Lifetime cost optimization of structures by a combined condition–reliability approach

Engineering Structures 31 (2009) 1572–1580

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Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Lifetime cost optimization of structures by a combined condition–reliabilityapproachDan M. Frangopol a, Alfred Strauss b,∗, Konrad Bergmeister ba Department of Civil and Environmental Engineering, ATLSS Center, Lehigh University, 117 ATLSS Dr., Bethlehem, PA 18015-4729, USAb Department of Civil Engineering and Natural Hazards, University of Natural Resources and Applied Life Sciences, Vienna, A-1190, Austria

a r t i c l e i n f o

Article history:Received 10 September 2008Received in revised form4 February 2009Accepted 13 February 2009Available online 17 March 2009

Keywords:Cost optimizationCondition stateReliability stateMaintenanceManagementLifetime performance

a b s t r a c t

Cost optimization for the determination of the most effective maintenance strategy of deterioratingstructures has recently been used by researchers and accepted by experts. The main reasons for this lie instrict requirements of budgetary efficiency and in reliability requirements for structures. Accessible datadirectly related to structural performance include the condition state, obtained from generally stipulatedvisual inspections, and the reliability state, derived from the inspection ormonitoring programs combinedwith numerical computations. In this paper, a novel approach to cost models for condition and reliabilityprofiles is described. Global cost optimization can be achieved by means of multi-objective optimizationor by means of a Cost-Optimized Condition-Reliability Profile (COCRP) approach. Since the assessmentof structures is under uncertainty, it is essential to embed the COCRP concept in a full probabilisticframework. Probabilistic COCRP computations can be performed, even for complex structures, within areasonable timebyusing advancedMonte CarloMethods such as LatinHypercube Sampling. TheproposedCOCRP approach allows realistic and efficient treatment of structures that involve uncertainties anddetects the parameters having the most significant effects on lifetime cost.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

For civil infrastructure systems, such as bridges, tunnels, andoffshore platforms, there is the vital requirement to provide theirpreservation and operation during the planned lifetime by ade-quatemaintenance-management strategies. These strategies mustenable optimumdecisionmakingwith respect to the timeof repair,rehabilitation, or replacement. Bridge management models suchas [1] are based on condition states. These models are currentlythe most widely adopted bridge management programs. They arebased primarily on visual inspection data and are relatively simpleto implement. The primary limitation of condition-state models isthat the safety is not adequately addressed. In fact, visual appear-ance does not always correlate to structural performance and accu-racy is lost due to a limited number of discrete condition states [2].In addition, recent research indicates that in some cases more than50% of bridges are being classified incorrectly [3]. In response tothese limitations, reliability-based life-cycle management modelswere proposed. In addition, Messervey and Frangopol [4] iden-tify and focus upon adoptions-in-concert between the parties in-volved in the design andmanagement of highway bridges. Specific

∗ Corresponding author. Tel.: +43 1476545254; fax: +43 1476545299.E-mail addresses: dan.frangopol@lehigh.edu (D.M. Frangopol),

alfred.strauss@boku.ac.at (A. Strauss), konrad.bergmeister@boku.ac.at(K. Bergmeister).

0141-0296/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2009.02.036

attention has to be focused on the inclusion of structural healthmonitoring (SHM) in bridge management programs and the needto coordinate, synchronize, and standardize key metrics, method-ologies, and communication tools to facilitate continued develop-ment of SHM applications to civil infrastructure. Nevertheless, de-cisions with respect to the maintenance of structures usually in-volve various expenses, such as repair expenses, long-term reha-bilitation expenses, and expenses for rebuilding. Such expensesare investigated and included in objective functions, based on thereliability profile [5]. In general, these approaches are based onconstruction cost, inspection costs, maintenance costs, user costs,failure costs, and reliability profile considerations. These types ofcosts are included as essential parameters in life-cycle cost (LCC)approaches for deteriorating structures [6–16].The time-dependent structural performance is expressed by

time variation of the condition state [17,18], reliability state [5],or both [19]. The condition state of structures or structuralcomponents can be expressed by the condition index CI . In thisstudy, based on results reported in [19], the condition index rangesbetween 1 and 4, where 1 and 4 indicate a very good and apoor condition, respectively. In general, the condition index can beobtained by using traditional inspection methods as specified inmanuals for bridge inspections. In general, the reliability state canbe expressed by the reliability index β which indicates the safetylevel. The reliability index is defined as β = (µR − µQ )/(σ

2R +

σ 2Q )1/2 where µR and µQ =mean resistance and mean load effect,

D.M. Frangopol et al. / Engineering Structures 31 (2009) 1572–1580 1573

Table 1Predefined parameters for optimization.

