30% 40% 50% 60%

<1850

1850-1900

1901-1950

1951-1975

1976-1995

>1995

Fig. 1. Breakdown of the national bridge stock into constructionperiods

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bridge inventory �nonscored bridges represent bridges under re-habilitation, new bridges…�. The target values of the quality in-dicators to be reached within 15 years �expressed as a percentageof the entire bridge stock deck area� are provided in the thirdcolumn of Table 2.

Maintenance and Rehabilitation Costs

Unit costs are issued from a previous analysis �Odent and Ber-thellemy 2002� carried out on a representative sample of dis-tressed bridges and extrapolated to the overall asset. The costsrepresent, for each IQOA score, the average financial needs forrestoring 1 m2 of bridge deck back to score 1 �Table 3�. Even ifit is not relevant to restore the whole asset in 1 year, the totalrehabilitation cost of the bridge stock will be used as asset valu-ation. This total rehabilitation cost indicates the efficiency of thepreservation strategy: it represents the average cost to move thefull asset from its actual condition to an “as new” condition �score1�. A reduction of this cost means that the quality of the bridgestock improves. Conversely, an increase means that the value ofthe asset is degrading. In 2006, the rehabilitation cost was as-sessed at €635 million whereas the annual budget of maintenance

Table 2. Target Values of the Quality Indicators �as Prescribed by theRoads Directorate in France�

Index Objective

Target valuewithin 15 years

�%�

I3=1+2 Routine maintenance to prevent repairs �55

I4=2E Specialized maintenance to prevent repairs �30

I5=3+3U Structural maintenance to prevent collapse �15

I6=3U Urgent structural operation to preventdisruption and to ensure safety of theroad network

�1

Table 3. Unit Repair Costs

ConditionCost per m2 �value 2006�

�€�

2 85

2E 249

3 335

3U 433b

DRa 2,265aDR=decommissioning-rebuilding.b

0%5%10%15%20%25%30%35%40%45%

1 2 2E 3 3U Notscored

Fig. 2. Breakdown of the national bridge stock into IQOA scores

For repairable bridges scored 3U.

BRIDGE ENGINEERING © ASCE / JANUARY/FEBRUARY 2011 / 45


was €45 million. One major objective for the Ministry of Trans-port is that the rehabilitation cost remains at least stable withtime.

Use of Markov Chains

Markov Process and Homogeneity Property

The objective is herein to assess the probability for a bridge tomove from a IQOA score to another one over a 1-year period. TheMarkovian approach is a classical approach to evaluate such tran-sition probabilities �Scherer and Glagola 1994; Corotis et al.2005�. The property of a Markov process is that the “state” �theIQOA score herein� at the inspection time i depends only on itsprevious “state” at the inspection time i−1. With this assumption,only the current state �score� is considered to determine the futureof the bridge �Bremaud 1999; Orcesi and Cremona 2006, 2009�.The access to the IQOA database enables to perform an analysisof the state transition sequences �STS� �Scherer and Glagola1994�. For a given three-state transition sequence �related to threeconsecutive condition states—past, present and future—for asame bridge�, one determines the number of times that this se-quence is identified in the IQOA database �state sequence occur-rence �SSO��. Then, similarly, one identifies the number of timesthat the sequence present/future occurs �two sequences occur-rence �TSO��. Considering a particular STS �i ,m , j�, the probabil-ity to move to score j knowing the scores i and m, is given byP�i ,m , j�=SSO /TSO. The validation of the Markovian hypoth-esis requires to check that this probability is the same for allpossible scores i �with assumption that there is no noisy data�.This means that the probability P�i ,m , j� is depending only on theprevious condition m. Table 4 presents some examples of STS.This suggests a relative independence from past states even ifsequences �1,2,2E� and �2,2,2E� are not exactly the same. It isnoted that some transition sequences cannot be analyzed sincethey do not occur in the IQOA database ��1,3,3U� or �2E,3,3U�for instance�.

