Technical Briefs. 1 Introduction. 876 / Vol. 129, AUGUST 2007 Copyright 2007 by ASME Transactions of the ASME

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1 Journal of Mechanical Design Technical Briefs Integration of Possibility-Based Optimization and Robust Design for Epistemic Uncertainty Byeng D. Youn 1 Assistant Professor Department of Mechanical Engineering and Engineering Mechanics, Michigan Technological University, oughton, MI bdyoun@mtu.edu Kyung K. Choi Roy J. Carver Professor kkchoi@ccad.uiowa.edu Liu Du Graduate Student liudu@ccad.uiowa.edu Department of Mechanical & Industrial Engineering, College of Engineering, The University of Iowa, Iowa City, IA 54 David Gorsich Director AMSTA-TR-N MS 63, U.S. Army National Automotive Center, Warren, MI gorsichd@tacom.army.mil 1 Corresponding author. Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECANICAL DESIGN. Manuscript received July 8, 005; final manuscript received May 4, 006. Review conducted by Zissimos P. Mourelatos. Paper presented at the 6th WCSMO World Congress of Structural and Multidisciplinary Optimization, 005. In practical engineering applications, there exist two different types of uncertainties: aleatory and epistemic uncertainties. This study attempts to develop a robust design optimization with epistemic uncertainty. For epistemic uncertainties, a possibilitybased design optimization improves the failure rate, while a robust design optimization minimizes the product quality loss. In general, product quality loss is described using the first two statistical moments for aleatory uncertainty: mean and standard deviation. owever, there is no metric for product quality loss defined when having epistemic uncertainty. This paper first proposes a new metric for product quality loss with epistemic uncertainty, and then a possibility-based robust design optimization. For numerical efficiency and stability, an enriched performance measure approach is employed for possibility-based robust design optimization, and the maximal possibility search is used for a possibility analysis. Three different types of robust objectives are considered for possibility-based robust design optimization: smaller-thebetter type (S-Type), larger-the-better type (L-Type), and nominalthe-better type (N-Type). Examples are used to demonstrate the effectiveness of possibility-based robust design optimization using the proposed metric for product quality loss with epistemic uncertainty. DOI: / Keywords: robust design, possibility, epistemic uncertainty, fuzzy, membership function 1 Introduction During the last three decades, extensive effort has been made in engineering analysis and design. Thus, design guidelines and/or standards have been modified to incorporate the concept of uncertainty into an early design stage. In response to these new design requirements, various methods have been developed to treat uncertainties in engineering analysis and, more recently, to carry out design optimization with reliability and robustness. In practical engineering applications, there are two different types of uncertainties: aleatory and epistemic uncertainties 1. Aleatory uncertainty is classified as objective and irreducible uncertainty with sufficient information on input uncertainty data, whereas epistemic uncertainty is a subjective and reducible uncertainty that stems from lack of knowledge on input uncertainty data. In general, a large amount of data is demanded to construct aleatory uncertainty for accurate uncertainty quantification. Often, it is very difficult to collect sufficient data for uncertainty quantification due to the restrictions of resources budgets, facilities, man-power, time, etc.. To handle epistemic uncertainty when modeling physical uncertainty with insufficient information, possibility-based or fuzzy set methods have recently been introduced in structural analysis and design 3,4. Accordingly, a possibility-based design optimization PBDO has been developed to consider epistemic engineering uncertainties in a design process in support of the operational framework of possibility theory 5 7. It has been reported that fuzzy operations is simpler than those required to use probability 7. When considering component-level, the possibility-based design provides a more conservative design than the probabilistic design in terms of a confidence level 7 9. A vertex method and a multilevel-cut method have been used for possibility analysis. owever, it has been reported that they could be inaccurate or expensive. To resolve their disadvantages, a new maximal possibility search MPS method 4,10,11 has been proposed for evaluating possibility constraints effectively. In the 876 / Vol. 19, AUGUST 007 Copyright 007 by ASME Transactions of the ASME Downloaded 01 Feb 008 to Redistribution subject to ASME license or copyright; see

design process, computational efficiency and stability have been improved using the enriched performance measure approach PMA+ 1, where two improvements are made over the original PMA: as an

