International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Pre-posterior optimization of sequence of measurement and intervention actions under structural reliability… Goulet, James A.; Der Kiureghian, Armen; Lin, Binbin Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Pre-Posterior Optimization of Sequence of Measurement andIntervention Actions Under Structural Reliability ConstraintJames-A. GouletPost-doc, Dept. of Civil and Environmental Engineering, University of California,Berkeley, USAArmen Der KiureghianTaisei Professor, Dept. of Civil and Environmental Engineering, University of California,Berkeley, USABinbin LiPh.D. Candidate, Dept. of Civil and Environmental Engineering, University ofCalifornia, Berkeley, USAABSTRACT: When, based on the available information, an existing structure has an estimated failureprobability above the admissible level, the default solution often is to either strengthen or replace it. Evenif this practice is safe, it may not be the most economical. In order to economically restore and improveour existing infrastructure, the engineering community needs to be able to assess the potential gains as-sociated with reducing epistemic uncertainties using measurements, before opting for costly interventionactions, if they become necessary. This paper provides a pre-posterior analysis framework to (1) optimizesequences of actions minimizing the expected costs and satisfying reliability constraints, and (2) quantifythe potential gain of making measurements in existing structures. Illustrative examples show that whenthe failure probability estimated based on the present state of knowledge does not satisfy an admissiblethreshold, strengthening or replacement interventions can be sub-optimal first actions. An example showsthat significant savings can be achieved by reducing epistemic uncertainties.1. INTRODUCTIONIt is common to assess the condition of an exist-ing infrastructure by reliability analysis using priorknowledge about capacities and demands. Whenan existing structure has an estimated failure proba-bility above an admissible level, pF > p{adm.}F , thedefault solution often is to perform a structural in-tervention action, such as strengthening or replace-ment. However, it is known that the prior informa-tion about capacities and demands of an existingstructure is characterized by epistemic uncertain-ties. By gathering additional information, it is oftenpossible to reduce these uncertainties and alter thefailure probability estimate. Therefore, in order toassess the true condition of an existing infrastructureand economically restore and improve it, the engi-neering community needs to be able to estimate thepotential gains associated with reducing epistemicuncertainties using information gathering actions,instead of directly opting for costly structural inter-ventions based on findings from prior knowledge.Uncertainties and their classification have receivedmuch attention from the scientific community, e.g.O’Hagan and Oakley (2004); Murphy et al. (2011);Der Kiureghian and Ditlevsen (2009). Uncertaintiesare most often classified as either aleatory or epis-temic, depending on whether they are attributed toinherent variability or to lack of knowledge. Accord-ing to this classification, epistemic uncertainties arereducible and aleatory uncertainties are not.112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015If we were able to precisely measure the properties(e.g., as-built dimensions, material constants, mem-ber capacities) of an existing structure, no uncertain-ties in these quantities would remain. Of course, itis not possible to accurately measure all structuralproperties. Nevertheless, any direct or indirect ob-servations about these quantities can serve to reducethe corresponding epistemic uncertainties. Note thatmeasuring a structural property may either increaseor decrease the estimated failure probability, depend-ing on the measurement outcome (Artstein and Wets(1993); Faber (2000)).Maintenance planning for structures has been ad-dressed in previous research related to structuralhealth monitoring, decision theory and reliabilitytheory. For instance, Faber (2000) proposed a gen-eral