Determination of the Transition Matrix

Considering the IQOA database with scores between years a0 andaf, the probability P�q1 ,q2� for 1 m2 of bridge deck area, to movefrom score q1 to score q2, is defined as the total deck area rated q1

at year i and q2 at year i+1 divided by the total deck area rated q1

Table 4. Example of State Transition Sequences

Sequence STS P�i ,m , j�

�1,2,2� 0.99

�2,2,2� 0.92

�1,2,2E� 0.010

�2,2,2E� 0.070

�1,2E,2E� 1.00

�2,2E,2E� 0.99

�2E,2E,2E� 0.97

�1,3,3U� 0.00

�2E,3,3U� 0.00

at year i, for i between a0 and af. This probability is expressed as



P�q1,q2� =

�i=a0

af−1 � �k=1

nq1,i→q2,i+1

Aq1,i→q2,i+1k �

�i=a0

af−1 ��k=1

nq1,i

Aq1,ik � �1�

where nq1,i=number of bridge decks rated q1 at year i;nq1,i→q2,i+1=number of bridge decks moving from score q1 toscore q2 between year i and year i+1; Aq1,i

k =area of bridge deck kscored q1 at year i; and Aq1,i→q2,i+1

k =area of bridge deck k movingfrom score q1 to score q2 between year i and year i+1. Such adirect method, to determine the transition matrix P, requires atleast two consecutive IQOA score records for a large number ofbridges at different condition levels �Lounis 2003�. Since IQOAinspections are performed every 3 years, the score of the lastinspection is always considered for years with no inspection. Thisis the case of the IQOA database. The obtained transition matrixfor a0=1996 and af =2005 is

P96-05 =�0.829 0.147 0.022 0.002 0.00

0.017 0.916 0.058 0.009 0.00

0.007 0.075 0.894 0.018 0.005

0.003 0.037 0.080 0.872 0.008

0.014 0.028 0.112 0.085 0.761� �2�

where the diagonal elements P96-05�k ,k� represent the probabilityfor 1 m2 of bridge deck area to remain in score k, and othermatrix elements P96-05�k , l� represent the probability for 1 m2 ofdeck area to move from score k to score l in 1 year �the new scorebeing worse if k� l and better on the contrary�. This matrix isused in the optimization process and assumed to model the cur-rent maintenance strategy. For example, the probability that 1 m2

of deck area scored 1 remains in 1 from one inspection to anotherone is 82.9% �see Eq. �2��. As mentioned previously, the IQOAscore is an overall score. This result may lead to a sort of over-scoring process, inspectors quoting more easily with score 2 abridge deck after several inspections for various reasons �for in-stance due to a lack of rainwater evacuation�. In turn, it is noticedthat the transition probability P96-05�2,2� remains very high �seeEq. �2��.

If the deck area fraction of the bridge stock related to eachscore is known at year i and collected in the vector

qi = �q1i q2

i q2Ei q3

i q3Ui � �3�

its evolution for year i+1 is given by the equation

qi+1 = qiP96-05 �4�

The breakdown of the deck area into IQOA scores can thereforebe determined if the transition matrix and the initial deck areafraction �i.e., qi�k�, k=1, . . . ,5, also denoted qk

i , k� �1,2 ,2E ,3 ,3U� are known.

Justification of the Homogeneous MarkovianAssumption

The hypothesis of the homogeneous nature of the Markov chainsimplies to check if transition probabilities are time invariant. Thisassumption can be analyzed by different estimations of the tran-sition matrix for different inspection periods. To perform thisstudy, transition matrices are calculated on the basis of a 6 years
inspection period �instead of the nine years used to determine the
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matrix P96-05 in Eq. �2��. As the IQOA inspection is performedevery three years, a 6-year period corresponds to two IQOA in-spections. Four periods are therefore considered: 1996–2002,1997–2003, 1998–2004, and 1999–2005. The results for the fullperiod 1996–2005 are compared with the results of these fourcases. They are given for the diagonal terms and the nondiagonalterms of the transition matrix P in Figs. 3�a and b� �terms areordered from the first to the last row of the transition matrix andby ordering terms of each row from the left to the right�. Figs. 3�aand b� show the influence of the calibration period on the differ-ent elementary probabilities of the transition matrix. These valuesdiffer according to the calibration period and it is rather difficultto argue for the homogeneity assumption of the Markovian pro-cess. It is considered that the discrepancy remains low enough toadopt the hypothesis in the rest of the paper. However, it is rec-ommended to update the transition matrix each year in order toget a more adapted transition matrix to the bridge stock evolution.

