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

Risk measures in engineering design under uncertainty Rockafellar, R. Tyrrell; Royset, Johannes O. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Risk Measures in Engineering Design under UncertaintyR. Tyrrell RockafellarEmeritus Professor, Dept. of Mathematics, Univ. of Washington, Seattle, USAJohannes O. RoysetAssociate Professor, Operations Research Dept., Naval Postgrad. School, Monterey, USAABSTRACT: Engineering decisions are made under substantial uncertainty about current and futuresystem cost and response. A risk-neutral decision maker would rely on expected values when comparingdesigns, while a risk-averse decision maker might adopt nonlinear utility functions or failure probabilitycriteria. The paper shows that these models for making decisions fall within a framework of risk measuresthat includes many other possibilities. The paper provides an overview of the framework, highlightsbenefits derived from certain risk measures, and gives a truss design example.In design and optimization of structures, engineersare faced with the challenge of assessing the ade-quacy of a system with uncertain performance orselecting the best among several uncertain candi-date systems. We consider the situations wherethe uncertain performance is captured by a randomvariable whose distribution is estimated by prob-abilistic models. A risk-neutral decision makerwould make assessments and ranking on the basisof expected values of such random variables. Tra-ditionally, a risk-averse decision maker would relyon expected utility theory, with a nonlinear utilityfunction, or consider the probability of exceeding athreshold, i.e., a failure probability. In this paper,we outline a framework based on risk measures forrisk-averse decision making that encapsulates theseapproaches, but also offers new possibilities. Vari-ous alternatives are illustrated in a design optimiza-tion problem for a truss structure.Since preferences of decision makers are highlysituational dependent, we avoid a discussion aboutwhether risk neutrality or risk averseness is moreappropriate; see Rockafellar and Royset (2015) andreferences therein. Here, we provide tools for han-dling risk averseness regardless of its source andmotivation.Risk measures as the basis for decision mak-ing are supported by a well-developed theoryand extensive use in financial engineering andincreasingly in other fields; see Dowd (2005);Commander et al. (2007); Rockafellar and Royset(2010); Minguez et al. (2013). Their connectionswith classical expected utility theory and risk-neutral decision making are revealing as discussedbelow. For other models of decision making, we re-fer to references in Rockafellar and Royset (2015).We proceed in Section 2 with a description andexamples of risk measures. Sections 3 and 4make connections with expected utility theory andrisk neutrality under distributional uncertainty, re-spectively. Section 5 discusses optimization ofrandom variables that depend on design param-eters. Sections 2-5 summarize the material byRockafellar and Royset (2015). Section 6 illus-trates the framework with a numerical example.1. RISK MEASURESA broad class of decision models that encapsulatesessentially all reasonable approaches rely on mea-sures of risk as defined next:A measure of risk is a functional R thatassigns to a random variable Y a numberR(Y ), which could be infinity, as a quan-tification of the risk in Y .The answer to the question of how “risky”is Y , is now simply defined to be R(Y ). Thecomparison of two choices Y and Y ′ then reduces112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015to comparing R(Y ) and R(Y ′). A requirementthat Y should be “adequately” ≤ b is interpretedas having R(Y ) ≤ b. The minimizing of R(Y )over a set of candidate random variables Y thenamounts to finding the lowest b such that there is aY “adequately” ≤ b. Here and throughout the paperwe assume that high values of Y are undesirable.For example, Y might be the (life-cycle) cost orresponse amplitude of a system. For technicalreasons and convenience, we limit the scope torandom variables with finite second moments. Weillustrate the breadth of