Predefined parameters Unit Notation ValueCondition Reliability Condition Reliability

Initial index at t0 (i.e. t = 0) C0I β0I 1.0 9.0Index at tH CH βH 2.4 3.5Maximum time of first maintenance year tPI,upper tPI,upper 60 60Minimum time of initiation of deterioration year tCI,lower tRI,lower 3 3Maximum time of initiation of deterioration year tCI,upper tRI,upper 25 25Duration of maintenance effect years tCD tRD 2.0 2.0Deterioration rate without maintenance year−1 αC αR 0.1 0.1Improvement in the index ∆γ ∆β 0.5 0.5Deterioration rate change after maintenance year−1 δC ∆α 0.05 0.05Lifetime (i.e., time horizon) years tH tH 60 60

Fig. 1. Optimum condition profiles associated with two initial condition indices.

respectively, and σR and σQ = standard deviation of the resistanceand the standard deviation of the load effect, respectively. If theresistance R and the load effect Q are normally distributed, theprobability of failure can be determined as Pf = Φ(−β) whereΦ = CDF of the standard normal distribution. Considering thatmany problems related to structural safety are non-Gaussian andnonlinear, improvement in the accuracy and efficiency of findingthe design point is important. Advanced first-order reliabilitymethods (FORM), second-order reliability methods (SORM) orsimulation techniques are used for this purpose. The calculationof β is a constrained optimization problem of finding the point onthe limit state surface with minimum distance to the origin of thestandard normal space [20].There are mainly two types of intervention: (a) interventions

that stop or slow down deterioration processes, and (b) interven-tions that cause an improvement in condition or reliability, or both.Since interventions are strongly related to costs and they can affectboth the condition and reliability of a structure or structural com-ponent, there is a requirement for a comprehensive developmentof combined cost optimizationmodels, based on condition and thereliability of deterioration profiles.Well-known cost models, supporting a cost optimization of

maintenance, have been developed for reliability profiles byFrangopol and Neves [19] and Kong and Frangopol [21]. However,a comprehensive cost optimization with respect to the structurallifetime requires the consideration of cost models associated withcondition and reliability profiles. Therefore, in addition to therequirements of existing models are the definition of condition(health) functions (i.e., condition profiles) and associated costmodels. The development of condition health functions andassociated cost models can be obtained by using the principlesof the previously mentioned reliability profile. There are severalways of combining the cost models of both profiles, such as (a) byusing expert opinions, (b) by multi-objective fitting of individual

optimizations [22] based on genetic algorithms, or (c) by functionalapproaches like Cost Optimized Condition-Reliability Profilefunctions (COCRP). The objectives of COCRP are the determinationof (a) optimum types and time of interventions, and (b) theinterrelation between the condition and reliability of a structure.The definitions of the descriptive elements of profiles and costmodels are combined with intrinsic uncertainties in the modelsand data. Therefore, a comprehensive cost optimization withrespect to structural lifetime has to be combinedwith probabilisticsimulation techniques, enabling the incorporation of uncertainties.The computational effort strongly depends on the number ofsimulations. This effort can be reduced by various advancedsimulation techniques, including sampling (e.g., importance anddirectional) such as Latin Hypercube Sampling (LHS). LHS permitsa small number of realizations and, consequently, producessolutions in reasonable time. These simulations under uncertaintyprovide useful insight into the parameters of interventions andcost. The COCRP methodology, less computation-intensive thanmulti-objective optimization approaches, including LHS, offersa user-friendly approach for minimum-cost and preservationplanning. The proposed method can be used for any deterioratingstructure requiring maintenance in the foreseeable future. Theinvestigations and developments presented in this paper aremostly based on research activities associatedwith cost-optimizedcondition profiles and reliability profiles, performed by Frangopoland Liu [2], Frangopol andNeves [19], andKong and Frangopol [21].The presented analytical formulations and optimization conceptsof this paper are strongly related to these studies.The major innovations in this research are (a) the probabilistic

treatment of the optimization problem based on a stratifiedMonte Carlo simulation technique, and (b) the optimized costbased probabilistic entanglement of the condition profile with thereliability profile. The cost functions are adopted from [21].

2. Cost optimization for condition or reliability

2.1. Condition profile

The condition profile is defined as the time variation of thecondition index. The main parameters describing the conditionprofile under maintenance were proposed in [23,19] as follows:(a) initial condition index C0I , (b) time of initiation of deteriorationtCI , (c) deterioration rate without maintenance αC , (d) time offirst maintenance tPI , (e) time of reapplication of maintenance tP ,(f) duration ofmaintenance effect tCD, (g) deterioration rate changeafter maintenance δC , and (h) improvement in the condition index∆γ . These parameters can be found in the two (a and b) conditionprofiles shown in Fig. 1, where the number of interventions N is 2and 4, respectively, and the required condition index CH at the timehorizon of 55–65 years, tH , is between 2.15 and 2.65. Tables 1 and 2indicate the predefined parameters and the design variables used

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Table 2Design variables.

Variables Unit Variable NotationCondition Reliability

Time of application of first maintenance year x1 TPI TPIInitiation of damage or conditionchange

year x2 TCI TRI

Duration of maintenance effect years x3 TCD TRDPeriod without maintenance effect years x4 TCA TRANumber of interventions x5 N N

Table 3Intervention-related cost functions (see [21]).