Optimization of the Maintenance Strategies

Prospective Scenarios

Three prospective scenarios are defined. They rely on the hypoth-esis that the number of bridges remains constant. This assumptionis obviously wrong since new bridges on new roads are always

(a)

1 7 13 19 250.7

0.75

0.8

0.85

0.9

0.95

1

Matrix components

Probability(1=100%)

1996-20021997-20031998-20041999-2005Full period

(b)

0 2 4 6 8 10 12 14 16 18 20 22 240

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Matrix components

Probability(1=100%)

1996-20021997-20031998-20041999-2005Full period

Fig. 3. Influence of the calibration period on the �a� diagonal terms;�b� nondiagonal terms of the transition matrix

built each year. Nevertheless, in absence of information regarding

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the asset increase, this hypothesis is considered. These scenariosgive priority either to preventive or corrective actions, or both,with the purpose to control the budget and to ensure the preser-vation of the asset. The first Scenario 1 is the “continuation of thecurrent policy.” The objective is to assess if the quality indicatorsand the rehabilitation costs fulfill the targeted values with thecurrent policy. Scenario 2 consists in fixing “constraints on thequality indicators and on the rehabilitation costs.” Annual budgetsare adjusted by giving priority to the preventive actions �score2E� and curative actions �score 3U� without increasing the totalrehabilitation cost. At last, Scenario 3 applies “constraints on thequality indicators and on the annual maintenance budget.” Theobjective of Scenario 3 is to determine optimal maintenance strat-egies associated with a constant annual budget.

Maintenance Strategies

Each maintenance strategy is associated with a transition matrix.For example, the current maintenance strategy, noted S1, is asso-ciated with the matrix

S1 = P96-05 �5a�

where P96-05 is defined in Eq. �2�. If the objective of anotherstrategy is systematically to upgrade scores i to j, the term �i , j� ofthe transition matrix P96-05 is fixed at 1 and other terms in the ithrow of the corresponding matrix are set to 0. The current main-tenance strategy that enhances maintenance actions on bridgesscored 2E and 3U is noted S2 and the associated transition matrixis

S2 =�0.829 0.147 0.022 0.002 0.00

0.017 0.916 0.058 0.009 0.00

0 1 0 0 0

0.003 0.037 0.080 0.872 0.008

0 1 0 0 0� �5b�

Cost Matrix

The values provided in Table 3 enable to perform a realistic eco-nomical study. The maintenance cost C�i , j� represents the cost tomove 1 m2 of bridge deck from score i to score j. Based on aprevious study of the Roads Directorate �Odent and Berthellemy2002�, Table 3 provides the average repair costs �i.e., the averagecosts to move a bridge deck from score j to score 1�. These valuesare therefore those the first column of the cost matrix C intro-duced in Eq. �6�. As mentioned previously, C�i , j� represents thecost to move 1 m2 of bridge deck from score i to score j. It isnoted that values in columns j�1 are unknown and, conse-quently, assessed by applying weighting factors on the terms ofthe first column �to assess an average cost�. Obviously, theweighting coefficients used herein �see Table 5� need further stud-

Table 5. Maintenance Cost per m2

Initial scoreMoving in 1

�cost /m2�Moving in 2

�cost /m2�Moving in 2E

�cost /m2�Moving in 3

�cost /m2�

2 85 — — —

2E 249 249�0.9

3 335 335�0.9 335�0.8 —

3U 2,265 �DR� 433 433�0.8 433�0.7

ies to be quantified in a more accurate way. They are chosen on



the basis of qualitative realistic engineering judgment by theRoads Directorate �Odent et al. 2008�. The proposed values en-able to grade the costs of the different types of maintenance ac-tions as follows:

C =�0 0 0 0 0

85 0 0 0 0

249 249 � 0.9 0 0 0

335 335 � 0.9 335 � 0.8 0 0

2265 433 443 � 0.8 443 � 0.7 0� �6�

Each strategy Sj �a strategy Sj is entirely defined by the associatedtransition matrix S j, as previously mentioned� is associated with acost vector CSj

= t�cSj,1cSj,2

cSj,2E cSj,3cSj,3U� where the kth ele-

ment of CSj�k� �1,2 ,2E ,3 ,3U� is

CSj�k� = ��1,k �2,k �2E,k �3,k �3U,k �Sj,

��1,k �2,k �2E,k �3,k �3U,k �C� �7�

and where S j = jth strategy matrix �j=1 or 2 herein�; . , .�=scalar product notation; and �l,k=Kronecker function. The glo-bal rehabilitation cost of the bridge stock CRe h �i.e., the necessarybudget to move of the scores of all bridges to score 1� is ex-pressed every year i as

CRe h�i� = AT�k=1

5

C�k,1�qi�k� �8�

where AT=total deck area ��3.91�106 m2 for the 9, 000 bridgesof this study�. In 2006, the percentage of bridge decks in eachIQOA score was q0= �10% 46% 30% 13% 1%�. By using Eq.�8�, the overall rehabilitation cost was estimated to be €635 mil-lion at year 2006.

General Framework

The objective is to determine the optimal annual combination ofthe different strategies to maintain the bridge stock in good con-dition with limited budgets. Each year i=1, . . . ,n−1, where n=number of years considered in the maintenance planning, a vec-tor XSj

i = t�xSj,1i xSj,2

i xSj,2Ei xSj,3

i xSj,3Ui � is associated with the strat-

egy Sj. The term XSji �k�=xSj,k

i represents the proportion of bridgesscored k for which strategy Sj is applied at year i. The vector qi+1

at year i+1 is obtained from that at year i as follows:

qi+1 = qiMi ∀ i = 1, . . . ,n − 1 �9a�

where

Mi = �j=1

m �xSj,1i 0

�

0 xSj,3Ui �S j �9b�

and m=number of strategies �m=2 in this paper�, with constraintthat

�j=1

m

xSj,ki = 1 ∀ i = 1, . . . ,n − 1, ∀ k � �1,2,2E,3,3U �9c�

The fractions xSj,ki are the variables in the optimization process.

They are determined in such a way that the condition of thebridge stock remains above a minimal threshold and that the costs

i
are as low as possible. The constraint on xSj,kensures that, taking


into account the fraction of bridges that are analyzed, the finalmatrix Mi verifies the property

�q=1

5

Mi�p,q� = 1 ∀ i = 1, . . . ,n − 1, ∀ p = 1, . . . ,5 �9d�

Scenario 1: Continuation of the Current Policy

The continuation of the current policy S1 means that only thetransition matrix S1 is considered in Eq. �9b�. There is, conse-quently, no optimization process in this first scenario. The corre-sponding annual maintenance budget is around €47 million at thebeginning of the maintenance planning and decreases during thenext fifteen years �Fig. 4�a��. It is shown that neither the qualityindicators �Fig. 4�b�� nor the rehabilitation costs �Fig. 4�c�� satisfy

(a)

0 2 4 6 8 10 12 1440

42

44

46

48

50

Time (years)

Maintenancecost(×106 €)

(b)

1 3 5 7 9 11 13 150

0.1

0.2

0.3

0.4

0.5

0.6

Time (years)

Percentage(1=100%)

1+22E3+3U3U

29.4% 35.2%

56.1% 54.3%

14.5%

10.5%1.3%1.0%

(c)