possibilities with examples.Expectation. The choice R(Y ) = E[Y ], theexpected value, is simple, but not sensitive to thepossibility of high values. Obviously, this choiceincorporates no level of risk averseness.Worst-case. The choice R(Y ) = supY , thesmallest value that Y exceeds only with probabilityzero, is conservative, usually overly so as it isinfinite for distributions such as the normal. Infact, the corresponding decision model ignoresall the information in the distribution of Y exceptits highest “possible” realization. Still, in someapplications there may be thresholds that shouldnot be exceeded.Quantile. For α ∈ (0,1), the α-quantile of a ran-dom variable Y , qα(Y ), is simply F−1Y (α) when thecumulative distribution function FY of Y is strictlyincreasing, with a slightly more complex formulain the general case. The choice of risk measureR(Y ) = qα(Y ) is widely used in financial engineer-ing under the name “value-at-risk” with typicallyan α of nearly one, and is equivalent to the failureprobability. We recall that the probability of failurep(Y ) = prob(Y > 0), where we assume that positiverealizations of Y are considered “failure.” The fail-ure probability is widely used in reliability analysis.It is clear thatp(Y ) ≤ 1−α if and only if qα(Y ) ≤ 0. (1)Consequently, the choice of a quantile as riskmeasure is equivalent to adopting a failure proba-bility criterion. There are two immediate concernswith these approaches. First, there may be twodesign with the same failure probability, but theirdistributions could be different, especially in thecritical upper tail. In fact, the failure probabilityis insensitive to the tail of the distribution andan exclusive focus on the corresponding decisionmodels may hide significant risks. The secondconcern when using the failure probability is itslack of convexity and smoothness as a functionof the design parameters. These deficiencies dra-matically increase the difficulty of solving designoptimization problems involving failure probabilityterms; see Rockafellar and Royset (2010) andRockafellar and Royset (2015) for details.Superquantile. The α-superquantile of Y at prob-ability α ∈ (0,1) is given byq¯α(Y ) = 11−α∫ 1αqβ(Y )dβ , (2)i.e., an α-superquantile is an average of quantilesfor probability levels α < β < 1. When the cumula-tive distribution function of Y has no discontinuityat the realization y = qα(Y ), we have the equiva-lent formula q¯α(Y ) = E[Y | Y ≥ qα(Y )], i.e., the α-superquantile is simply the conditional expectationof Y above the α-quantile. Despite its somewhatcomplicated definition, convenient expressions fa-cilitate the computation of superquantiles makingthem almost as accessible as an expectation. If Y isnormally distributed with mean µ and standard de-viation σ , then q¯α(Y ) = µ+σφ(Φ−1(α))/(1−α),where φ and Φ are the probability density (pdf)and cumulative distribution functions for a standardnormal random variable. IfY follows a discrete dis-tribution with realizations y1 < y2 < ... < yn andcorresponding probabilities p1, p2, ..., pn, thenq¯α(Y )=∑nj=1 p jy j for α = 011−α[(∑ij=1 p j −α)yi +∑nj=i+1 p jy j]for∑i−1j=1 p j < α ≤ ∑ij=1 p j < 1yn for α > 1− pn.We note that the realizations are sorted, without lossof generality, to simplify the formula. Generally,q¯α(Y ) = mincc+ 11−αE[max{0,Y − c}]212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015i.e., a superquantile is the minimum value of a one-dimensional optimization problem with variablec. A risk measure that focuses primarily on theimportant upper tail of the distribution of Y is thenthe superquantile risk measureR(Y ) = q¯α(Y ); alsocalled conditional value-at-risk. A superquantilerisk measure depends on the parameter α thatrepresents the degree of risk averseness of thedecision maker. For α = 0, q¯α(Y ) = E[Y ] andtherefore corresponds to the risk-neutral situation.An α = 1 gives q¯α(Y ) = supY and thereforecorresponds to the ultimate risk-averse decisionmaker. The superquantile risk measure leadsto a number of benefits in subsequent analysis.For example, if Y depends on a set of designparameters, then the risk remains convex in theparameters as long as the parameterization isconvex; see Rockafellar and Royset (2015) for adetailed discussion. The correspondence betweena failure probability constraint p(Y ) ≤ 1−α andthe quantile condition qα(Y ) ≤ 0 is given in (1).Analogously, a superquantile condition q¯α(Y ) ≤ 0corresponds to the condition p¯(Y ) ≤ 1−α , wherep¯(Y ) is the buffered failure probability of Y definedas the probability 1− α that satisfies q¯α(Y ) = 0.We refer to Rockafellar and Royset (2010) for adiscussion of the advantages that emerge fromreplacing a failure probability by a buffered failureprobability.A measure of riskR is regular if it satisfiesR(Y ) = c when Y ≡ c (constant equivalence);R((1− τ)Y + τY ′) ≤ (1− τ)R(Y )+ τR(Y ′)for all Y,Y ′ and τ ∈ (0,1) (convexity);{Y |R(Y ) ≤ c} is a closed set for everyconstant c (closedness);R(Y ) > E[Y ] for nonconstant Y (averseness).The first condition is natural as it simply asserts thata random variable that always takes on the samevalue, has risk equal to that value. The secondcondition insists that a linear combination of tworandom variables has a risk that is no larger thanthe linear combination of the individual risks. Thiscondition is also natural as it promotes diversifica-tion. The third condition is mostly technical as itsimply asserts that a risk measure should have acertain continuity property. The last condition as-serts that the risk should be greater than the expec-tation of a random variable as long as the randomvariable is not a deterministic constant. The choiceR(Y ) = E[Y ] is therefore not regular, which is rea-sonable as it does not capture any degree of riskaverseness. Of the examples above, the worst-caserisk, and the superquantile risk measures satisfy theconditions.2. RISK MEASURES AND UTILITY FUNCTIONSAlthough a utility function u from classical ex-pected utility theory by von Neumann and Morgen-stern leads to a “quantification” E[u(Y )] of a ran-dom variable Y , it is not natural to call this quantitya measure of risk. First, the orientation is flipped,with high values preferred to low ones. Second, theutility function distorts even a deterministic con-stant and therefore regularity cannot be achievedexcept in trivial cases. Still, important connectionsexist as we see next.To avoid the awkward inconsistency between ourorientation concerned with high values of Y andthat of utility theory, concerned with low values, wedefine an analogous concept to a utility function.A measure of regret is a functional V that as-signs to a random variableY a number V (Y ), whichmay be infinity, as a quantification of the displea-sure with the mix of possible realizations of Y . Itcould correspond to a utility function u throughV (Y ) = −E[u(−Y )], (3)but we ensure that it is anchored at zero. Hence,we insist that V (0) = 0 and V (Y ) > E[Y ] whenY is not the constant zero. The correspondenceis therefore with relative utility. Analogouslyto the regularity of risk measures, we say thata measure of regret is regular if it satisfies theclosedness, convexity, and the two above condi-tions. If the random variable is not discrete, anadditional technical condition might also be re-quired; see Rockafellar and Uryasev (2013) for de-tails. An example of a measure of regret is V (Y ) =11−αE[max{0,Y}], with α ∈ (0,1), where nega-tive realizations of Y are assigned zero regret, but312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015positive realizations are viewed increasingly “re-gretable,” with the increase being linear. This ex-pression corresponds to a piecewise linear utilityfunction with a kink at zero.Major advantages derive from the following fact(see Rockafellar and Uryasev (2013)): A regularmeasure of riskR can be constructed from a regularmeasure of regret V through the one-dimensionaloptimization problemR(Y ) = mincc+V (Y − c). (4)For example, R(Y ) = q¯α(Y ) derives from themeasure of regret V (Y ) = 11−αE[max{0,Y}],which leads to the already claimed expression(1). A large number of other measures of riskcan be constructed in a similar manner; seeRockafellar and Uryasev (2013). With the connec-tions between regret and relative utility, this im-plies that every utility function u, with u(0) = 0 andu(y) > y for y 6= 0, is in correspondence with a reg-ular measure of risk through (3) and (4).The trade-off formula (4) provides important in-terpretations of a regular measure of risk as the re-sult of a two-stage decision process involving a reg-ular measure of regret (and therefore also a relativeutility function). As an example, suppose that Ygives the damage cost of a system and the measureof regret V (Y ) quantifies our displeasure with thepossible damage costs. In (4), view c as the moneyput aside today to cover future damage costs andY − c as the net damage cost in the future. Then,c+V (Y − c) becomes the total cost consisting ofthe sum of the money put aside today plus the cur-rent displeasure with future damage costs. The riskR(Y ) is then the smallest possible total cost one canobtain by selecting the amount to put aside todayin the best possible manner. Consequently, a riskmeasure probes deeper than a measure of regret asit considers how one can mitigate displeasure.With the close connection between regret andrisk, one may be led to believe that a decision modelbased on regret would be equivalent to one based onthe corresponding risk measure. Section 6 showsthat this conclusion is incorrect.3. RISK NEUTRALITY AND UNCERTAINTYRegular measures of risk have alternative “dual”expressions; see Rockafellar and Uryasev (2013).Specifically, every regular measure of risk that ispositively homogeneous (i.e.,R(λY ) = λR(Y ) forall λ ≥ 0, which implies scale invariance) can beexpressed in the formR(Y ) = the max of E[YQ] across all Q ∈Q, (5)where Q is a random variable that is taken froma set Q of random variables called a risk enve-lope associated with the risk measure. For exam-ple, ifR(Y ) = q¯α(Y ), thenQ consists of those ran-dom variables with realizations between zero and1/(1−α) and that has expectation one. An exam-ple illustrates the formula.We consider the simple situation where the ran-dom variable Y of a system takes the value 1 withprobability 0.1 and the value 0 with probability0.9, with expected value 0.1. A risk-neutral deci-sion maker centered on the expectation would use0.1 in numerical comparisons with other systemsand requirements. Next, we consider a risk-aversedecision maker that has adopted the superquantilerisk measure with α = 0.8. Since qβ(Y ) = 0 forβ ≤ 0.9 and qβ(Y ) = 1 for β > 0.9, the formula(2) gives that R(Y ) = 0.5. A risk-averse decisionmaker with this decision model would use 0.5 incomparison with other designs. We now considerthe dual expression. In this case, with the scaling1/(1−α) = 5, (5) simplifies toR(Y ) = maximum value of 0.9 ·0 ·q1 +0.1 ·1 ·q2such that 0 ≤ q1,q2 ≤ 5,0.9q1 +0.1q2 = 1,which has the optimal solution q1 = 5/9 andq2 = 5. The maximum value then becomes0.9 · 0 · 5/9 + 0.1 · 1 · 5 = 0.5 that confirms theprevious calculation of R(Y ). More interestinglyhowever, the expression can be interpreted as theassessment made by a risk-neutral decision makerthat has a nominal distribution with probabilities0.9 and 0.1 for the realizations 0 and 1, respec-tively, but that is uncertain about the validity ofthis distribution. To compensate, she allows theprobabilities to be scaled up with a factor of atmost 5, while still making sure that they sum to412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015one, in a manner that is the least favorable. Thisrisk-neutral decision maker then makes the exactsame assessment of the situation as the risk-aversedecision maker.The alternative formula (5) helps explain asource of risk averseness: lack of trust in proba-bilistic models. In fact, this insight can help quan-tify the exact benefit of better probabilistic models.4. DESIGN OPTIMIZATIONIn design, the random variable of interest is param-eterized by a vector x = (x1, ...,xn) of design vari-ables. The goal might then be to select x such thatthe risk of the random variable is minimized, usu-ally subject to constraints on x. This leads to thedesign optimization problemminimizeR(Y (x)) subject to x ∈X ,where R is a regular risk measure applied to a re-sponse or cost random variable Y (x) depending onthe design vector x. For example, Y (x) = g(x,V),with g a (limit-state) function parameterized by thedesign vector x and a random vector V. The setX specifies constraints on x, which we for