Design Optimization with Imprecise Random Variables

2009-01-0201

Jeffrey W. Herrmann University of Maryland, College Park

ABSTRACT using precise probability distributions or representing uncertain quantities as fuzzy sets. Haldar and Design optimization is an important engineering design Mahadevan [7] give a general introduction to reliability- activity. Performing design optimization in the presence based design optimization, and many different solution of uncertainty has been an active area of research. The techniques have been developed [8, 9, 10]. Other approaches used require modeling the random variables approaches include evidence-based design optimization using precise probability distributions or representing [11], possibility-based design optimization [12], and uncertain quantities as fuzzy sets. This work, however, approaches that combine possibilities and probabilities considers problems in which the random variables are [13]. Zhou and Mourelatos [11] discussed an evidence described with imprecise probability distributions, which theory-based design optimization (EBDO) problem. are highly relevant when there is limited information They used a hybrid approach that first solves a RBDO to about the distribution of a random variable. In particular, get close to the optimal solution and then generates this paper formulates the imprecise probability design response surfaces for the active constraints and uses a optimization problem and presents an approach for derivative-free optimizer to find a solution. solving it. We present examples for illustrating the approach. The amount of information available and the outlook of the decision-maker (design engineer) determines the INTRODUCTION appropriateness of different models of uncertainty. No single model should be considered universally valid. In Design optimization is an important engineering design this paper, we consider situations in which there is activity in automotive, aerospace, and other insufficient information about the random variables to development processes. In general, design optimization model them with precise probability distributions. determines values for design variables such that an Instead, imprecise probability distributions (described in objective function is optimized while performance and more detail below) are used to capture the limited other constraints are satisfied [1, 2, 3]. The use of information or knowledge. In the extreme case, the design optimization in engineering design continues to imprecise probability distribution may be a simple increase, driven by more powerful software packages interval. This paper presents an approach for solving and the formulation of new design optimization problems design optimization problems in which the random motivated by the decision-based design (DBD) variables are described with imprecise probability framework [4, 5] and the corresponding idea of design distributions because there exists limited information for market systems [6]. about the uncertainties.

Because many engineering problems must be solved in IMPRECISE PROBABILITIES the presence of uncertainty, developing approaches for solving design optimization problems that have uncertain In traditional probability theory, the probability of an variables has been an active area of research. The event is defined by a single number between in the approaches used require modeling the random variables range [0, 1]. However, because this may be inappropriate in cases of incomplete or conflicting

The Engineering Meetings Board has approved this paper for publication. It has successfully completed SAE’s peer review process under the supervision of the session organizer. This process requires a minimum of three (3) reviews by industry experts. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of SAE. ISSN 0148-7191 Positions and opinions advanced in this paper are those of the author(s) and not necessarily those of SAE. The author is solely responsible for the content of the paper. SAE Customer Service: Tel: 877-606-7323 (inside USA and Canada) Tel: 724-776-4970 (outside USA) Fax: 724-776-0790 Email: [email protected] SAE Web Address: http://www.sae.org Printed in USA information, researchers have proposed theories of DESIGN OPTIMIZATION WITH IMPRECISE imprecise probabilities. For these situations, PROBABILITIES probabilities can be intervals or sets, rather than precise numbers [14, 15, 16]. The theory of imprecise In the imprecise probability design optimization (IPDO) probabilities, formalized by Walley [15], uses the same problem, there is a set of deterministic design variables fundamental notion of rationality as the work of de Finetti for which the designer chooses values and a set of [17, 18]. However, the theory allows a range of random variables, which may be manufacturing errors, indeterminacy—prices at which a decision-maker will not uncertain engineering parameters, or other sources of enter a gamble as either a buyer or a seller. These in uncertainty. Unlike other work, this formulation does not turn correspond to ranges of probabilities. include in the model “random design variables.” Such variables are typically those in which the designer Imprecise probabilities have previously been considered chooses the mean, but the actual value is random. In in reliability analysis [19, 20, 21] and engineering design the IPDO formulation presented here, each such [22, 23, 24]. Aughenbaugh and Herrmann [25, 26, 27] quantity is modeled with two quantities: a deterministic have compared techniques using imprecise probabilities design variable and a random parameter that represents to other statistical approaches for making reliability- the error of that variable. This does not limit the scope based design decisions. The work described in this of the model. For instance, suppose we have a “random paper builds upon these previous results. design variable” X that is a dimension of a part. The mean of X, denoted μ , is chosen by the designer, but In many engineering applications, the relevant random X the dimension is a normally distributed random variable variables (e.g., parameters or manufacturing errors) are with a standard deviation of σ . Examples of this type of continuous variables. One common way to represent the imprecision in the probability distribution of such a variable have been considered in Zhou and Mourelatos random variable is a probability box (“p-box”) that is a [12] and elsewhere. In this formulation, we replace the set of cumulative probability distributions bounded by an variable X with X = dZXX+ , where the first term, which upper distribution F and a lower distribution F . These corresponds to the mean, is a deterministic design bounds model the epistemic uncertainty about the variable, and the second term is a random variable that probability distribution for the random variable. Of is normally distributed with a mean of 0 and a standard course, a traditional precise probability distribution is a deviation of σ . special case of a p-box, in which the upper and lower bounds are equal. The general IPDO is formulated as follows:

There are multiple ways to construct a p-box for a minVf⎣⎦⎡⎤(dZ , ) random variable [28]. In some cases, the type of d distribution is known (or assumed) but its parameters s.t. Pg{}ii()dZ ,≤≤ 0 p i = 1,… , n (1) are imprecise (such as an interval for a mean). In other dddLU≤≤ cases, the distribution is constructed from sample data.

Additionally, one can create a p-box from a Dempster- k Shafer structure, in which intervals (not points) within the In this formulation, d ∈ R is the vector of deterministic range of the random variable are assigned probabilities. design variables, and Z ∈ Rr is the vector of random For more about p-boxes and the link between p-box variables that have imprecise probability distributions. representation and Dempster-Shafer structures, see The probabilistic constraints are functions of the Ferson et al. [29]. deterministic design variables and the random variables.

We want gi (dZ,0) ≥ (which is the “safe region”) but will Functions of random variables that have imprecise be satisfied if the upper bound on the failure probability probability distributions also have imprecise probability is less than the target p . We choose the upper distributions. Methods exist for calculating these i convolutions [30, 31, 32, 33]. Wang [34] proposes a probability in order to be conservative. new interval arithmetic that could be used as well. The function f is the system performance, which may be Therefore, p-boxes are a very general way to represent random, in which case the function V is a moment of that uncertainty. For computational purposes, in the random performance, such as the upper limit for the approach below, we will convert a p-box into a mean; thus V is a deterministic function of d . In many “canonical” Dempster-Shafer structure [28], which will cases, the objective is specified as a function of only the necessarily be bounded. deterministic design variables, in which case we get the following formulation:

min f ()d SOLUTION APPROACH d s.t. Pg{}ii()dZ ,≤≤ 0 p i = 1,… , n (2) To solve the IPDO, we will use a sequential approach similar to that of Du and Chen [35] and Zhou and dddLU≤≤ Mourelatos [12]. First, this formulation has the usual difficulty of A key part of the approach is to solve the following computing the failure probability for each constraint. deterministic optimization problem P given values for the Analytically evaluating the failure probability is possible ik() only in special cases. An additional complication is the random variables in each constraint in S2 . Let Z be imprecision of the random variables, which makes specific values for the random variables in constraint applying standard RBDO techniques difficult. iS∈ 2 in iteration k.