1 3 5 7 9 11 13 15640

642

644

646

648

650

Time (years)

RehabilitationCost(×106€)

Fig. 4. Profiles, associated with Scenario 1, of the �a� annual main-tenance cost; �b� quality indicators; and �c� rehabilitation cost

the constraints since: �1� there is less than 55% of bridge surface

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in score 1 or 2, and more than 30% in score 2E after 15 years �thetarget values after fifteen years are given in Table 2 and are pre-scribed by the Roads Directorate in France in their annual perfor-mance plan�, and �2� the rehabilitation cost increases during thenext 15 years whereas it is expected to stay at today’s level. Inorder to satisfy each constraint, the proposed methodology is toslightly modify the current maintenance strategy by combiningmaintenance strategies S1 and S2 �i.e., matrices S1 and S2, respec-tively� in Eq. �9b�. New scenarios are introduced, the first onecontrolling the rehabilitation cost, the second one targeting theannual maintenance cost.

Scenario 2: Constraints on Quality Indicators and onthe Rehabilitation Costs

The annual cost

Ca�i� = AT�j=1

m

�k�K

XSj

i �k�CSj�k�qk

i �10�

is the sum of all the costs from the different strategies for eachyear i, where K= �1,2 ,2E ,3 ,3U; XSj

i �k�=fraction of bridges withscore k concerned by strategy Sj at year i; CSj

�k�=cost of thestrategy Sj for 1 m2 of bridge deck area scored k �see Eq. �7��;and qk

i =percentage of the bridge deck area scored k at year i. Theproblem detailed in Eqs. �11a� and �11b� consists in minimizingCa�i� every year i of the planning. It is reminded that the Ministryof Transport has to provide each year a set of performance objec-tives for the next year and an annual performance report for thepast year. In this context, optimal maintenance strategies are de-termined annually, which justifies the year by year minimizationproposed herein. An optimization over the whole planning periodis possible and was proposed in a preliminary analysis on thestock of reinforced concrete bridges �Orcesi and Cremona 2009�.Constraints are applied on the quality indicators and on the reha-bilitation cost that has to remain at least constant �see Eq. �11c��.It is noted that the profile of the constraint I6 �on the fraction ofdeck areas scored 3U� is linear and decrease by 1% in 15 years.The optimization problem is expressed as

Find XSj

i ∀ j = 1 ¯ m, ∀ i = 1 ¯ n − 1 �11a�

To minimize Ca�i� ∀ i = 1 ¯ n − 1 �11b�

Such that CRe h�i� � CRe h�0�I3�i� � 55%

I4�i� � 30%

I5�i� � 15%

I6�i� � I6�0� −�I6�0� − 1%�

ni� i = 1 ¯ n

�11c�

where I3, I4, I5, and I6=quality indicators described in the firstcolumn of Table 2, and n=number of years in the planning. Be-sides, the annual index

�k,lSx�i� =

AS1Sx

k,l �i� − AS1

k,l�i�

AS1

k,l�i�� 100 ∀ i = 1, . . . ,n − 1 �12�

represents the relative percentage of bridge deck area to movefrom score k to score l at year i compared to the current strategy

k,l k,l
�Scenario 1�, where AS1�i� and AS1Sx
�i�=total deck area that

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moves from score k to score l at year i when only strategy S1 isconsidered �current strategy� and when strategies S1 and Sx arecombined, respectively.