framework for assessment of existing structuresbased on reliability theory considering evidencesobtained during inspection. Engineering decisionanalysis can be made in three stages (Benjamin andCornell (1975); Faber (2005); Faber and Stewart(2003)): prior decision analysis, posterior decisionanalysis and pre-posterior decision analysis. Thispaper deals with pre-posterior decision analysis,where the planning of information gathering actionsis made based on the prior probabilistic model ofuncertainties. In this scheme, the consequences (e.g.costs) of the possible outcomes of measurementor other information gathering actions are weighedwith their probabilities of occurrence.This paper presents a pre-posterior framework foroptimizing sequences of actions minimizing the ex-pected costs and satisfying reliability constraints foran existing structure. This framework is intendedto: (1) provide optimized sequences of informationgathering and intervention actions, and (2) quan-tify the potential gains of measuring structures in-stead of directly opting for costly strengthening andreplacement interventions. Section 2 presents theformulation for assessing the reliability of an exist-ing structure, Section 3 presents the mathematicalframework for the pre-posterior decision analysisfor sequences of actions, and Section 4 presents anillustrative application of the proposed methodol-ogy. The current paper summarizes a journal paperto appear in Structural Safety (Goulet et al. (2015)).Readers are invited to consult the full paper for moredetails.2. ASSESSING THE RELIABILITY OF AN EXIST-ING STRUCTUREThe safety and serviceability of an existing struc-ture is usually assured by verifying that, given theavailable knowledge, the structure has a failure prob-ability (complement of reliability) lower or equalto an admissible value, i.e. pF ≤ p{adm.}F . LetV = [V1,V2, · · · ,Vn]T denote the set of random vari-ables defining the state of the structure and fV(v)represent its joint probability density function (PDF).The failure probability is defined aspF =∫ΩfV(v)dv (1)whereΩ≡ {v|∪k∩i∈CkGi(v)≤ 0} (2)is the failure domain. This formulation is writtenin terms of unions of intersections of componentalfailure events. The ith component is defined in termsof a limit state function Gi(V) with {Gi(V) ≤ 0}indicating its failure. The union operation is overmin cut sets Ck, k = {1,2, · · ·}, where each min cutset represents a minimal set of components whosejoint failure constitutes failure of the structure. Theintersection operations are over components withineach min cut set. Special cases of this formulationare series structural systems, when each min cut sethas a single component, parallel structural systems,when there is only one cut set, and structural com-ponent, when there is only one min cut set with asingle component (Der Kiureghian (2005)).The limit-state functions Gi(V) defining the com-ponent states are usually made up of sub-modelsrepresenting component capacity and demand val-ues. Such a sub-model typically has the formR(X,ε) = Rˆ(X)+ ε (3)where Rˆ(X) represents an idealized mathematicalmodel and ε is the model error, which is usually con-sidered to have the Normal distribution. The additiveerror model is based on an assumption of normality,which is usually satisfied by an appropriate trans-formation of the model, see Der Kiureghian (2008).212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Physics-based models of structural components aregenerally biased so that the mean of ε , µε , can benonzero. The standard deviation, σε , represents ameasure of quality of the model. The vector V col-lects random variables X and ε for all sub-models.In addition, it may include any uncertain parametersΘ involved in the definition of the distributions of Xand ε for the various sub-models.At the outset of our analysis, the PDF of V repre-sents our prior state of knowledge about the structureand its future loads. We designate this by using thenotation f {0}V (v). The corresponding estimate of thefailure probability is denoted p{0}F . If p{0}F ≤ p{adm.}F ,the reliability constraint (p{adm.}F ) is satisfied and nofurther action is necessary. When p{0}F > p{adm.