Cost Unit Cost functionCondition Reliability

Improvement in theindex

$/m2 C◦γ = Cγ 0 + p · (∆γ )q C◦β = Cβ0 + p · (∆β)

q

Improvement in thedeterioration rate

$/m2 C◦δ = Cδ0 + g · (δC )h C◦α = Cα0 + g · (∆α)

h

Cost parameter $/m2 λCI = p1CI2+ p2CI+ p3 λβ = p1β2 + p2β + p3

in the optimization, respectively. Optimization can be performedwith regard to several aspects, such as (a) minimizing the numberof interventions for a specified lifetime tH and a required conditionindex CH , and (b) minimizing the overall cost for maintaining thecondition in a state that will not violate a prescribed thresholdindex during the specified lifetime.The optimization of the condition profile by overall cost

minimization needs the definition of cost functions. Suitableformulations of cost functions regarding the reliability profilehave already been proposed by Kong and Frangopol [21]. Theseformulations can be transferred without restrictions to conditionprofiles. Table 3, for instance, shows the cost function thatrepresents the cost to be expected for the improvement of thecondition index by ∆γ . The function consists of a fixed costparameter Cγ 0, a proportional factor p, and an exponent q. Thecost function for the change in the rate of deterioration δC is alsopresented in Table 3. The parameters and factors of these functionsare assumed as Cγ 0 = 300, p = 313, q = 2.0 and Cδ0 = 150,g = 5000, h = 2.0 for the following examples, in accordancewith interventionmeasures reported in [21]. Since themeasures ofimprovement in the condition index∆γ are not independent of thelevel of condition index CI, the magnitude of CI has to be includedin the cost formulation. A polynomial formulation, as shown inthe last line of Table 3, takes into account the level of CI by usingthe factor λ. The coefficients of the polynomial formulation areassumed as p1 = 1/9, p2 = −8/9 and p3 = 25/9, accordingto Kong and Frangopol [21]. The basis for the cost models used inthis study is the time variant condition index (i.e. condition profile)which can be obtained as

CI = C0I + [x2 − x1] · αc + [∆γ − x3 · (αc − δc)− x4 · αc ] · x5. (1)

This equation consists of threemainparts: the initial part describedby C0I and the time without deterioration x2; the period of

Table 5Constraints on condition, reliability, and number of intervention.

Parameter Notation ConstraintsLower Upper

Condition index at tH CH CH − 0.25 CH + 0.25Reliability index at tH βH βH − 0.5 βH + 0.5Number of interventions N 0

deterioration, described by x1, x2 and αC ; and the period affectedby interventions and reapplication of interventions described byx3, x4 and x5. Eq. (1) can be expressed as:

CI = C0 + (C1 + C2 + C3) · x5 = C0 +∆C · x5 (2)

where

C0 = C0I + [x2 − x1] · αc (3)C1 = ∆γ (4)

C2 = − [x3 · (αC − δC )] (5)C3 = − [x4 · αC ] (6)

C0 is not influenced by the number of interventions. However, C1,C2 and C3 are directly related to the number of interventions. Usingthis formulation, the cost parameter

λCI =

N=x5∑i=1

p1 · (C0 +∆C · (i− 1))2

+ p2 · (C0 +∆C · (i− 1))+ p3 (7)

can be used as a global coefficient. Applying this coefficient to thecondition cost function, as shown in Table 3, yields

Cγ =[Cγ 0 + p · (∆γ )q

]· λCI (8)

Cδ =[Cδ0 + g · (δc)h

]· x5. (9)

The cost Cδ due to the interventions for reducing the deteriorationrate δC is not dependent on CI. The addition of the two costs,shown in Eq. (8) and Eq. (9), leads to a cost model suitable for theoptimization of a condition profile as follows:

CCI = Cγ + Cδ. (10)

The optimization of CCI requires theminimization of themultivari-ate function

CCI = f (TPI , TCI , TCD, TCA,N) (11)

where TPI , TCI , TCD, TCA,N are defined in Tables 4 and 5. This taskcan be expressed as

min CCI = minXf (X). (12)

The optimization in this study was performed by a line searchmedium-scale method [24]. To ensure the feasibility of thecondition profile, the design variables associated with Eq. (11)must be subjected to constraints. The constraints on time,condition index, reliability index, and number of interventions, areprovided in Tables 4 and 5.

Table 4Time constraints.

Parameter Notation ConstraintsLower Upper

Initiation of condition change (year) TCI tCI,lower tH − (TCD + TCA)N; tCI,upperInitiation of reliability change (year) TRI tRI,lower tH − (TRD + TRA)N; tRI,upperTime of first maintenance for condition (year) TPI TCI tH − (TCD + TCA)N; TCI + (CH − C0I )/αC ; tPI,upperTime of first maintenance for reliability (year) TPI TRI tH − (TRD + TRA)N; TRI + (β0I − βH )/αR; tPI,upperDuration of maintenance for condition (years) TCD 0 (tH − TPI )/N − TCADuration of maintenance for reliability (years) TRD 0 (tH − TPI )/N − TRAPeriod without maintenance effect on condition (years) TCA 0Period without maintenance effect on reliability (years) TRA 0Lifetime (years) tH tH − 5 tH + 5

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Fig. 2. Optimum reliability profiles associated with two initial reliability indices.

Fig. 3. Normalized values for optimum condition profiles vs. initial condition indexfor CH = 2.4± 0.25.