It is noted that a nonlinear constrained approach based onKuhn-Tucker equations, is used herein to minimize a nonlinearobjective function under constraints �Bazaraa et al. 1992�. Theresults of the optimization problem are illustrated by the annualmaintenance cost Ca�i� �Fig. 5�a��, the quality indicators �Fig.5�b��, and the rehabilitation cost �Fig. 5�c��. It is shown that allconstraints on quality indicators are satisfied each year of theplanning �Fig. 5�b�� and that the rehabilitation cost decreases �Fig.5�c��. This result predicts an increase of the bridge stock overallquality. The annual percentage XS2

i �2E� of deck area concerned byS2

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 1440

45

50

55

60

Time (years)


(b)

1 3 5 7 9 11 13 150

0.2

0.4

0.6

0.7

Time (years)

Percentage(1=100%) 1+2

2E3+3U3U

0.9%

56.1% 60.0%

1.3%

14.5%

29.4% 30.0%

10.0%

(c)

1 3 5 7 9 11 13 15600

610

620

630

640

650

Time (years)

Rehabilitationcost(×106 €)

Fig. 5. Profiles, associated with Scenario 2 when CRe h remains atleast stable, of the �a� annual maintenance cost; �b� quality indicators;and �c� rehabilitation cost

the strategy S2 and the annual index �2E,2�i� are provided in Figs.



6 and 7, respectively. It is noted that �3U,3S2 �i�=0, which means that

only the current maintenance strategy �strategy S1� is applied forbridges scored 3U in this scenario.

The influence of the rehabilitation cost constraint on the main-tenance planning is tested for a linear decrease in CRe h by 40, 50,60, 70, and €80 million, respectively, over 15 years �this newconstraint replaces the first one in Eq. �11c��. The percentagesXS2

i �2E� and XS2i �3U� are shown in Figs. 8�a and b�, respectively.

High level of constraint on the rehabilitation cost means a strongproactive policy to increase the overall quality of the bridge stockand requires important financial funds. Indeed, performing moremaintenance actions on 2E and 3U bridges leads to an increase offinancial needs. Such strategies require large budgets. These re-sults show that the model can be easily applied even with highconstraint levels.

Scenario 3: Constraints on the Quality Indicators andon Annual Maintenance Budget

The optimization problem �Eqs. �13a�–�13c�� consists in minimiz-ing each year the difference between the sum of all the costs fromthe different strategies and a fixed annual budget. Constraints areapplied on the quality indicators and the annual maintenance bud-get Ba, the objective being to avoid chaotic annual budgets. Thenew optimization problem is expressed as follows:

Find XSj

i ∀ j = 1 ¯ m, ∀ i = 1 ¯ n − 1 �13a�

To minimize �Ca�i� − Ba�i�� ∀ i = 1, . . . ,n − 1 �13b�

1 2 3 4 5 6 7 8 9 10 11 12 13 140

0.01

0.02

0.03

0.04

Time (years)

Percentage(1=100%)

Fig. 6. Profile of XS2i �2E� associated with Scenario 2 when CRe h

remains at least stable

1 2 3 4 5 6 7 8 9 10 11 12 13 141

1.5

2

2.5

3

3.5

Time (years)

Percentage(%)

Fig. 7. Profile of �2E,2S2 associated with Scenario 2 when CRe h re-

mains at least stable



Such that I3�i� � 55%; I4�i� � 30%

I5�i� � 15%; I6�i� � I6�0� −�I6�0� − 1%�

ni �

i = 1 ¯ n �13c�

Eq. �13b� merits some comments. The objective function is theabsolute difference between the target maintenance budget andthe optimal one. This objective function enables to determine theset of maintenance activities associated with the annual mainte-nance budget �when the difference between the annual cost andthe annual budget is 0� or at least with a maintenance cost as closeas possible to the targeted budget. It could seem appropriate toconsider the annual budget in the constraint set. However, theremight be some convergence problems when annual budget andquality indicators are considered simultaneously as constraints.The formulation proposed in Eqs. �13a� and �13c� enables both totarget the annual budget and always provide a solution to theoptimization problem.