Now, Z has an imprecise joint probability distribution, min f (d ) which can be considered as a set H of precise joint d probability distributions. For any precise joint probability s.t. Fii()bpiS()d ≤ i ∈ 1 (4) distribution Fj ∈ H , let Pgji{}()dZ,0≤ be the g dZ,0i(k) ≥∈ iS probability of violating constraint i when Z has that i () 2 precise joint probability distribution. Then, we could dddLU≤≤ reformulate the IPDO as the following RBDO: i(k) In the space of the design variables, the gi (dZ,0) ≥ min f ()d d constraints move the boundaries of the “safe region” (by s.t. Pgji{}()dZ ,≤≤ 0 pi i = 1,… , nF , j ∈ H (3) making it smaller) in order to reduce the probability of LU failure. However, it is still necessary to determine the ddd≤≤ probability of failure and compare it to the target. If it is too large, then we have to move that constraint some Unfortunately, because of the large number of more. constraints, this reformulation is not helpful unless the set H is limited to a reasonable number of “extreme” Given these preliminaries, the complete approach distributions that can be used as surrogates for the follows: entire set. Research on this topic is ongoing and may provide a way to increase the computational efficiency of 1. Let k = 0 . For iS∈ , let each component of Z ik() IPDO in the future. 2 equal a value within the range of its expected value. Due to these difficulties, we will pursue a numerical approach. To do this, we will first partition the 2. Solve P to get the solution d (1)k + . constraints gi ()dZ,0≥ into two sets. Set S1 includes (1)k + any constraint that can be rearranged so that 3. For iS∈ 2 , evaluate Pg{}i ()dZ,0≤ . If gahbdZ, Z d , where a is a positive iiii()=−() ()() i (1)k + Pg{ ii(dZ,0) ≤ } ≤ p for all iS∈ 2 , then the design scalar. Note gi ()dZ,0≤ if and only if hbii()Zd≤ ( ) . point is feasible; stop. Otherwise, for all constraints The constraints that cannot be rearranged in this way iS∈ where PgdZ(1)k + ,0≤≤ p, set ZZik(1)+ = ik (). are placed in set S2 . 2 { ii( ) } For the others, find Z ik(1)+ by solving the following For each constraint in S1 , we will perform the problem: convolution needed to get the imprecise distribution of (1)(1)kik++ hi ()Z by combining the Dempster-Shafer structures for ming dZ , ik(1)+ i ( ) the relevant random variables. Because the upper Z (5) cumulative probability distribution will be a discontinuous (1)kkik+++ (1)(1) s.t. Pg{} ii()(dZ ,≤ g dZ , )= p i function, we will approximate it with F xPh≈≤Z x. Therefore, we can replace each ii() {}() This yields a “very bad” (but not worst-case) value of those random variables used in that constraint. (A of the constraints in S1 by Fii()bp()d ≤ i. technique for solving this problem is described below.)

4. kk= +1. Repeat steps 2 and 3 until a feasible design point is found.

At this point we have no proof that the algorithm will ⎣⎡−+11.5(kk − 1/99,0.51.5)() − + − 1/99⎦⎤ , for k = 1,… ,100 converge, and the approach may fail on problems with irregular objective functions and constraints. Further (each interval has a probability of 0.01). Therefore, they analysis and experimentation is needed to study this can range from -1 to 1. The distribution of random aspect of the method. parameter z3 is characterized by the intervals ⎣⎡1+− 0.5(kk 1)() / 99,1.5 +− 0.5 1 / 99⎦⎤ , for k = 1,… ,100 We use the following reliability analysis technique to (each interval has a probability of 0.01). This random (1)k + ik(1)+ determine if Pg{}ii()dZ,0≤≤ p and to find Z . parameter ranges from 1 to 2. This reliability analysis technique corresponds roughly to solving the inverse “most probable point” problem [35] or For z3 , we will approximate its upper cumulative finding the “shifting vector” [12]. Given d (1)k + , set probability as follows:

p f = 0 Without loss of generality, we assume that ⎧ 0if x < 1 g dZ(1)k + , is a function of m random variables ⎪ i () Pz{}3 ≤= x⎨2-1() x if 1 ≤≤ x 1.5 (7) ⎪ Z1,,… Zm . The Dempster-Shafer structure of Zi is ⎩ 1if x > 1.5 represented by ni equally likely intervals. Let * The IPDO solution approach begins with z and z both Nnnn= 12 m . Let NpN= ⎣⎦⎢⎥i be the number of values 1 2 2(0) and intervals to save. Consider each of the N set to zero; that is, Z = (0,0) : combinations of intervals for the random variables. For each combination, let each random variable range over min dd12+ min (1)k + d its interval and find gi , the minimum of gi ()dZ, ⎧⎫4 min s.t. Pz−≤≤ 0 0.02 for that combination. If g ≤ 0 , add 1 (the probability ⎨⎬3 (8) i N ⎩⎭d1 of that combination) to p f . As the N combinations of 2 dd12≥ 20 * intervals are checked, keep the N smallest minima dddLU≤≤ found along with the values of Z1,,… Zm that yield those *(1) minima. The final value of p f is used to estimate This yields d = (3.9604,1.2751) , but the upper Pg dZ(1)k + ,0≤ . We set Z ik(1)+ equal to the values probability of violating the second constraint is too high. {}i () Our reliability analysis technique estimates that of Z ,,… Z that yield the largest of the N* smallest 2 1 m Pdzdz20−+*(1) *(1) +≤= 0 0.7830 . Because minima found. { ()()1212} there are 10,000= 1002 combinations of intervals, we EXAMPLES keep the worst 200 interval lower bounds. 2(1) 2(1) This section presents three examples to illustrate the ( zz12,) =−() 0.5303, − 0.9697 gives the best of these IPDO solution method. The first example has two worst. (The first superscript refers to the second design variables and three random variables: constraint, the second to the iteration number.)

min dd12+ Now we solve by adding the shifting vector to the d second constraint: ⎧⎫4 s.t. Pz⎨⎬3 −≤≤ 0 0.02 ⎩⎭d1 (6) min dd12+ d 2 Pd{}()()11++−≤≤ z d 2 z 220 0 0.02 ⎧⎫4 s.t. Pz⎨⎬3 −≤≤ 0 0.02 LU d (9) ddd≤≤ ⎩⎭1 2 ()()dd12−−≥0.5303 0.9697 20 The bounds d L = 0.01,0.01 and dU = 10,10 . Note () () dddLU≤≤ that the first constraint is in set S1 , whereas the second *(2) one is in set S2 . This yields d = (3.9604,2.6696) . Our reliability analysis technique estimates that All three random variables have imprecise probability 2 Pdzdz20−+*(2) *(2) +≤= 0 0.0655 . Thus, the distributions. The random errors z1 and z2 have the { ()()1212} same distribution, each characterized by the intervals upper probability of violating the second constraint is lower but still too high. We also determine that 2(2) 2(2) ()zz12,=−() 0.7727, − 0.9545 gives the best of the worst for this design point.

Now we solve with the new shifting vector:

min dd12+ d ⎧⎫4 s.t. Pz⎨⎬3 −≤≤ 0 0.02 ⎩⎭d1 (10) 2 ()()dd12−−≥0.7727 0.9545 20 dddLU≤≤

This yields d*(3) = ()4.1927,2.6645 . The first constraint is not active, but the upper probability of violating the Figure 1. The p-box for z (and z ). second constraint is now acceptable. Our reliability 1 2 analysis technique estimates that First, we will solve the deterministic optimization problem *(3)2 *(3) with both random variables replaced by 0. We get Pdzdz20−+()()1212 +≤= 0 0.0194 . So the {} d = (3.1139,2.0626) , the same optimal solution as Zhou solution is feasible, and the algorithm stops. and Mourelatos [12]. The objective function value is 5.1765. The second example is the mathematical example from Zhou and Mourelatos [12]. In our version, the problem i(0) Next, we let Z = (0,0) for all iS∈ 2 and solve problem has two design variables d = ()dd12, and two random P. This yields the solution d (1) = (3.1139,2.0626) . variables, the error for each one: Z = ()zz12, . The Pg dZ(1) ,0≤ is greater than 0.02 for the first two bounds are 010≤≤di for both design variables. The { i ( ) } objective is to minimize the sum of the design variables. constraints but equals zero for the third constraint. In the terms of our general IPDO, we have the following relationships: When evaluating this probability, we have to compare 10,000 combinations, in which each combination has an