2 design process, computational efficiency and stability have been improved using the enriched performance measure approach PMA+ 1, where two improvements are made over the original PMA: as an efficient possibilistic feasibility check and as a new numerical method for possibility analysis. In addition to the reliability or possibility of safety, another design requirement, product quality, is commonly addressed in many engineering applications. ence, various methods have been developed to estimate product quality loss for robust design, such as worst-case method arithmetic sum 13, root sum square using a Taylor series 14, Monte Carlo Simulation, experimental design techniques or Taguchi s method 15, a variability function method 16, etc. In general, product quality loss is described using the first two statistical moments of robust objective: mean and standard deviation. It has been reported in Ref. 17 that these methods have difficulties in estimating the quality loss accurately and efficiently because the order difference between mean and variance is not considered. To overcome those difficulties, the performance moment integration PMI method was proposed for three different types of robust objectives, such as smaller-thebetter type S-Type, larger-the-better type L-Type, and nominalthe-better type N-Type 18,19. owever, the PMI method is not applicable for engineering design problems with epistemic uncertainty. Therefore, this paper proposes a new metric for product quality loss in three different types of robust objectives. PBDO is then integrated to a robust design optimization with the new formulation of product quality loss for epistemic uncertainty. The MPS method and PMA+ are employed for more effectively estimating possibilistic constraints and conducting the design optimization, respectively. Two examples are used to show the feasibility of possibility-based robust design with epistemic uncertainty. Possibility-Based Design Optimization Models 1.1 General Formulation of Possibility-Based Robust Design Optimization. In general, the possibility-based design optimization 4,10,11 can be formulated as minimize CZ;d subject to G i Z;d 0 ti, i =1,...,np d L d d U, d R ndv and Z R nrv where CZ;d is the objective function, the design vector d =mz is the most likely value of fuzzy random vector, Z is the fuzzy random vector, and G i Z;d0 ti is the possibilistic constraint for the performance function G i Z;d with a target possibility of failure ti, while np, ndv, and nrv are the number of possibilistic constraints, design variables, and fuzzy random variables, respectively. For robustness of the design, the cost function in Eq. 1 is defined as CZ;d = C m Z;d + C ql Z;d where Z;d is a robust response associated with a product quality, C m Z;d is the material cost, and C ql Z;d is the quality loss cost defined as the loss that the product costs society from the time the product is released for shipment e.g., rework cost, scrap cost, maintenance cost. Redefining the possibility of the design safety using PMA with a robust objective, possibilitybased robust design optimization can be rewritten as minimize CZ;d = C m Z;d + C ql Z;d subject to G i = G i Z;d 0 ti, i =1,...,np 3 d L d d U, d R ndv and Z R nrv where G i is the ith possibilistic constraint. Fig. 1 Temporary-PDF using normal distribution fitting Z. Generation of Membership Function for Fuzzy Variables. Generating the input membership functions using the available limited set of data is a very important step for the possibility analysis and PBDO. Several methods have been proposed for creating the membership functions 8,9. This paper employs two procedures introduced in Refs. 4,5,10,11. It encompasses two steps: 1 generating the temporary-probability density function temporary-pdf of the fuzzy variable from the available data and generating the membership function of the fuzzy variable from the temporary-pdf...1 Generation of Temporary-Probability Density Function (Temporary-PDF) [4]. There are four different cases to consider in generating the temporary probability density function: 1 the upper and lower bounds, the judgment of experts subjective with the most likely value and the interval, 3 random sample data without known distribution type, and 4 random sample data with known distribution type. A more detailed description on generating a temporary-pdf is given in Ref. 4. In this paper, the fourth case is used to build the temporary-pdf using the most likelihood estimation method, as shown in Fig. 1. For uncertain variable Z, the histogram with the randomly generated fifty samples are generated using a normal distribution and the parameters of =0.6 and =0.01. Using the data, the temporary-pdf can be generated using the most likelihood estimation method... Generation of Membership Function [4]. To generate the membership function of the fuzzy variable from the temporary-pdf, two principles are used in this paper. The probability-possibility consistency principle 5 and the least conservative principle 4,9,11 are used to generate the membership function from the temporary-pdf. There are three methods to create the membership function. It depends on the degree of conservativeness to be achieved 4. In this paper, the first case the least conservative case is employed. The membership function is created from the temporary-cumulative distribution function F z z of a fuzzy variable. The membership function of the fuzzy variable satisfying both principles is defined as Z z =1 F z z 1 = F Zz z z:f Z z 0.5 F Z z z z:f Z z The membership function of the uncertain variable Z is generated using two principles in Eq. 4, as shown in Fig.. Journal of Mechanical Design AUGUST 007, Vol. 19 / 877 Downloaded 01 Feb 008 to Redistribution subject to ASME license or copyright; see

Fig. Membership function from Temporary-PDF of Z. a Quality improvement of product with aleatory uncertainty. b Quality improvement of product with epistemic uncertainty. 3 Quality Loss Function 3.