simplic-ity drop below. This formulation can be expandedto include multiple random variables and multiplemeasures of risk with few complications.A key property of regular measures of risk is thatthe canonical formulation is a convex optimizationproblem whenever Y (x) is an affine function of xfor every realization, possibly except for an eventwith probability zero, and X is a convex set. If Ris monotone, i.e., R(Y ) ≤R(Y ′) whenever Y ≤ Y ′with probability one, then linearity can be relaxedto convexity. The value of convexity of an opti-mization problem cannot be overestimated as it dra-matically improves the ability of algorithms to ob-tain globally optimal solutions efficiently. In theabsence of convexity, a globally optimal solution isusually inaccessible unless x only involves a smallnumber of variables and a huge computational ef-fort is employed.The trade-off formula (4) allows a simplifica-tion of the canonical formulation into the follow-ing equivalent form: minimize c0 +V (Y (x)− c0),where V is a regular measures of risk corre-sponding to the regular risk measure R through(4) and c0 is an auxiliary design variable to beoptimized unconstrained. This equivalent formis computationally beneficial as expressions forregret are usually simpler than those for risk.For example, if R(Y (x)) = q¯α(Y (x)), i.e., us-ing a superquantile risk measure, then V (Y (x)) =11−αE[max{0,Y (x)}] and the design optimizationproblem takes the following equivalent formminimize c0 +11−αE[max{0,Y (x)− c0}]which simply involves an expectation. If Y (x) =g(x,V) for some function g and the distribution ofV is discrete with realizations v1, ...,vJ and prob-abilities γ1, ...,γJ , then the formulation simplifiesfurther tominimize c0 +11−αJ∑j=1γjc jsubject to g(x,v j)− c0 ≤ c j, for all j = 1, ...,J0 ≤ c j, for all j = 1, ...,J,with c j, j = 1, ...,J, being auxiliary design vari-ables. The reformulation involves additional con-straints and variables, but this is outweighed by theremoval of all complicating expressions with theexception of the unavoidable function g. In fact,the formulation resembles the corresponding one inthe absence of uncertainty. Consequently, designoptimization under uncertainty using a superquan-tile risk measure is in some sense only marginallyharder than the corresponding design optimizationproblem without uncertainty.5. DESIGN EXAMPLEWe consider the simply supported truss in Figure1. Let Vk be the yield stress of member k, k =1,2, ...,7. Members 1 and 2 have lognormally dis-tributed yield stresses with mean 100 N/mm2 andstandard deviation 20 N/mm2. The other mem-bers have lognormally distributed yield stresseswith mean 200 N/mm2 and standard deviation 40N/mm2. The yield stresses of members 1 and 2 arecorrelated with correlation coefficients 0.8. How-ever, their correlation coefficients with the other512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 201576543218.66m10m10mV8Figure 1: Design of Truss−200 −180 −160 −140 −120 −100 −80 −60 −40 −20 000.0020.0040.0060.0080.010.0120.0140.016responsedensity  RegretFixedFigure 2: Pdf for fixed and regret-optimized designs.yield stresses are 0.5. Similarly, the yield stresses ofmembers 3-7 are correlated with correlation coeffi-cients 0.8. The truss is subject to a random load V8in its mid-span. V8 is lognormally distributed withmean 1000 kN and standard deviation 400 kN. Theload V8 is independent of the yield stresses. We usea joint lognormal distribution and the above corre-lation coefficients to approximate the joint distribu-tion of V = (V1,V2, ...,V8).The design vector x = (x1,x2, ...,x7), where xk isthe cross-section area (in 1000 mm2) of member k.The truss is constrained by the setX = {x | 0.5 ≤xk ≤ 2, k = 1,2, ...,7, x1 + x2 + ...