fdd()d =+12 interval from the p-box for z1 and an interval from the p- 2 box for z2 . For each combination, we must find the gdzdz11122()()()dZ,20=+ +− minimal value of gi over those values of z1 and z2 . gdzdzdZ,4=+++− 52 (11) 21122()( ) Based on the mathematical analysis of the constraints, it 2 is possible to develop simple rules to identify the values ++−−−()dzdz11 2 212 − 120 2 of z1 and z2 that give the minimum for that combination. gdzdz31122()dZ,75=− ( + )( − 8 + ) Let ⎡zzmin, max ⎤ and ⎡zzmin, max ⎤ be the intervals that Both random variables have the same imprecise ⎣ 11⎦ ⎣ 22⎦ probability distribution, which is characterized by the form the combination. For the first constraint, the min intervals ⎣⎦⎡⎤−+11.5()kk − 1/99,0.51.5 − + () − 1/99, for minimum is found at zz22= , and z1 is either an k = 1,… ,100 (each interval has a probability of 0.01). endpoint of the interval or −d1 . For the second Therefore, they can range from -1 to 1. constraint, the minimum is found at one of the following min min min min five points: (zz12, ) , ()6.4−−dd1222 0.6 − 0.6 zz , , We cannot separate any of the constraints, so S = ∅ 1 max min max max ( zz12, ) , ( zddz1121,1.6−−− 0.6 0.6 ) , or and S2 = {}1, 2, 3 . The two random variables have the max max same imprecise probability distribution, which is ( zz12, ) . For third constraint, the minimum is found approximately an imprecise uniform distribution with a max lower bound in the range [-1, 0.5] and the upper bound at zz22= , and z1 is one of the endpoints of its in the range [0.5, 1]. Figure 1 shows the actual p-box. interval.

1(1) 1(1) We set p = 0.02 for i = 1,… , 5 . The bounds on the From this algorithm we set ()zz12,0.7121,1=−() − and i design variables are the following ranges: 524≤ R ≤ , 2(1) 2(1) ()zz12,=−() 0.8939, 0.8182 . We will use these two 10≤ L ≤ 48 , and 0.25≤ t ≤ 2 . The last three constraints, vectors in the first two constraints as we try to find a which form set S1 = {3, 4, 5} , can be rearranged as feasible solution in the next iteration. For the third follows: constraint, which was already feasible, we let zz3(1),0,0 3(1) = . ()12 () ghzb3313(dZ,2()) =−( ( d)) =−−230zLRt1 ++− The second iteration of the problem yields the solution ()1 ()2 (13) (2) d = ()3.5836,3.4255 . At this point, the objective ghzbzRt44141()dZ,()=−=−−+− () d ( 12 )

(1) ghzbztR55151()dZ,()=−=−− ()( d 5 ) function equals 7.0091. Pg{}i ()dZ,0≤ is again greater than 0.02 for the first two constraints but equals Therefore, S = {1, 2} . Each of the three random zero for the third constraint. We set 2 variables has an imprecise probability distribution. The 1(2) 1(2) and 2(2) 2(2) . ()zz12,=−() 1, − 0.5909 ()zz12,=−() 0.3939, 1 imprecise probability distribution of the internal pressure For the third constraint, zz3(2),0,0 3(2) = (). P is approximately an imprecise normal distribution. The ()12 imprecise mean has a range of [975, 1025]. The standard deviation is precisely 50. The imprecise The third iteration of the problem yields the solution probability distribution of the material yielding strength Y d (3) = ()3.6113,3.5240 . At this point, the objective is also approximately an imprecise normal distribution. The imprecise mean has a range of [253500, 266500]. (1) function equals 7.1353. Pg{}i ()dZ,0≤ is less than The standard deviation is precisely 13000. 0.02 for the first two constraints and equals zero for the The actual p-boxes used for these two random variables third constraint, so the solution is feasible. are constructed as follows: for k = 1,… ,100 , let