}F , ac-tions are necessary to reduce the failure probabilityestimate.As we take actions to modify the structure, learnabout the random variables, or improve the mod-els, the distribution of V changes. We show this bychanging the superscript {0}. Specifically, f {a1:i}V (v)denotes the distribution of V after an ordered setof actions {a1:i} = {a1, · · · ,ai}. The correspond-ing failure probability estimate is denoted p{a1:i}F .Our aim is to find an optimal sequence of futureactions Aopt = {a1, · · · ,an} that minimizes the ex-pected costs, while assuring that p{a1:i}F ≤ p{adm.}F .3. OPTIMIZATION FRAMEWORKAs mentioned in Section 2, when p{0}F > p{adm.}F , ac-tions are necessary to reduce the failure probabilityestimate. Let A = {a1, · · · ,ai} denote an orderedset of candidate actions so that action ai can takeplace only after actions {a1, · · · ,ai−1} have beencompleted. Example actions include replacementor strengthening of the structure, measurement ofcomponent capacities, measurement of variables in-volved in the capacity or demand models, prooftesting of the structure, etc. Each action ai will alterour state of knowledge about one or more of therandom variables so that f {a1:i−1}V (v) will change tof {a1:i}V (v) after action ai is taken. If action ai is astructural intervention, e.g., replacement or strength-ening, the new distribution f {a1:i}V (v) is that of thenew or strengthened structural design. If the actionis one of information gathering, f {a1:i}V (v) is derivedfrom f {a1:i−1}V (v) by conditioning on the observa-tions, while accounting for possible measurementerrors. However, since the analysis is performedbefore observations are made, one needs to considerall possible realizations of the observations withtheir corresponding prior probabilities. This aspectrequires pre-posterior analysis. The correspondingprobability estimate p{a1:i}F should be regarded as theconditional probability of failure, given the observa-tions. Thus, it is a function of the future observations.In the outcome space of these observations, the do-main where p{a1:i}F ≤ p{adm.}F constitutes the eventthat actions {a1, · · · ,ai} will lead to satisfaction ofthe reliability constraint. The next sub-section elab-orates on the characterization of this domain andspecification of the probability of success in satisfy-ing the reliability constraint.Our task is to identify an optimized sequence offuture actionsAopt = {a1, · · · ,an} so thatAopt mini-mizes the expected costs subject to p{Aopt}F ≤ p{adm.}F .For intervention actions (e.g., strengthening or re-placement), the probability of satisfying the reliabil-ity constraint based on the prior state of knowledgeis either zero or one. This is because for any strength-ening or replacement design, the estimated failureprobability is either greater than, or less than the ad-missible value. The only reason for contemplatingintervention actions with zero probability of satis-fying the reliability constraint is to consider themsubsequent to other actions that may improve ourstate of knowledge. For example, a partial retrofitmay become a viable option after measurementshave shown that the capacity is likely to be greaterthan initially estimated.3.1. Actions involving measurementsAssume that the first action a1 of a sequence A con-sists in measuring a structural property or structuralresponse. Given a measurement outcome m{a1} ∈ Rand the conditioned PDF f {a1}V (v), the failure prob-ability conditional on the measurement outcome isp{a1}F . Because the outcome of the measurement isunknown a-priori, it is treated as a random variableM{a1}. The subset of outcomes of M{a1} leading to312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015satisfaction of the reliability constraint isM {a1} = {m{a1} : p{a1}F ≤ p{adm.