2.2. Reliability profile

The reliability profile can be created in a similar way as thecondition profile [25–27]. The parameters of the reliability pro-file are as follows [23]: (a) initial index β0I , (b) time of initiationof deterioration tRI , (c) deterioration rate without maintenance αR,(d) time of first maintenance tPI , (e) time of reapplication mainte-nance tp, (f) duration of maintenance effect tRD, (g) deteriorationrate change after maintenance∆α, and (h) improvement in the re-liability index ∆β. These predefined parameters and design vari-ables can be found in Tables 1 and 2. Two cost-optimized profiles aand b, developed from these parameters and variables, are shownFig. 2. The profiles vary between an initial index β0I of 4.0 and 9.0and are restricted to βH = 4.5 ± 0.5 at the lifetime tH = 60 ±5 years.The reliability profile consists of the following parts: (a) initial

period of the reliability profile, described by the index β0I andthe time without deterioration x2, (b) period of deterioration,described by x1, x2 and αR, and (c) period of interventionsdominated by x5:

β = β0I − [x2 − x1] · αR+ [∆β − x3 · (αR −∆α)− x4 · αR] · x5 (13)

or

β = β0 + (β1 + β2 + β3) · x5 = β0 +∆B · x5 (14)

where∆B = β1 + β2 + β3 and

β0 = β0I − [x2 − x1] · αR (15)β1 = ∆β (16)

β2 = − [x3 · (αR −∆α)] (17)β3 = − [x4 · αR] . (18)

Fig. 4. Normalized values for optimum reliability profiles vs. initial reliability indexfor βH = 4.5± 0.5.

A cost-increasing factor can be used, as follows:

λβ =

N=x5∑i=1

p1 · (β0 +∆B · (i− 1))2

+ p2 · (β0 +∆B · (i− 1))+ p3. (19)Finally, using Eq. (19) and the cost-reliability function in Table 3results inCβ =

(Cβ0 + p ·∆βq

)· λβ . (20)

The cost term for the improvement in the deterioration rate isindependent of the magnitude of the reliability index and can,therefore, be expressed as:

Cα =(Cα0 + g ·∆αh

)· x5. (21)

The two derived cost terms result in the overall maintenance costCβH = Cβ + Cα. (22)To obtain the optimum cost requires the minimization of themultivariate functionCβH = f (TPI , TRI , TRD, TRA,N) (23)as follows:min CβH = min

Xf (X). (24)

The constraints of the multivariable function for the feasibility ofthe reliability profile are shown in Tables 4 and 5.

2.3. Numerical examples

The goal of the first example is to find the optimized cost byusing the objective function formulated in Eq. (12) consideringthe initial condition index, C0I , as variable. C0I was selectedbetween 1 and 3, best and worst condition, respectively. Theprofile parameters listed in Table 1, the intervention-related costfunctions shown in Table 3 and the constraints in Tables 4and 5 served as computation basis. The start vector x0 =[x1, x2, x3, x4, x5] of the optimization process (see Table 2) wasset to [23 15 12 4 1], and the vector lb = [0.1 0.1 0.1 0.1 0.1]was used for the lower bound conditions. The constraints on thecondition at the end of the lifetime tH were CH = 2.4 ± 0.25.The input parameters for the condition and reliability profiles andfor the cost functions have been adopted from maintenance dataof existing bridges reported in [21,19]. Two of the cost-optimizedcondition and reliability profiles (a) and (b), which are discussedsubsequently, are in Figs. 1 and 2. The associated normalizedcharacteristics tCD, tPI ,N, COST of optimized condition profiles andtRD, tPI ,N, COST of optimized reliability profiles are presented inFigs. 3 and 4, respectively. The profiles (a) and (b) shown in Figs. 1and 2 serve as upper and lower bounds for the initial conditionindex C0I in Fig. 3 and for the initial reliability index β0I in Fig. 4.

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Table 6Optimum profile parameters.

Variables Unit Condition ReliabilityVariable Profile a Profile b Variable Profile a Profile b

Time of application of first maintenance year TPI 23.2 26.8 TPI 28.9 15.3Initiation of damage or condition change year TCI 15.2 25.3 TRI 2.9 7.8Duration of maintenance effect year TCD 12.4 5.3 TRD 14.2 8.7Period without maintenance effect year TCA 4.9 1.8 TRA 13.1 1.9Number of interventions N 2 4 N 1 4Cost $/m2 956 1536 654 2253

The cost-optimized interventions (Table 2) were obtained asfunctions of the initial condition indexes C0I , see Figs. 1 and 3. Thequantities tCD,max = 12.4 years, tPI,max = 26.8 years,Nmax = 4, andCOSTmax = 1662 $/m2 shown in Fig. 3 are the maximum valueswithin the investigated range of the initial condition index.Fig. 3 shows two (a, b) of a group of cost-optimized condition

profiles CI. The initial condition index C0I (e.g. excellent andpoor for profiles a and b, respectively) affects the number ofnecessary interventions N to fulfill the requirement at the end ofthe lifetime (i.e., 2.15 ≤ CH ≤ 2.65). This fact demonstratesthe close dependency between the initial condition C0I and thenecessary number of interventions N . The costs associated withthe optimized profiles a and b are 956 $/m2 and 1536 $/m2,respectively. Table 6 indicates the optimum parameters of the twooptimum profiles in Figs. 1 and 2.The solutions associated with the optimized profiles in Fig. 3