Fig. 9�a� shows the result of the optimization process when theannual budget Ba is fixed at €50 million, €52 million, and €54million, respectively. The first two cases are underestimated bud-gets since the budget needs to be increased during several years tosatisfy constraints on the quality indicators I3, I4, I5, and I6. Aperfectly stable budget is obtained with €54 million, which rep-resents an increase of €7 million each year compared to the cur-rent annual budget. The results of the optimization problem for abudget of €54 million are illustrated by the profiles of the qualityindicators �Fig. 9�b�� and the annual rehabilitation cost �Fig. 9�c��.All constraints on the quality indicators are satisfied at the end ofthe planning and it is observed that the rehabilitation cost de-

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 140

0.02

0.04

0.06

0.08

0.1

Time (years)

Percentage(1=100%)

∆Ba=-40M€

∆Ba=-50M€

∆Ba=-60M€

∆Ba=-70M€

∆Ba=-80M€

(b)

1 2 3 4 5 6 7 8 9 10 11 12 13 140

0.2

0.4

0.6

0.8

1

Time (years)

Percentage(1=100%)

∆Ba=-40M€

∆Ba=-50M€

∆Ba=-60M€

∆Ba=-70M€

∆Ba=-80M€

Fig. 8. Profiles, associated with Scenario 2 when CRe h decreaseslinearly by €40 million, €50 million, €60 million, €70 million, and€80 million over the next 15 years, respectively, of �a� XS2

i �2E�; �b�XS2

i �3U�

creases, even if there is no constraint on this parameter.

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Besides, the annual percentages XS2i �2E� and XS2

i �3U�, shownin Figs. 10�a and b�, respectively, and the annual indices �2E,2

S2 �i�and �3U,3

S2 �i�, shown in Figs. 11�a and b�, respectively, quantify theapplication of the additional strategy, and, consequently, the ad-ditional effort of the stakeholder to satisfy each constraint.

Conclusions

A Markovian-based method for the determination of optimalmaintenance strategies at the bridge stock level is described andtested. The Markovian hypothesis to model the aging of thebridge stock is discussed and justified. Also, the homogeneity of

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 1449

50

51

52

53

54

55

Time (years)


Ba=50M€ B

a=52M€ B

a=54M€

(b)

1 3 5 7 9 11 13 150

0.2

0.4

0.6

0.7

Time (years)

Percentage(1=100%) 1+2

2E3+3U3U

0.3%

56.1%

29.4%

14.5%

1.3%

8.8%

26.4%

64.8%

Ba=54 M€

(c)

1 3 5 7 9 11 13 15560

580

600

620

640

660

Time (years)

Rehabilitationcost(×106 €)

Ba=54 M€

Fig. 9. Profiles, associated with Scenario 3, of the �a� annual main-tenance cost when Ba is fixed at €50 million, €52 million, and €54million; �b� quality indicators when Ba is fixed at €54 million; and �c�rehabilitation cost when Ba is fixed at €54 million

the database allows using homogeneous Markov chains in the

JOURNAL OF


prediction model. Then, the transition matrix of the bridge stockis estimated. This prediction model enables to carry out an eco-nomical study and determine the optimal maintenance strategies.

The originality of this approach is �1� to consider global qual-ity indicators and �2� to apply Markov chains for the overall

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 140

0.02

0.04

0.06

0.08

Time (years)

Percentage(1=100%)

(b)

1 2 3 4 5 6 7 8 9 10 11 12 13 140

0.1

0.2

0.3

0.4

0.5

Time (years)

Percentage(1=100%)

Fig. 10. Profiles, associated with Scenario 3 and an annual budget of€54 million, of �a� XS2

i �2E�; �b� XS2i �3U�

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 140

1

2

3

4

5

6

7

Time (years)

Percentage(%)

(b)

1 2 3 4 5 6 7 8 9 10 11 12 13 140

0.05

0.1

0.15

0.2

Time (years)

Percentage(%)

Fig. 11. Profiles, associated with Scenario 3 and an annual budget of€54 million, of �a� �2E,2

S2 ; �b� �3E,3S2



bridge stock. The use of quality indicators can provide bridgemanagers with a snapshot of the bridge stock quality. By combin-ing the current strategy with a new one, stakeholders can deter-mine which maintenance actions need to be performed in additionto those currently applied. Several scenarios are tested and theoptimal maintenance planning is determined in each case. In ad-dition to the constraints on the quality indicators, the rehabilita-tion cost is also introduced, representing the preservation of theinitial capital. It is noted that the optimization process is per-formed year by year in this paper since the budget of the Ministryof Transport is annually based. The optimization over the wholeplanning period is, consequently, not considered by the RoadsDirectorate as a scenario to test and analyze. Nevertheless, suchanalysis was proposed by Orcesi and Cremona �2009� on theFrench reinforced concrete bridge stock.