}F } (4)Thus, the probability that after taking action a1 thereliability constraint will be satisfied isp{a1}succeed =∫M {a1}fM{a1}(m{a1})dm{a1} (5)in which fM{a1}(m{a1}) is the PDF of the measure-ment outcome M{a1}. The conditional probabilityof failure as a function of the measurement m{a1},p{a1}F , the subset of measurement outcomes M{a1},its complement M{a1}, and the probability of meet-ing the reliability constraint, p{a1}succeed are illustratedin Figure 1. For a1 there is a probability 1− p{a1}succeedFigure 1: Outcome space of a measurement: a) failureprobability conditioned on the measurement outcomep{a1}F , b) PDF of the measurement outcome M{a1} withthe shaded area showing probability p{a1}succeed of meetingthe reliability constraint.that the measurement action will not satisfy the re-liability constraint. Therefore, for all measurementoutcomes m{a1} ∈M{a1}, it is necessary to plan forat least one additional measurement or interventionaction. Figure 2 shows the outcome space of twosuccessive measurements. Figure 2(a) depicts thesubset of successful outcomes of the first measure-ment M {a1} (shaded area). Figure 2(b) depicts the(a) (b)Figure 2: Outcome space of two successive measure-ment outcomes: a) subset of successful first measure-ment outcomes, b) subset of successful second mea-surement outcomes conditioned on first unsuccessfulmeasurement outcomesubset of successful outcomes of the second mea-surement, conditional on a first unsuccessful mea-surement, M {a1:2} (darkly shaded area). The bound-ary of M {a1:2} is nonlinear because of interactionbetween the previous unsuccessful measurement out-come and the new measurement. (A previous un-successful measurement far from the boundary ofsuccess requires a more favorable outcome of thesecond measurement to assure success.)More generally, for any subsequent measurementaction ai ∈A , i = 2, · · · ,n, the subset of successfulmeasurement outcomes M {a1:i} ⊆ Ri isM {a1:i} ={m{a1:i} : p{a1:i}F ≤ p{adm.}F∧m{a1:i−1} /∈M {a1:i−1}} (6)In Eq.(6), the subset of successful measurement out-comes M {a1:i} is obtained while excluding the pre-vious subset of successful measurement outcomesM {a1:i−1}. Measurement outcomes m{a1:i−1} ∈M {a1:i−1} are excluded because ai would only betaken if all previous measurement actions were un-successful. The conditional probability of successof the ith measurement action given no success upto the (i−1)th action is,p{ai}succeed =1c{a1:i}·∫M {a1:i}fM{a1:i}(m{a1:i})dm{a1:i}(7)412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015where fM{a1:i}(m{a1:i}) is the joint PDF of the i mea-surements andc{a1:i} =∫M{a1:i−1}fM{a1:i}(m{a1:i})dm{a1:i} (8)is a normalization constant. Equation 7 is obtainedby dividing the probability of intersection of no suc-cess in the first i−1 measurements and success inthe ith measurement by the probability of no successin the first i−1 measurements.3.2. Expected costs for sequences of actionsIn order to compute the expected costs, actions mustbe added to the setA until the sequence of n actionshas a cumulative probability p{A }c,succeed = p{a1:n}c,succeed =1. When p{A }c,succeed = 1, it is certain that the sequenceof actions planned are sufficient to satisfy the relia-bility constraint. The cumulative probability that asequence of actions {a1, · · · ,ai}, i ∈ {2, · · · ,n} willresult in meeting the reliability constraint is givenbyp{a1:i}c,succeed = p{a1:i−1}c,succeed + p{a1:i}succeed (9)where the probability of satisfying the reliability con-straint using a sequence of actions {a1, · · · ,ai}, i ∈{2, · · · ,n} is,p{a1:i}succeed = p{ai}succeed× (1− p{a1:i−1}c,succeed) (10)Note that c{a1:i} presented in Eq. 8 is identicalto 1− p{ai−1}c,succeed. Figure 3 presents an example ofthe probability mass function p{a1:i}succeed and the cor-responding cumulative probability mass functionp{a1:i}c,succeed, plotted against the cumulative cost of ac-tions C({a1:i}). As illustrated in Figure 3, it is likelythat a subset of A will reach a p{a1:i}c,succeed close toone, so that in most cases, performing only the firstfew actions in A will be sufficient to satisfy thereliability constraint.In decision theory, optimal decisions are those thatmaximize the expected value of a utility functionVon Neumann and Morgenstern (1947). Accord-ingly, the optimization problem at hand consists infinding a sequence of actions Aopt so thatAopt = argminA{E[C(A )]|p{A }c,succeed = 1} (11)Figure 3: Probability mass function p{a1:i}succeed andcorresponding cumulative probability mass func-tion p{a1:i}c,succeed against the cumulative cost of actionsC({a1:i}).in which E[C(A )] is the expected cost for a se-quence of measurement and intervention actions Aobtained asE[C(A )] =n∑i=1(p{a1:i}succeed×C ({a1:i}))(12)where p{a1:i}succeed is the probability of occurrence of asequence of i actions leading to success.Decision makers may adopt optimized manage-ment policies by planning to perform actions se-quentially as defined in Aopt until p{a1:i}F ≤ p{adm.