+ x7 ≤ 9}, wherethe first restriction limits each member to be be-tween 500 mm2 and 2000 mm2 and the last restric-tion limits the total cross-section area.For each member, we compare load effect withcapacity through gk(x,v) = v8/ζk − vkxk, k =1,2, ...,7, where ζk is a factor given by the geometryand loading of the truss. From Figure 1, we deter-mine that ζk = 1/(2√3) for k= 1,2, and ζk = 1/√3for k = 3,4, ...,7. If gk(x,v) is positive the load ef-fect is larger than the capacity of the member. A−10 −5 0 5 10 15 20 25 30 35 4000.20.40.60.81 x 10−4responsedensity  RegretFixedFigure 3: Tails of pdf for fixed and regret-optimizeddesigns.random variable of concern might then be the re-sponse Y (x) =maxk=1,...,7 gk(x,V), which gives thehighest difference between load effect and capac-ity across all the members. In the following, weapproximate the distribution of V by the empiricaldistribution generated by an independent sample ofsize 100,000. This approximation facilitates com-putations.We initially consider the “fixed” design x =(9,9, ...,9)/7 that exactly satisfies the cross-sectionarea budget. The third row labelled “Fixed” in Ta-ble 1 gives the mean E[Y (x)] and probability offailure p(Y (x)) = prob(Y (x) > 0). The full pdf ofY (x) is given by the solid line in Figure 2. Here,and below, we smooth the discrete data using alogconcave exponential epi-spline (see Royset et al.(2013)) to construct estimates of the pdf of Y (x).Figure 3 highlights the corresponding upper tail.Although, the response is typically negative, highvalues might occur.The quality and risk of the fixed design is as-sessed using various quantifiers. First, we considerthe regret V (Y (x)) = (1/(1−α))E[max{Y (x),0}],with α = 0. This expression gives the average ca-pacity exceedance. It corresponds to a piecewiselinear utility function. The expression can also beinterpreted as the expected cost under the assump-tion that no load exceedance has a zero cost and aload exceedance has a cost proportional to the de-gree of exceedance. The last column, second rowof Table 2 gives the regret of Y (x) as V (Y (x)) =0.01156.Second, we consider the measuresof risk R(Y (x)) = q¯α(Y (x)) for α =0,0.5,0.9,0.99,0.999; see the last column of612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Design Size of member (in mm2) Mean p(Y (x))1 2 3 4 5 6 7Fixed 1286 1286 1286 1286 1286 1286 1286 -93.5 0.00044Regret 1220 1231 1316 1297 1315 1298 1323 -87.8 0.00053α = 0 1804 1805 1079 1077 1078 1080 1078 -125.9 0.00162α = 0.5 1755 1756 1099 1097 1096 1100 1097 -125.5 0.00145α = 0.9 1650 1649 1142 1138 1139 1141 1141 -122.1 0.00114α = 0.99 1465 1467 1212 1222 1210 1219 1204 -109.7 0.00060α = 0.999 1292 1288 1290 1263 1281 1296 1289 -93.9 0.00043Table 1: Designs of trussRisk measure Optimized FixedRegret 0.01118 0.01156α = 0 -125.9 -93.49α = 0.5 -98.21 -72.82α = 0.9 -65.41 -49.82α = 0.99 -28.89 -24.17α = 0.999 8.232 8.308Table 2: Regret and risk in optimized and fixed designsTable 2. We recall that q¯α(Y (x)) is essentiallythe conditional expectation of Y (x) given Y (x)is no smaller than its α-quantile. Consequently,q¯α(Y (x)) gives the average of the (1− α)100%worst responses. All these averages are well belowzero except for α = 0.999; the average of the 0.1%worst responses exceeds zero. In comparison, theaverage load exceedance is only slightly abovezero at 0.01156. The choice of α depends on thedegree of risk averseness of the decision maker.We next turn to optimization of the design. First,we minimize the regret V (Y (x)) subject to the con-straint x ∈ X . The resulting optimization prob-lem is a linear program solved in the GeneralAlgebraic Modeling System (GAMS) Distribution24.1.3, with the CPLEX 12.5.1 solver, on a laptopcomputer with 4 GB of RAM and 2.6 GHz proces-sor running Windows 7. The solver time is 0.78seconds. The optimal design, only marginally dif-ferent than the previous design, is given in row fourof Table 1. We note that both the mean and proba-bility of capacity exceedance are worst for the opti-mized design relative to the fixed design. However,the regret is 0.01118 and slightly better; see the sec-ond row of Table 2. Although similar, the pdf of theregret-optimized response is different than that forthe fixed design as seen by comparing the dottedand solid lines in Figures 2 and 3. It is interestingto note that the optimized design gives up averageresponse and worsens the probability of capacityexceedance to ensure slightly lower likelihood forhigh realizations and therefore a slightly improvedregret. The reduction in likelihood is too small tobe visible in Figure 3.Using the same computational platform, wesecond minimize the superquantile