have the following properties: (a) The number of necessaryinterventions N is strongly correlated with the cost, (b) thecost increases with increasing the initial condition C0I , and(c) the durations of maintenance tCD are reduced with increasing(worsening) the initial condition C0I .The goal of the second example is to find the optimized cost

by using the objective function formulated in Eq. (24) consideringthe initial reliability β0I as a variable. β0I was selected between 9and 4. The profile parameters listed in Table 1, the interventionrelated cost functions of Table 3, and the constraints in Tables 4and 5 served as basis. The start vector x0 and the vector lb forthe lower bound conditions are identical to those used for thecondition profile. The constraint on the reliability at the end of thelifetime tH is βH = 4.5± 0.5. The quantities tRD,max = 14.2 years,tPI,max = 28.9 years, Nmax = 4, and COSTmax = 2253 $/m2represent the maximum values within the examined region ofthe initial reliability index. Fig. 2 shows two (i.e., a and b) of agroup of cost-optimized reliability profiles β . The initial reliabilityindex β0I and the number of necessary interventions N interactto fulfill the requirement at the end of the lifetime (i.e., 4.0 ≤βH ≤ 5.0). As for the condition profile, there is a strong correlationbetween the initial reliability β0I and the necessary number ofinterventions N . The costs associated with the optimized profilesa and b are 654 $/m2 and 2253 $/m2, respectively. The optimumparameters for these two profiles can be found in Table 6. Fig. 4shows the normalized values for the optimum reliability profilesvs. initial reliability index. The results in this figure show that:(a) the number of necessary interventions N affects the optimumcost, which can be derived from the simultaneous increase of bothnormalized parameters u = N/Nmax and t = COST/COSTmaxwith decreasing β0I , (b) the cost increases with decreasing initialreliability β0I , and (c) the time of first maintenance applicationtPI and the duration of maintenance effect tRD are reduced withdecreasing β0I .

3. Probabilistic approach to cost-optimized interventions

The independent deterministic cost-optimized interventions ofthe condition profile and the reliability profile presented in Figs. 1–4, do not meet all intervention requirements since the interaction

between condition and reliability is not taken into account. Forthis reason, it is necessary to develop a unified approach in whichthe interaction between condition and reliability is incorporated.In this unified approach, the uncertainties associated with thecondition and reliability profiles can be treated by using advancedMonte Carlo simulation techniques, such as LHS.

3.1. Probabilistic models

The probabilistic considerations require the extension of theMATLAB routines, which are used for the deterministic cost-optimization of the condition and reliability profiles. Theseroutines are enhancedwith FREeT. FREeT [28] is a software packagethat offers advanced statistical tools and the LHS technique. TheLHS programmed in MATLAB is used for comparison purposes.FREeT, which is called automatically by the MATLAB routine,allows a user-friendly creation of the probabilistic models in formof probability density functions (PDFs). All parameters and designvariables in Tables 1 and 2 are now considered as random withthe means and standard deviations indicated in Table 7 for bothcondition and reliability. The values of the standard deviation forthe initial condition index C0I and the initial reliability indexβ0I areassociated with coefficients of variation of 3% and 5%, respectively.Uncertainties in the initial condition and reliability indices arereported in [19].The constraints in Tables 4 and 5 and cost functions in

Table 3 are used in the optimization process performed with LHSand FREeT. The initial reliability index and condition index areconsidered as 9.0 and 1.0, respectively. For a selected group of1000 samples, Figs. 5a and 5b show the results of the optimizationprocess (i.e. optimum cost) vs. αR and tCD, respectively. Thedeveloped routine performs the optimization of both profilessimultaneously. A sensitivity analysis performed using FREeT,allows a realistic assessment of the effect of the random variableson the optimum cost. A user-friendly full probabilistic cost-optimization of the condition profile and the reliability profile wasmade, by developing aMATLAB routinewith improved capabilities,in order to: (a) define all input parameters as randomvariables, and(b) allow an interactive optimization.Define all input parameters as random variables. As an example,

all parameters anddesign variables in Tables 1 and2 are consideredas random variables with prescribed mean values, standarddeviations, and PDFs (see Table 7). Figs. 6a and 6b show theresults of an optimization process considering 1600 samples, forcondition and reliability, respectively. The mean values of thecondition index CH were varied between 2.4 and 3.8 and thoseof the reliability index βH between 3.8 and 5.8. From the resultsshown in Figs. 6a and 6b, it is possible to quantify the tendency ofincrease of the cost with improving (i.e., decreasing) the conditionindex at lifetime (Fig. 6a) or increasing the reliability index atlifetime (Fig. 6b). The normalized optimum cost at lifetime forcondition, CUC , or reliability, CUR, represents the ratio betweenthe mean optimum cost at lifetime, resulted from sampling andthe maximum optimum cost associated with CH (i.e., COSTmax =1221 $/m2) orβH (i.e., COSTmax = 7014 $/m2), respectively. Fig. 6a

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Table 7Description of random variables.