}F ,i ∈ {1, · · · ,#Aopt}. By following this procedure, thecost of taking actions will, on average, be equal toE[C(Aopt)]. In implementation, each time an actionis taken, the subsequent sequence of future actionscan be re-optimized. Doing this, the expected costis likely to be smaller than E[C(Aopt)].4. ILLUSTRATIVE EXAMPLEThe example investigates the reliability of the centralcolumn supporting a two-span bridge against buck-ling (component reliability). The required level ofreliability is set at p{adm.}F = 0.0013, which is equiv-alent to reliability index β = 3. We first determineif the column(s) meets this requirement based on theavailable information. Since the requirement is notmet, we develop an optimal plan for a sequence ofactions to undertake to assure satisfaction of the reli-ability constraint, while minimizing expected costs.512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 4 shows the layout of the considered struc-ture. It is known a-priori that the columns heightis H = 9 m and that its rectangular section has adepth of d = 3 m and a width of w = 0.25 m. Thecolumn is made of reinforced concrete with its elas-tic modulus E having a lognormal distribution withprior mean µ{0}E = 33.5 GPa and standard deviationσ{0}E = 3 GPa (corresponding to distribution param-eters λ {0} = 3.51 and ζ {0} = 0.0894). The contribu-Continuous beamsPinned rollingsupportPinnedsupportFigure 4: Example two span bridge where the compo-nent studied is the central column.tion of the reinforcement to the flexural stiffness isneglected. The top end of the column is pinned andthe bottom end is partially fixed by a concrete slablying on the ground. The effective length coefficientK is represented by a uniform distribution within theinterval (0.7,1.0). The buckling capacity model forthis slender column is given by Rˆ = pi2EI(KH)2 , whereI = dw3/12 is the moment of inertial in the weakdirection of the column. The true log-capacity isdefined by lnR = ln Rˆ+ ε , where the model error εis a Gaussian random variable having mean µε andstandard deviation σε . It is known that the standarddeviation is σε = 0.05. However, the model bias µεis unknown and our prior information is that it isnormally distributed with prior mean µ{0}µ = 0.05and standard deviation σ{0}µ = 0.05. Here, the priormean of the mean error is greater than zero (themodel is conservatively biased), representing theconservative nature of the design model. This couldbe due to, e.g., the effect of neglecting the contribu-tion of the reinforcement to the section moment ofinertia. The total dead load supported by the columnis known to be D = 4000 kN. The column weight isneglected. The maximal live load, L, applied on thecolumn is described by a lognormal distribution withprior mean µ{0}L = 600 kN and standard deviationσ{0}L = 50 kN. The set of random variables defin-ing this problem is V = {E,K,L,ε,µε}. The firstfour random variables are assumed to be statisticallyindependent, and ε depends on µε .The column failure is represented by the limit statefunction G(V) = R(E,K,ε)−D− L. Reliabilityanalysis with the prior information yields the esti-mated failure probability p{0}F = 0.0088 (β u 2.37).Since this is greater than the admissible failure prob-ability p{adm.}F = 0.0013, actions must be undertakento satisfy the reliability constraint. The subsequentsections define the candidate actions considered anddetermine the optimal sequence of actions that willreduce the estimated failure probability below theadmissible threshold, while minimizing the expectedcosts.4.1. Management actionsTable 1 lists a summary of the considered actionsand their costs and effects. We assume that limitingthe allowable live load or increasing risk acceptancehave costs higher than replacing the structure andare not considered as viable actions.Table 1: Summary of management actions and theircosts and effects.Action UnitscostsEffectReplace, (aCI1) 500Replaces the column with one thatsatisfies the reliability constraintStrengthen, (aCI2) 200Increases capacity by increasingcolumn moment of inertiaLoad test, (aME1) 5If test passes, guarantees that R >D+L{adm.