The third example that we consider is the optimization of fkk =−(21/200) , which therefore ranges from 0.005 to a thin-walled pressure vessel. Our formulation is based 0.995. The k-th interval in the p-box for the internal on the RBDO formulation of Zhou and Mourelatos [11]. pressure P is ⎡975+Φ 50−−11f ,1025 +Φ 50 f ⎤ , and The problem was originally introduced by Lewis and ⎣ ()kk ()⎦ Mistree [36]. The problem has three design variables: the k-th interval in the p-box for the material yielding the radius R, the mid-section length L, and the wall strength Y is thickness t. The objective is to maximize the volume of ⎡253500+Φ 13000−−11()f ,266500 +Φ 13000 ()f ⎤ . the pressure vessel. Five constraints ensure that the ⎣ kk⎦ design is strong enough to resist the internal pressure (with a safety factor of 2) and meets geometric The imprecise probability distribution of z1 , the requirements. manufacturing error of the radius, is based on data given by Zhou and Mourelatos [11], who assume that we have In our formulation there are three random variables: the 100 sample points for the error, but the data are given manufacturing error of the radius ( z1 ), the internal only in bins as follows: 3 points are in the range [-4.5, - pressure P, and the material yielding strength Y. 3], 45 points are in the range [-3, 0], 49 points are in the range [0, 3], 2 points are in the range [3, 4.5], and 1 In the terms of our general IPDO, we have d=(RLt,,) point is in the range [4.5, 6]. Figure 2 shows the corresponding p-box for z1 and the curve we use for and Z = zPY,, and the following relationships: ()1 approximating the upper bound of this p-box. An approximation is created for each part of the p-box. For fRRL()d =+4 ππ32 ⎡ LU⎤ 3 x ∈ ⎣zz, ⎦ where the lower left corner of the upper 1 gtYPRzt()dZ,22=− ++ LL 11()2 bound is ( zF, ) and the upper right corner of the 2 gRzttY()dZ,2=++ ( ) UU 21() upper bound is ( zF, ) , the approximation 2 2 −22PRz()() ++++ 2 Rztt (12) 2 ()11 ⎛⎞xz− L Fx=− FUUL F − F 1 − . Figure 3 shows i () ()⎜⎟UL gLRzt31()dZ,602=−+() ( + ) + 2 ⎝⎠zz−

gRzt41()dZ,12=−++ ( ) the corresponding p-box for −z1 and the curve we use for approximating the upper bound of this p-box. gRzt51()dZ,5=+−

Because F ()xPh≥≤()Z x, the approximation (1) ii{} shows that, for all iS∈ 2 , Pg{}i ()dZ,00≤ = , so the reduces the feasible region. If a solution is feasible with solution is feasible, and the algorithm stops. respect to the approximation, then it is feasible with respect to the original p-box. COMPARISON TO RBDO

The IPDO addresses situations in which probability distributions are not precise. An alternative approach is to use a traditional RBDO approach while varying the probability distributions of the random variables. The basic idea is to loop over different combinations of the distributions for the random variables. For each combination, we solve a traditional RBDO problem to get a solution. This procedure will yield a set of solutions and gives the designer some idea of where good solutions lie. But it is not clear how a designer should select a solution from this set.

Another alternative is to remove the imprecision. For instance, one can replace each imprecise probability

distribution by the maximum entropy probability distribution that fits within the p-box. This yields an Figure 2. The p-box for z1 and the approximation for its upper bound. RBDO problem. For the first example in Section 5, we can model z1 , z2 , and z3 with uniform distributions.

The range for z1 and z2 is [-1, 1], and the range for z3 is [1, 2]. Solving the RBDO problem yields the solution d = (4.2227,2.5132) . The objective function value is 6.7359, which is better than that of the more conservative IPDO solution, but the probability of failure of this new solution is greater than the desired target for some of the probability distributions in the p-boxes of the random variables.