Condition profilea Reliability profilea

Notation Units Mean value Standard deviation Notation Units Mean value Standard deviation

C0I 1 0.03 β0I 9 0.45∆γ 0.5 0.2 ∆β 0.5 0.03αC year−1 0.1 0.005 αR year−1 0.1 0.01δC year−1 0.05 0.005 ∆α year−1 0.05 0.005CH 2.4 0.07 βH 5.9 0.30tCD year 2 0.06 tRD year 2 0.10Cγ 0 300 9 Cβ0 300 15q 2 0.06 p 313 15.65p 313 9.39 q 2 0.10Cδ0 150 4.5 Cα0 150 7.5h 2 0.06 g 5000 250g 5000 150 h 2 0.10p1 0.11 0.005 p1 0.11 0.01p2 −0.89 0.03 p2 −0.89 0.04p3 2.78 0.08 p3 2.78 0.14tPI year 23 0.69 tPI year 15 0.75tP year 12 0.36 tRI year 5 0.25tCI year 15 0.45 tP year 8 0.40tCA year 4 0.12 tRA year 2 0.10N 1 0.03 N 1 0.05COSTCI $/m2 1000 0.30 COSTRE $/m2 1000 50a All random variables have a log-normal distribution.

Fig. 5a. Optimum cost vs. duration ofmaintenance effect of the condition index for1000 samples.

Fig. 5b. Optimum cost vs. deterioration rate of the reliability index for 1000samples.

shows 1600 samples of optimum lifetime cost, divided in eightgroups according to the mean value of CH (i.e., 2.4, 2.6, 2.8, 3.0,3.2, 3.4, 3.6, 3.8). Each group of 200 samples is shown projectedon the associatedmean value of CUC . Fig. 6b shows 1600 samples ofoptimum lifetime cost, divided in eight groups according themeanvalue ofβH (i.e., 3.8, 4.1, 4.4, 4.7, 5.0, 5.3, 5.6, 5.8). Again, each groupof 200 samples is shown projected on the associated mean value

Fig. 6a. Condition index at lifetime, CH vs. normalized optimum cost; 1600samples.

Fig. 6b. Reliability index at lifetime, βH vs. normalized optimum cost; 1600samples.

of CUR. Fig. 6c shows the 200 samples associated with the largestmean value of βH considered in Fig. 6b (i.e., 5.8). They result in themean value of optimum cost of 0.86× (7014 $/m2) = 6032 $/m2.Their projection on CUR is indicated in Fig. 6b at CUR = 0.86.Finally, Fig. 6d shows the increase in the standard deviations of thenormalized optimum cost σ(CUR) and σ(CUC ) with increasing CURfor both lifetime condition CH and lifetime reliability βH . Using the

1578 D.M. Frangopol et al. / Engineering Structures 31 (2009) 1572–1580

Fig. 6c. Samples of the reliability index with mean = 5.8 at lifetime, βH , vs.normalized optimum cost.

Fig. 6d. Standard deviations of normalized optimum cost for condition andreliability vs. normalized optimum cost.

information shown in Figs. 6a–6d, the decision-maker can specifythe level of both the condition index and reliability index at theend of lifetime in a more realistic framework by quantifying thecost associated with each decision.Allow an interactive optimization. A probabilistic model for cost

associated with the condition profile has to be combined with aprobabilisticmodel for cost associatedwith the reliability profile toallow an interactive optimization. Combining the two performancefunctions representing condition and reliability in a single costobjective allows an interactive economic intervention planning.The significance of this optimized intervention is the simultaneousminimization of costs for interventions which are necessary toprovide a desired condition and reliability performance. It is notedthat the costs associated with the assurance of the condition orreliability performance can be very different. In this study, theCost-Optimized Condition-Reliability Profile (COCRP) concept isproposed. The main advantage of COCRP is providing a unifiedoptimization approach for taking into account the interactionbetween the condition and reliability profiles. This combinationyields to the Cost-Optimized Condition-Reliability Profile (COCRP)concept. Combining the two performance functions representingcondition and reliability in a single cost objective allows aninteractive economic intervention planning. The significance ofthis optimized intervention is the simultaneous minimization ofcosts for interventions which are necessary to provide a desiredcondition and assurance of reliability performance. It is noted thatthe individual costs associated with the assurance of the conditionor reliability performance can be very different. In this study, theCost-Optimized Condition-Reliability Profile (COCRP) concept isproposed. The main advantage of COCRP is providing a unifiedoptimization approach for taking into account the interactionbetween the condition and reliability profiles.

Fig. 7. Reliability index βH and condition index CH at lifetime vs. normalizedoptimum costs.

3.2. Global cost optimization

Global cost optimization (i.e. simultaneous cost optimizationof both condition and reliability) requires the normalization ofthe optimum costs associated with the condition and reliabilityprofiles. There are efficient strategies for the combination of theoptimum costs of the condition profile and the reliability profileconsidering probabilistic descriptors of CH and βH . One of themis explained herein. At first the mean values of the optimum costassociated with the condition profile CH and the reliability profileβH at the end of the lifetime have to be computed (see Figs. 6a and6b). The higher maximum mean cost is selected as the dominantcost (for the considered example this is the cost associated withβH ). The sum of the maximum cost COSTmax (i.e., dominant cost)and theminimumCOSTmin serves as the denominator for the globalnormalizationOptimum COST associated with CH