The quality loss function developed by Taguchi 15 is simply the cost of deviating from the target nominal value h t, which can be approximated as 0 C ql X;d = k h t 5 where X is the random vector, k

3 Fig. Membership function from Temporary-PDF of Z. a Quality improvement of product with aleatory uncertainty. b Quality improvement of product with epistemic uncertainty. 3 Quality Loss Function 3.1 Quality Loss Function for Aleatory Uncertainty. The quality loss function developed by Taguchi 15 is simply the cost of deviating from the target nominal value h t, which can be approximated as 0 C ql X;d = k h t 5 where X is the random vector, k is a proportionality constant, and h t is the target nominal value of the robust response vector. Different quality loss functions for aleatory uncertainty were defined to describe different quality characteristics as 19 N-Type: C ql = h t 0 h t0 + 0 S-Type: C ql = sgn 0 L-Type: C ql 1 = sgn 1/ where sgn is the signum function of =1 or 1 multiplied to properly minimize the S-Type robust objective. Using a performance moment integration PMI method 19, the mean and variation are effectively estimated through two reliability analyses at =± 3, as shown in Eq. 7. The reason to select two reliability levels has clearly been explained in Ref. 19. f hdh 1 6 h = h h =+ 3 k f hdh 1 6 h = 3 h h =+ 3 h 7 The coefficients in Eq. 7 represent the weights for numerical integration to estimate the mean and variation of the robust response. In reliability-based robust design optimization, the PMI Fig. 3 Quality improvement of product with aleatory and epistemic uncertainties method holds two major advantages against existing methods: no essian second-order sensitivity information required and computational efficiency independent of the number of random variables. 3. Quality Loss Function for Epistemic Uncertainty. The statistical moments e.g., such as mean and standard deviation are not defined for epistemic uncertaint, so that the quality loss function in Eq. 6 cannot be used for epistemic uncertainty. Thus, the quality loss function must be defined for epistemic uncertainty to further conduct possibility-based robust design optimization. Two similar metrics can be developed to define the quality loss function for epistemic uncertainty using the analogy between a probability theory and a possibility theory given in Table 1. In Table 1, m and s are, respectively, the most likely value and the variability of robust objective in a possibility theory that are statistically equivalent to the mean and variation, respectively, in a probability theory, where h r L and h r U are the respective minimum and maximum values of robust objective for a given possibility level r. These two metrics are graphically illustrated to describe the quality loss function for both aleatory and epistemic uncertainties in Fig. 3. As the quality with aleatory uncertainty can be maximized by controlling the mean and standard deviation of robust objective, the one with epistemic uncertainty can be maximized in a similar manner. Similar to Eq. 6, different quality loss functions are defined for different types of robust objective to describe different quality characteristics as N-Type: S-Type: m 0 h t0 C ql = m h t + s s 0 C ql = sgnm m m 0 + s s 0 Statistically equivalent means under the condition that probability-possibility consistency and least conservative principles are satisfied. 878 / Vol. 19, AUGUST 007 Transactions of the ASME Downloaded 01 Feb 008 to Redistribution subject to ASME license or copyright; see

Table 1 Analogy between aleatory and epistemic uncertainty Probability theory for aleatory uncertainty Possibility theory for epistemic uncertainty Mean = 1 6 h = 3 + 4 6 h+ 1 6 h Most likely =+ 3