CONCLUSION

This paper introduced the imprecise probability design optimization (IPDO) problem, in which there is a set of deterministic design variables for which the designer chooses values and a set of imprecise random Figure 3. The p-box for −z1 and the approximation for variables, which may be manufacturing errors, uncertain its upper bound. engineering parameters, or other sources of uncertainty. This paper presented a sequential approach for solving First, we will solve the deterministic optimization problem this problem. To avoid unnecessary calculations, the with all three random variables replaced by constants: approach partitions the constraints into two sets. By Z = ()zPY1,, = (0, 1000, 260000). d = ()RLt,, = (11.75, exploiting their special structure, the cumulative 36, 0.25) is the optimal solution that we found. The probability distributions for constraints in the first set are pressure vessel volume equals 22,410. Note that the calculated only once and then replaced with an probability of failure for constraints 4 and 5 equals the approximation. After this, the approach solves a series probability that −z1 is less than equal to zero, which is of deterministic optimization problems and shifts imprecise but can be quite large, so it is not a feasible selected constraints in each iteration in order to reduce solution to the IPDO problem. the probability of failure. We have used examples to illustrate the usefulness of Next, we let Z=Z1(0) 2(0) = 0,1000, 260000 and solve () the approach. The results show that the proposed IPDO problem P. This yields the solution approach finds high-quality feasible solutions, though d (1) = ()6.7606,36.4396,0.3186 . (The optimization the computational effort is increased because of the required 348 function evaluations.) The pressure vessel computational effort of the reliability analysis technique volume equals 6,527. The reliability analysis technique and the iterations needed to converge to a solution.

Although this work was motivated by problems in which 5. Renaud, J.E., Gu, X. Decision-Based Collaborative only imprecise probability distributions are available, the Optimization of Multidisciplinary Systems. In approach’s use of Dempster-Shafer structures makes it Decision Making in Engineering Design, W. Chen, K. compatible with other approaches within the domain of Lewis, and L.C. Schmidt, editors, ASME Press, New evidence theory as well [37]. In the examples given York, 2006. here, all but one of the imprecise random variables have 6. Shiau, Ching-Shin Norman, Michalek, Jeremy J. known distributions with interval parameters. However, (2009) Should Designers Worry About Market as shown in the last example, the approach can be used Systems? Journal of Mechanical Design, 131(1). for problems with any type of imprecise random variable; 7. Haldar, Achintya, Mahadevan, Sankaran, the computational effort will not change. Probability, Reliability, and Statistical Methods in Engineering Design, John Wiley & Sons, Inc., New Many RBDO and similar studies have developed York, 2000. sequential approaches, and we followed this practice as 8. Liang, Jinghong, Mourelatos, Zissimos P., well because it reduces the computational effort needed Nikolaidis, Efstratios (2007). A Single-Loop to find a solution. As mentioned earlier, we cannot prove that the approach will converge, which is a Approach for System Reliability-Based Design general disadvantage of this strategy. In addition, our Optimization. Journal of Mechanical Design, 129. approach may be too conservative because it uses the 9. Tu, J., Choi, K.K., Park, Y.H. (1999). A New Study upper probabilities in the constraints. Additional errors on Reliability-Based Design Optimization. Journal of may be introduced by using approximations for the Mechanical Design, 121. upper probabilities. 10. Youn, Byeng D. (2007). Adaptive-Loop Method for Non-Deterministic Design Optimization. Proceedings The scalability of the proposed approach depends of the Institution of Mechanical Engineers, Part O: primarily upon the convolutions needed to determine the Journal of Risk and Reliability, 221. 11. Zhou, Jun, Mourelatos, Zissimos P. Design under imprecise distributions of the hi ()Z . If there are many Uncertainty Using a Combination of Evidence imprecise random variables that interact in the same Theory and a Bayesian Approach. Third convolution, the reliability analysis will require more International Workshop on Reliable Engineering effort. Computing (REC), NSF Workshop on Imprecise Future work will focus on improving the computational Probability in Engineering Analysis & Design, 2008. efficiency and stability of the approach by considering 12. Zhou, Jun, Mourelatos, Zissimos P. (2008). 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