COSTmax+ COSTmin

+Optimum COST associated with βH

COSTmax+ COSTmin= A+ B = 1 (25)

where Optimum COST associated with CH and Optimum COSTassociated with βH are the optimum costs associated withprescribed values of CH and βH , respectively. If an optimumintervention strategy provides the Optimum COST associated witha prescribed level of βH , the global normalization expressed inEq. (25) can be rearranged as follows:

Optimum COST associated with CH = (COSTmax+ COSTmin− Optimum COST associated with βH) . (26)

This optimum COST associated with CH together with simulationresults shown in Fig. 6a serves to create the relation betweenCH and βH . Therefore, the significance of the cost-optimizednormalization is the constraint defined by Eq. (25) expressingthe simultaneous consideration of both optimized condition andreliability profiles. The procedure applied to a range of values ofβH allows the creation of a combined (CH , βH ) profile based onthe normalized optimum cost. The results presented in Figs. 6aand 6b, leads to the global normalized optimum cost CUG of CH(i.e., represented by notation A in Eq. (25)) and βH (i.e., representedby B in Eq. (25)) as shown in Fig. 7.The formulations presented in Eqs. (25) and (26) are strictly

valid (i.e., CUG = 1) for cases when the condition profiles interactwith the reliability profiles (e.g., see region a in Figs. 7 and 8). Inregions with no interaction (e.g., see regions b and c in Figs. 7 and8) CUG is smaller than one.Region a (see Fig. 7), characterized by a global optimum cost of

7785 $/m2 = 7014 $/m2 + 771 $/m2, serves for the combinationof the profiles according to Eq. (25). In region b (see Fig. 7), which is

D.M. Frangopol et al. / Engineering Structures 31 (2009) 1572–1580 1579

Fig. 8. Global normalized optimum cost.

associated with the condition cost, the global optimum cost variesbetween 4195 $/m2(CUG = 0.54) and 7785 $/m2(CUG = 1.00),and in region c (see Fig. 7) the global optimum cost was 1602 $/m2(CUG = 0.21) and 4195 $/m2 (CUG = 0.54). Note that in bothregions b and c the cost CUG associated with the condition CH = 1is decreasing with decreasing βH (see Fig. 8). According to theseresults a maintenance program in region c has to be performed.However, in this region, the reliability index at lifetime βH cannotbe larger than 5.3.Fig. 7 together with Eq. (25) are both necessary for computing

the Cost-Optimized Condition-Reliability Profile (COCRP) shown inFig. 8. The COCRP indicates the optimal relationship, based on theoptimum costs, between the condition index CH and the reliabilityindex βH at the end of the lifetime tH . The three aforementionedregions (a, b, c, in Fig. 7) are also assigned to the optimum COCRP.Probabilistic descriptors of the global optimum profile can also beobtained considering this strategy.The merging of the optimum cost profiles of condition and

reliability into COCRP, as demonstrated in Figs. 7 and 8, creates thenecessary correlation between the parameters of the condition andreliability profiles. Therefore, COCRP offers the advantage that theintervention planning based on the condition profile can be usedfor the intervention planning based on the reliability profile or viceversa.In its present form, the first line of application of the method

is for highway bridges. However, the method can be used for anystructure, or group of structures, requiring maintenance in theforeseeable future. The concept is suitable for application to bothnew and existing structures under variousmaintenance strategies.

4. Conclusions

Based on the results presented and discussed, the followingconclusions are drawn.1. In order tomaintain both condition and reliability of a structureover a planned lifetime requires interventions, which arerelated to cost.

2. Advanced cost models regarding the condition and the reliabil-ity of a structure have to be used for global cost-optimizationstrategies. There are advanced cost models under uncertaintyreported in the literature regarding the reliability of a structure,which can be extended to structural condition.

3. Since the assessment of structures is under uncertainty,it is essential to embed the optimization concepts in afull probabilistic framework. The approach presented in thispaper allows the probabilistic treatment of the condition andreliability profile parameters and, in consequence, a realisticassessment of optimum cost to be expected at the end of thelifetime.

4. A global optimization has to allow the merging of theprobabilistic models associated with the condition profile andthe reliability profile. This is reached by the Cost-OptimizedCondition- Reliability Profile (CORCP) concept andmethodologydiscussed in this paper.

5. COCRP is a computationally intensive approach, however, it isless expensive than most existing multi-objective optimizationapproaches. It offers togetherwith LHS a user-friendly approachfor lifetime minimum-cost planning of maintenance interven-tions. Nevertheless multi-objective optimization under uncer-tainty guarantees the Pareto optimum.

Acknowledgments

The support of the National Science Foundation throughgrants CMS-0217290 and CMS-0509772 to the University ofColorado, Boulder, and grants CMS-0638728 and CMS-0639428 toLehigh University is gratefully acknowledged. The opinions andconclusions presented in this paper are those of the authors and donot necessarily reflect the views of the sponsoring organization.

References

[1] PONTIS . User manual and product web site; Cambridge systematics.Availale through the AASHTO products office at http://www.camsys.com/tp_inframan_pontis.htm [accessed 2.11.2007]; 2007.

[2] Frangopol DM, Liu M. Maintenance and management of civil infrastructurebased on condition, safety, optimization, and life-cycle cost. Struct InfrastructEng 2007;3(1):29–41.