4 Table 1 Analogy between aleatory and epistemic uncertainty Probability theory for aleatory uncertainty Possibility theory for epistemic uncertainty Mean = 1 6 h = h+ 1 6 h Most likely =+ 3 value = 1 6 h = 3 h Variation h Variability =+ 3 h m =hz m S =h L r hz m +h U r hz m 1 L-Type: C ql 1 = sgnm 1/ m m 1 0 +s 1 s 1 0 Just like the PMI method for aleatory uncertainty, the product quality loss can be evaluated by estimating the minimum and maximum values of robust objective for a given possibility level through two possibility analyses. It is suggested to have a small possibility level, which is equivalent to the probability level; i.e., =± 3. Thus, the variability of the membership function of robust objective can be estimated; in this paper, the level r is set to = Formulation of Possibility-Based Robust Design Optimization. For different types of robust objectives, possibility-based robust design optimization can be formulated as minimize CZ;d = C m Z;d + C ql Z;d subject to G i Z;d 0, i =1,...,np d L d d U, d R ndv and Z R nrv 9 where different types of quality loss function C ql Z;d in the cost function can be selected from Eq Results of Possibility-Based Robust Design Optimization Two examples are used to show the feasibility of possibilitybased robust design with epistemic uncertainty. 4.1 Mathematical Example [19]. A possibility-based robust design optimization is formulated as minimize C ql Z;d 8 subject to G i Z;d 0 ti, i = 1,,3 and Z = Z 1 8 Z 3, G 1 Z =1 Z 1 Z /0 G Z =1 Z 1 + Z 5 30 Z 1 Z 1, G 3 Z = Z Z +5 where the membership functions for fuzzy random variables are generated from temporary-pdfs using Eq. 4 4,11, which are assumed to follow N 0 i,=0.3 for i=1,; 0 1 =.0 and 0 =8.0; and ti =0.1. Three different types of the robust objective are considered in this example. For N-Type robust objective, the nominal value h t of is obtained by h t =h r L +h r U /. At the initial design, fuzzy variables are modeled using Eq. 4 for possibility-based robust design optimization, as shown in Fig. 4. The most likely values of fuzzy random variables are considered as the design parameters. Without requiring a second-order design sensitivity, the possibility-based robust design optimizations with three different types of robust objectives are successfully carried out using the robust metric for epistemic uncertainty. Results of possibilitybased design optimization show similar trend to those of reliability-based robust design optimization 19. The robust objectives for N-, S-, and L-Type are minimized, while all possibilistic constraints become feasible and active using the PMA+. The optimum design for the S-Type is the same as the one for the N-Type, since both N- and S-Type objectives are minimized at the same optimum design point. owever, as depicted in Figs. 5 7, the design optimization path is shown to be rather different, because different sensitivities of N- and S-Type objectives make two Fig. 5 Optimization history of possibility-based robust design Fig. 4 Membership function of fuzzy random variables Z 1 and Z for N-Type Journal of Mechanical Design AUGUST 007, Vol. 19 / 879 Downloaded 01 Feb 008 to Redistribution subject to ASME license or copyright; see

Table 3 Temporary-PDF for fuzzy random variables Fuzzy random variables Z i Distribution Mean Standard deviation 1: Surface roughness of the ring µm Normal 5.0 0.

5 Table 3 Temporary-PDF for fuzzy random variables Fuzzy random variables Z i Distribution Mean Standard deviation 1: Surface roughness of the ring µm Normal : Surface roughness of the liner µm Normal : Young s modulus GPa Normal : ardness BV Normal Fig. 6 Optimization history of possibility based robust design for S-Type different paths for design optimization. The L-Type robustness provides a different optimum design from the other types, as shown in Fig. 7. Detail results of possibility-based robust design optimization are displayed for only S-Type robust objective in Table. The robust objective is substantially decreased from.0 to 0.093, while G and G 3 become active and G 1 is well feasible. In Table, NFEl and NFE refer to numbers of function evaluations for robust and possibility analyses parts, respectively. 4. Piston-Ring/Cylinder-Liner System [1,]. In this example the piston-ring/cylinder-liner assembly is considered for possibility-based robust design optimization. The ring/liner assembly simulation takes as inputs the surface roughness of the ring and the liner and the Young s modulus and hardness and computes power loss due to friction. The root mean square of the asperity height is used to represent asperity roughness. The membership functions for all fuzzy random variables are generated from temporary-pdfs using Eq. 4, as shown in Table 3. The most likely values of fuzzy random variables are considered as the corresponding design parameters. There are four mechanical interests in the ring/liner system: liner wear rate, blow-by, oil consumption, and power loss due to friction. The first three are considered as possibility constraints, and the last as the robust objective. Accordingly, the possibilitybased robust design optimization is formulated as min C ql : power loss due to friction = sgn s.t. liner wear rate m 3 /s t blow-by kg/s t oil consumption kg/h t 1 m d 1,d 10 m 80 GPa d GPa Fig. 7 Optimization history of possibility-based robust design for L-Type 150 BV d 4 40 BV 11 and t =0.1. The robust objective of power loss due to friction is taken as S-Type, since smaller power loss is regarded as better ring/liner product in terms of objective function product quality. Detailed descriptions of the problem can be found in. Detail results of possibility-based robust design optimization are displayed in Table 4. Even with a highly nonlinear robust objective, the possibility-based robust design optimization is suc- Table Results of possibility-based robust design S-Type : Mathematical problem Iter. Cost m s d 1 d G 1 G G 3 NFE1 NFE Opt Feasi. Act. Act / Vol. 19, AUGUST 007 Transactions of the ASME Downloaded 01 Feb 008 to Redistribution subject to ASME license or copyright; see