Future improvements require taking into account uncertaintiesof the transition matrix elements in a more accurate way. Newinspection results should be included in the database each year toinsure that the transition matrix always includes the up-to-dateinformation of the bridge stock. Moreover, transition matrices canbe detailed for different types of structures �such as reinforcedconcrete bridges, prestressed concrete bridges or compositebridges� and for different geographical areas to include the effectof the environment in the prediction model.

The purpose of the proposed approach is to provide stakehold-ers with a global vision of the condition of their bridge stock andto allocate the maintenance funds in an optimal way. The qualityindicators are used to define a long-term program for maintenanceand repair. Finally, the simulations allow planning the mainte-nance actions such that the bridge stock remains in a good overallcondition. The proposed methodology has been applied on thenational bridge stock in France on behalf of the Roads Director-ate. The conclusions obtained from the different simulations showthat the procedure is effective, meaningful and provides pertinentresults in terms of costs calculations and optimal solutions. Theseresults have been used to prepare the first round table and discussextra financial needs with the Ministry of Finance. They will befollowed by more long run detailed investigations in terms ofbridge management.



Acknowledgments

The writers are grateful to the Roads Directorate and to SETRAfor providing data from the inventory database. They want inparticular to thank Mrs. Nathalie Odent, Mr. Michaël Toriel, andMr. Guy Poirier, certified Engineers, for their invaluable help andcooperation.

References

Bazaraa, M. S., Sherali, H. D., and Shetty, C. M. �1992�. Nonlinearprogramming: Theory and algorithms, Wiley, Hoboken, N.J.

Bremaud, P. �1999�. Markov chains, Gibbs fields Monte Carlo simulationand queues, Springer, New York.

Corotis, R. B., Ellis, J. H., and Jiang, M. �2005�. “Modeling of risk-basedinspection, maintenance and life-cycle cost with partially observableMarkov decision processes.” Struct. Infrastruct. Eng., 1�1�, 75–84.

ITSEOA. �1979�. Technical guidelines for surveillance and inspection ofengineering structures, Ministry of Transportation, Paris �in French�.

Lounis, Z. �2003�. “Identification of environmental categories for Mar-kovian deterioration models of bridge decks.” J. Bridge Eng., 8�6�,353–361.

Odent, N., and Berthellemy, J. �2002�. “Bridges asset ageing processmodel and theoretical simulation of a maintenance budget policy onbridge condition in national heritage.” Bridge Maintenance, Safetyand Management, Proc., IABMAS’02 Conf., CIMNE, Barcelona,Spain, 67–68.

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Orcesi, A. D., and Cremona, C. F. �2006�. “Optimization of the reinforcedconcrete bridges maintenance by using Markov chains.” Advances inBridge Maintenance, Safety Management, and Life-Cycle Perfor-mance, Proc., IABMAS’06 Conf., Taylor & Francis, London, 105–106.

Orcesi, A. D., and Cremona, C. F. �2009�. “Optimization of managementstrategies applied to the national reinforced concrete bridge stock inFrance.” Struct. Infrastruct. Eng., 5�5�, 75–84.

Orcesi, A. D., and Cremona, C. F. �2010�. “Optimal maintenance strate-gies for bridge networks using the supply and demand approach.”Struct. Infrastruct. Eng., in press.

Scherer, W. T., and Glagola, D. M. �1994�. “Markovian models for bridgemaintenance management.” J. Transp. Eng., 120�1�, 37–51.

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Optimization of Maintenance Strategies for the Management of the National Bridge Stock in France

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