[3] Catbas NF, Zaurin R, Frangopol DM, Susoy M. System reliability-basedstructural health monitoring with FEM simulation. In: Proceedings of the3rd international conference on structural health monitoring of intelligentinfrastructure 2007.

[4] Messervey TB, Frangopol DM. Integration of health monitoring in assetmanagement in a life-cycle perspective. In: Proceedings of the fourthinternational conference on bridge maintenance and safety 2008.

[5] Kong JS, Frangopol DM. Evaluation of expected life-cycle maintenance cost ofdeteriorating structures. J Struct Eng ASCE 2003;129(5):682–91.

[6] Chang SE, Shinozuka M. Life-cycle cost analysis with natural hazard risk. JInfrastruct Syst ASCE 1996;2:118–26.

[7] Ang AH-S, De Leon D. Determination of optimal target reliabilities andupgrading of structures. Struct Saf 1997;19(1):91–103.

[8] Frangopol DM, Lin K-Y, Estes AC. Life-cycle cost design of deterioratingstructures. J Struct Eng ASCE 1997;123(10):1390–401.

[9] FrangopolDM,Gharaibeh ES, Kong JS,MiyakeM.Optimal network-level bridgemaintenance planning based on minimum expected cost. J Transp Res Board,Transp Res Record 2000;1696(1):26–33.

[10] Ang AH-S, Frangopol DM, Ciampoli M, Das PC, Kanda J. Life-cycle costevaluation and target reliability for design. Struct Saf Reliab 1998;1:77–8.

[11] Das PC. New developments in bridge management methodology. Struct EngInt IABSE 1998;8(4):299–302.

[12] Maunsell Ltd. Transport Research Laboratory. Strategic review of bridgemaintenance costs: Report on 1998 review. Final report. London: TheHighways Agency; 1999.

[13] Enright MP, Frangopol DM. Maintenance planning for deteriorating concretebridges. J Struct Eng ASCE 1999;125(12):1407–14.

[14] Frangopol DM. Life-cycle cost analysis for bridges. In: Frangopol DM, editor.Bridge safety and reliability. ASCE; 1999. p. 210–36 [Chapter 9].

[15] Frangopol DM,Das PC.Management of bridge stocks based on future reliabilityand maintenance costs. In: Das PC, Frangopol DM, Nowak AS, editors. Currentand future trends in bridge design, construction, and maintenance. TheInstitution of Civil Engineers, Thomas Telford; 1999. p. 45–58.

[16] Wallbank EJ, Tailor P, Vassie PR. Strategic planning of future maintenanceneeds. In: Das PC, editor.Management of highway structures. London: ThomasTelford; 1999. p. 163–72.

[17] Hawk H, Small EP. The BRIDGIT bridge management system. Struct Eng IntIABSE 1998;8(4):303–14.

[18] Thompson PD. The Pontis bridge management system. In: Carr WP, editor.Proc., pacific rim transtech conf.: International ties, management systems,propulsion technology, strategic highway research program, Vol. II. Seattle:ASCE; 1993. p. 500–6.

[19] Frangopol DM, Neves LC. Probabilistic performance prediction of deterioratingstructures under different maintenance strategies: Condition, safety and cost.In: Frangopol DM, Brühwiler E, Faber MH, Adey B, editors. Keynote paper inlife-cycle performance of deteriorating structures: Assessment, design andmanagement. Reston (VA): ASCE; 2004. p. 9–18.

1580 D.M. Frangopol et al. / Engineering Structures 31 (2009) 1572–1580

[20] ShinozukaM. Basic analysis of structural safety. J Struct EngASCE1983;109(3):721–40.

[21] Kong JS, Frangopol DM. Cost-reliability interaction in life-cycle cost op-timization of deteriorating structures. J Struct Eng ASCE 2004;130(11):1704–1712.

[22] Neves LC, Frangopol DM, Cruz PJS. Probabilistic lifetime-oriented multiobjec-tive optimization of bridgemaintenance: Singlemaintenance type. J Struct EngASCE 2006;132(6):991–1005.

[23] Frangopol DM, Kong J, Gharaibeh E. Reliability-based life-cycle managementof highway bridges. J Comp Civ Eng ASCE 2001;15(1):27–34.

[24] Coleman TF, Li Y. An interior, trust region approach for nonlinearminimizationsubject to bounds. SIAM J Optim 1996;6:418–45.

[25] Thoft-Christensen P. Reliability profiles for concrete bridges. In: FrangopolDM,Hearn G, editors. Structural reliability in bridge engineering. New York:McGraw-Hill; 1996. p. 239–44.

[26] Estes AC, Frangopol DM. Life-cycle reliability-based optimal repair planningfor highway bridges: A case study. In: Frangopol DM, Hearn G, editors.Structural reliability in bridge engineering. New York: McGraw-Hill; 1996.p. 54–9.

[27] Estes AC, Frangopol DM. Repair optimization of highway bridges using systemreliability approach. J Struct Eng ASCE 1999;125(7):766–75.

[28] Vořechovský M, Novák D. Statistical correlation in stratified sampling. In: 9thInt. conf. on applications of statistics and probability in civil engineering –ICASP 9. 2003. p. 119—24.