6 Table 4 Results of possibility-based robust design S-Type : Ring/liner problem Iter. Cost m h s h d 1 d d 3 d 4 G 1 G G 3 NFE1 NFE Opt Feasi. Feasi. Act cessfully carried out to improve the quality of the power loss due to friction, while satisfying all possibilistic constraints. The product quality is improved by 8.%, and it mainly results from minimizing the most likely value of the power loss. To minimize the power loss, the ring surface roughness goes to the lower bound to minimize a friction, while the liner surface roughness remains at 6.53 m to maintain an optimal oil thickness. The oil consumption becomes active at the optimum. In Table 4, NFE1/NFE refer to numbers of function evaluations for robust/possibility analyses, respectively. 5 Conclusions This paper addresses the issue of robust design optimization in areas where input random data is insufficient to model random variables precisely. It is often found in practical engineering applications. The analogy between the probability and possibility theories is used to define the product quality loss for epistemic uncertainty. For the epistemic uncertainty, the most likely value and equivalent variation are employed to define the new metric for the product quality loss in three different types of robust objectives. Possibility-based design optimization was then successfully integrated to a robust design optimization, called a possibilitybased robust design optimization. The proposed possibility-based robust design optimization was successfully conducted with two examples: a mathematical example and a piston-ring/cylinderliner system. It was found that the new metric of the product quality loss for epistemic uncertainty enables us to improve product quality for different types of robust objectives. Acknowledgment Research is partially supported by the STAS contract TCN- 051 sponsored by the U.S. Army TARDEC. Nomenclature CZ;d cost function for possibility-based robust design optimization C m Z;d material cost at a given design C ql Z;d quality loss cost due to manufacturing variability d design parameter; d=d 1,d,...,d n T F X x cumulative distribution function of X f X x probability density function PDF of the random parameter GZ performance function; the design is considered fail if GZ0 G Z;d possibilistic constraint,h t robust response vector and its target nominal vector h r L and h r U minimum and maximum values of robust objective for a given r, respectively k proportionality constant for quality loss function m most likely value of for epistemic uncertainty s variability of for epistemic uncertainty np number of possibilistic constraints ndv,nrv numbers of design variables and fuzzy random variables, respectively Z fuzzy random vector epistemic; Z=Z,Z,...,Z n T z realization of Z; z=z 1,z,...,z n T X random vector aleatory, X=X 1,...,X n T, x realization of X, x=x 1,...,x n T U,u standard normal random vector, U=U 1,...,U n T, and its realization t, r target possibility of failure and possibility levels for robust objective mean of random vector X; = 1,,..., n T, mean and standard deviation of output response, respectively z z membership function of fuzzy variable Z References 1 elton, J. 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7 Optimization Under Uncertainty Using Inverse Reliability Strategy, ASME J. Mech. Des., 164, pp Youn, B. D., Choi, K. K., and Yi, K., 005, Performance Moment Integration PMI Method for Quality Assessment in Reliability-Based Robust Design Optimization, Mech. Based Des. Struct. Mach., 33, pp Chandra, M. J., 001, Statistical Quality Control, CRC Press, Boca Raton, FL, Chap Youn, B. D., Choi, K. K., Kokkolaras, M., Papalambros, P. Y., Mourelatos, Z., and Gorsich, D., 004, Techniques to Identify Uncertainty Propagation for Probabilistic Design of Multilevel Systems, 10th AIAA/ISSMO Symposium on MAO, AIAA , Albany, NY, Aug. 30 Sept. 1. Chan, K. Y., Kokkolaras, M., Papalambros, P., Skerlos, S. J., and Mourelatos, Z., 004, Propagation of Uncertainty in Optimal Design of Multilevel Systems: Piston-Ring/Cylinder-Liner Case Study, Proceedings of SAE World Congress, Detroit, MI, March 8 11, Paper No / Vol. 19, AUGUST 007 Transactions of the ASME Downloaded 01 Feb 008 to Redistribution subject to ASME license or copyright; see

A New Robust Concept in Possibility Theory for Possibility-Based Robust Design Optimization

7 th World Congresses of Structural and Multidisciplinary Optimization COEX Seoul, May 5 May 007, Korea A New Robust Concept in Possibility Theory for Possibility-Based Robust Design Optimization K.K.