risk q¯α(Y (x))under the same constraints and obtain the de-signs of rows 5-9 in Table 1 using α =0,0.5,0.9,0.99,0.999. The solver times vary be-tween 4 and 218 seconds. As α increases, themean response worsens, but the probability of ca-pacity exceedance decreases. The resulting pdf aregiven in Figure 4, with upper tails given in Figure 5,where we leave out the case α = 0.5, which is sim-ilar to that with α = 0. We see from Table 2 thatthe optimized designs have, usually, substantiallylower risk than those of the fixed design. The effectof optimization diminishes as α increases simply todue to the fact that the fixed design happens to be abetter design in those cases.As seen in Table 1, the minimization of regretyields a rather different design than the minimiza-tion of risk. (We note that the choice of α has nobearing on the minimum-regret design as the fac-tor 1/(1−α) simply scales the regret.) Hence, al-though the measure of regret V is the foundationof the measures of risk q¯α, the latter measures ex-amine “deeper” the random variable in question by712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015−200 −180 −160 −140 −120 −100 −80 −60 −40 −20 000.0020.0040.0060.0080.010.0120.0140.016responsedensity  a=0a=0.9a=0.99a=0.999Figure 4: Pdf for risk optimized designsalso considering how to best mitigate the displea-sure of high responses. For example, the minimum-regret design in Table 1 is a substantially inferiordesign compared to the minimum-risk design, asmeasure by the corresponding q¯α. The minimum-risk design is better in this sense as it more easily al-lows mitigation of risk through an intelligent choiceof c in the trade-off formula (1).We note that a risk neutral decision maker wouldselect the design in row 5 of Table 1. In viewof the discussion in Section 4, a risk-neutral deci-sion maker that is uncertain about the underlyingprobability distribution might select one of the de-signs in the lower rows of that table. As statedabove, we approximate V by a discrete randomvariable with 100,000 possible realizations, eachwith probability 10−5. In this case, the discus-sion of Section 4 takes the following form. If thedecision maker believes that she could have esti-mated the probabilities incorrectly with a factor of1/(1−α) = 1/(1−0.99) = 100, i.e., the probabil-ity of each realization can be any number between0 and 10−5 · 100 = 10−3, then she would have se-lected the design of row 8 of Table 1. This resultsin a design that is significantly worse on average(−109.7 vs −125.9). Hence, the average worsen-ing of the response due to incomplete informationabout the distribution is−109.7−(−125.9)= 16.2.Analysis of this kind might help justifying efforts toimprove probabilistic models.ACKNOWLEDGEMENTThe authors acknowledges financial supported fromAFOSR and the second author also from DARPA.−10 −5 0 5 10 15 20 25 30 35 4000.20.40.60.81 x 10−4responsedensity  a=0a=0.9a=0.99a=0.999Figure 5: Tails of pdf for risk optimized designs6. REFERENCESCommander, C., Pardalos, P., Ryabchenko, V., Uryasev,S., and Zrazhevsky, G. (2007). “The wireless networkjamming problem.” Journal of Combinatorial Opti-mization, 14, 481–498.Dowd, K. (2005). Measuring market risk. Wiley, NewYork, NY, 2. edition.Minguez, R., Castillo, E., and Lara, J. (2013). “Itera-tive scenario reduction technique to solve reliabilitybased optimization problems using the buffered fail-ure probability.” Proceedings of ICOSSAR, G. Deo-datis, ed.Rockafellar, R. and Royset, J. (2010). “On buffered fail-ure probability in design and optimization of struc-tures.” Reliability Engineering & System Safety, 95,499–510.Rockafellar, R. and Royset, J. (2015). “Engineering de-cisions under risk-averseness.” ASCE-ASME Journalof Risk and Uncertainty in Engineering Systems, PartA: Civil Engineering.Rockafellar, R. and Uryasev, S. (2013). “The fundamen-tal risk quadrangle in risk management, optimizationand statistical estimation.” Surveys in Operations Re-search and Management Science, 18, 33–53.Royset, J., Sukumar, N., and Wets, R. J.-B. (2013).“Uncertainty quantification using exponential epi-splines.” Proceedings of ICOSSAR, G. Deodatis, ed.8

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