}Measure elasticmodulus, (aME2)10Reduces epistemic uncertainty inthe estimate of column elastic mod-ulusCalibrate capacity-model error, (aME3)200Reduces epistemic uncertainty inthe estimate of model bias (mean er-ror)Refine capacitymodel, (aMR)10Reduces epistemic uncertainty inthe estimate of effective length co-efficient4.2. Optimization algorithm and numerical reso-lutionAs described above, there are n = 6 candidate ac-tions considered for this example. The optimizedsequence of actions Aopt is obtained using a greedy612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015optimization algorithm (Cormen (2001)) that isadapted to this problem. Although the greedy al-gorithm is known for occasionally leading to sub-optimal solutions, it is chosen for its simplicity andfast convergence. It is noted that the proposed frame-work is independent of the specific optimizationalgorithm selected and that other algorithms capa-ble of solving this problem are available (Cormen(2001)).With the greedy algorithm, the optimized sequenceof actions is constructed iteratively over n loops. Foreach loop k = 1, · · · ,n, the optimized sequence of kactions isAopt,k =Aopt,k−1∪ argminaiE[C(Ak,i)] (13)in which Aopt,0 is an empty set. Essentially, ineach step, the algorithm looks for the next best ac-tion in the sequence. In order to compute the ex-pected cost E[C(Ak,i)], we must have p{Ak,i}c,succeed = 1,i.e., the set of actions must assure satisfaction ofthe reliability constraint. When this condition isnot satisfied, an optimized upper bound of the ex-pected cost is computed for the sequence Ak,i ={Aopt,k−1,ai,aCON}, where the concluding actionaCON is such that p{Ak,i}c,succeed = 1. In this example,aCON = aCI1 is selected because the latter is theonly action that guarantees satisfaction of the re-liability constraint. The optimization procedure isrepeated until p{Aopt,k}c,succeed = 1 and the expected cost isthen computed for the optimized sequence of actionsAopt =Aopt,k. Note that if k = n and p{Aopt,k}c,succeed < 1,additional alternative actions must be considered.In this example, Monte Carlo simulations are usedto compute the conditional probability of failure andthe probability of success of for each sequence ofactions. The coefficient of variation (c.o.v.) for com-puting a probability p by Monte Carlo simulationis δpˆ =√1−pˆN·pˆ , where pˆ is the estimated probability.The minimum number of samples required to obtaina c.o.v. smaller than 0.05 for pˆ{a1:i}c,succeed and pˆF areabout 5000 and 3×105, respectively. The numbersof samples used in this example are greater thanthese minima.0 100 200 300 400 500 600 700 80000.20.40.60.81costsprobability  NoactionFigure 5: Probability mass function p{a1:i}succeed and cumu-lative probability p{a1:i}c,succeed for the costs of the optimizedsequence of actions.4.3. Minimization of the expected costs of se-quences of actionsFigure 5 presents the probability mass functionp{a1:i}succeed and the cumulative probability mass func-tion p{a1:i}c,succeed against the costs of the optimized se-quence of actions Aopt. This figure shows that thereis a high probability that the low-cost model refine-ment and measurement actions will be sufficient toreduce the estimated failure probability below theadmissible threshold. The overall expected cost forthe optimal sequence of actions is E[C(Aopt)] = 91,which is substantially lower than the cost of strength-ening or replacing the structure. This is a result ofthe likely favorable outcomes of the low-cost can-didate actions of refining the model and measuringthe elastic modulus, which together have the successprobability p{aMR,aME2}c,succeed = 0.81.Results of the analysis indicate that performing aload test as a first action has a zero probability oflowering the failure probability below the admissiblelevel. This is because the initial estimate of thefailure probability of the column is large so thatthe admissible live load, back-calculated from theadmissible failure probability, is limited to 241 kN,which is below the mean value. Therefore, for anynon-zero cost, a load test is a sub-optimal first actionbecause it is certain that at least one additional actionwill be required to satisfy the reliability constraint.When performed after having refined the capacitymodel, the probability of satisfying the reliabilityconstraint with a load test increases to p{aME1}succeed =712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150.12. Despite this low probability of success, thisaction is expected to be a more efficient fourth actionthan the other alternatives because of its low cost.By adopting the optimized management strategyAopt, there is a probability 0.67 that only refiningthe model will be sufficient to satisfy the reliabilityconstraint. There is a probability lower than 0.08that replacing the structure will become necessaryafter having performed all information gatheringactions.5. SUMMARY AND CONCLUSIONSThis paper provides a pre-posterior framework foroptimizing sequences of information gathering andintervention actions and for quantifying the poten-tial gain of measuring structures instead of choosingcostly strengthening and replacement options. Theillustrative example shows that (1) when a structuredoes not satisfy an admissible failure probability,strengthening or replacement interventions can besub-optimal first actions, and (2) that significantsavings can be achieved by reducing the epistemicuncertainty in existing structures before costly inter-ventions are made to assure sufficient reliability. Interms of future work, the proposed framework opensnew opportunities for enhancing network-level in-frastructure management.6. ACKNOWLEDGEMENTSThis paper summarizes a journal paper to appear inStructural Safety (Goulet et al. (2015)). The firstauthor thanks the Swiss National Science Founda-tion and the Quebec Research Fund for Nature andTechnology for supporting this research. Additionalsupport was provided by the U.S. National ScienceFoundation under Grant No. CMMI-1130061.7. REFERENCESArtstein, Z. and Wets, R. J. (1993). “Sensors and infor-mation in optimization under stochastic uncertainty.”Mathematics of Operations Research, 18(3), 523–547.Benjamin, J. R. and Cornell, C. A. (1975). Probability,statistics and decision for civil engineers. McGraw-Hill, New York.Cormen, T. (2001). Introduction to algorithms. The MITpress, Cambridge, MA.Der Kiureghian, A. (2005). First- and second-order reli-ability methods. (E. Nikolaidis, D. M. Ghiocel and S.Singhal, Ed.) CRC Press, Boca Raton, FL, Chapter 14in Engineering design reliability handbook.Der Kiureghian, A. (2008). “Analysis of structural reli-ability under parameter uncertainties.” ProbabilisticEngineering Mechanics, 23(4), 351–358.Der Kiureghian, A. and Ditlevsen, O. (2009). “Aleatoryor epistemic? does it matter?.” Structural Safety, 31(2),105–112.Faber, M. H. (2000). “Reliability based assessment of ex-isting structures.” Progress in Structural Engineeringand Materials, 2(2), 247–253.Faber, M. H. (2005). “On the treatment of uncertain-ties and probabilities in engineering decision analysis.”Journal of Offshore Mechanics and Arctic Engineering,Transactions of the ASME, 127(3), 243–248.Faber, M. H. and Stewart, M. G. (2003). “Risk assessmentfor civil engineering facilities: critical overview anddiscussion.” Reliability Engineering & System Safety,80(2), 173–184.Goulet, J., Der Kiureghian, A., and Li, B. ((in press)2015). “Pre-posterior optimization of sequence ofmeasurement and intervention actions under structuralreliability constraint.” Structural Safety, 52, Part A.Murphy, C., Gardoni, P., and Harris, Jr, C. E. (2011).“Classification and moral evaluation of uncertaintiesin engineering modeling.” Science and engineeringethics, 17(3), 553–570.O’Hagan, A. and Oakley, J. (2004). “Probability is per-fect, but we can’t elicit it perfectly.” Reliability Engi-neering & System Safety, 85(1-3), 239–248.Von Neumann, J. and Morgenstern, O. (1947). The theoryof games and economic behavior. Princeton universitypress.8

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