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

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1 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. Choi *, Liu Du, Ikin Lee 3, and David Gorsich 4 3 Department of Mechanical & Industrial Engineering College of Engineering The University of Iowa Iowa City, IA 54, U.S.A. kkchoi@engineering.uiowa.edu liudu@engineering.uiowa.edu ilee@engineering.uiowa.edu 4 U.S. Army RDECOM/TARDEC AMSRD-TAR-N, MS East Mile Road Warren, MI , U.S.A. gorsichd@tacom.army.mil. Abstract Whereas the robust design concept has been well established in the probability theory, it has not been developed in the possibility theory. For problems where accurate statistical information for input data is not available, a possibility-based (or fuzzy set) robust design concept is proposed in this paper by investigating the similarity between the membership function of the fuzzy variable and the cumulative distribution function of the random variable. Based on the probability-possibility consistency principle, a random variable that corresponds to the fuzzy variable is introduced in this paper in order to define the robust design concept for the fuzzy variable. For the system with input fuzzy variables, the robustness measure of the output performance is computed using the performance measure integration (PMI)-like method, while the integration points are obtained from the inverse possibility analysis by using the maximal possibility search method with interpolation (MPS). The proposed robust design concept in the possibility theory is used to formulate a possibility-based robust design optimization (PBRDO) method. Several numerical examples are used to verify the robust design concept in the possibility theory and the PBRDO formulation.. Keywords: Possibility theory; robustness measure; maximal failure search method; possibility-based robust design optimization. 3. Introduction Structural analysis and design optimization have recently been extended to use stochastic approach to consider various types of input uncertainties. For problems where it is not possible to produce accurate statistical information for input data, the possibility-based (or fuzzy set) methods have recently been developed in structural analysis and design optimization [-5]. The maximal possibility search method with interpolation (MPS) [4] was developed to perform the inverse possibility analysis for possibility-based design optimization (PBDO) [3-4]. When the system has both uncertainties with sufficient input data and uncertainties with insufficient input data, the mixed (random and fuzzy) variable design optimization (MVDO) [5] problem was formulated using the conditional probability theory. The maximal failure search (MFS) [5] method was developed to solve the inverse analysis problem for MVDO. Due to the competitive market, manufacturers are also trying to improve the quality of their product designs. For the system with input random variables with sufficient statistical information, the quality is measured by the output variance. To improve both reliability and quality, the reliability-based robust design optimization (RBRDO) [6-] was formulated and solved using the root sum squares (RSS) method [6,7,]; performance measure integration (PMI) [8,]; dimension reduction method (DRM) [3,4,]; and percentile difference method (PDM) [9-]. Comparison of PMI, DRM, and PDM was presented in []. owever, the robust design concept in the possibility theory has not been defined in the literature [5]. In the

2 probability theory, the variance is defined as the mean-centered integration, or centered second moment. This definition takes advantage of the probability theory, i.e., summation to unit. Thus, the probability density is similar to the weight, and the integration is averaged in the probability sense. Unfortunately, instead of the property of summation to unit, the possibility theory has the property of maximum to unit, and the multiplication is defined as the minimum [6-9]. Thus, the possibility integration [9] cannot be used for the robust design concept, or variation. In this paper, we investigate the similarity of the membership function of the fuzzy variable and the cumulative distribution function (CDF) of the random variable. Using the boundedness and convexity of the unit membership function, we inversely generate a CDF from the membership function of the fuzzy variable. Based on the probability-possibility consistency principle, we construct a random variable corresponding to the fuzzy variable. We propose to define the robustness measure of a fuzzy variable as the variance of the corresponding random variable. For the system with input fuzzy variables, a PMI-like method can be developed to compute the robustness measure of the output performance fuzzy variable through the corresponding input random variable. Instead of the inverse reliability analysis for PMI, the inverse possibility analysis by MPS algorithm is used to determine the Gaussian integration points. A method to provide conservative estimation of the robustness for problems with lack of input uncertainty information is developed. For design optimization, the sensitivity of the output robustness measure is also investigated in this paper. 4. Robust Concept in Possibility Theory In this section, we first briefly introduce the possibility theory. In order to propose the robust design concept in the possibility theory, similarity between the membership function and the cumulative distribution function (CDF) is investigated, and the random variable that corresponds to the fuzzy variable is introduced. 4.. Possibility Theory Possibility is a subective measure that expresses the degree to which a person considers that an event can occur [0]. The possibility measure Π should satisfy the following axioms: boundary requirement, monotonicity, and union measure. The membership function of the fuzzy variable represents the grade of membership. The membership function Π ( y ) of the fuzzy variable X should be bounded between 0 and, and Π ( A) = max{ Π ( y)}, A R () y A A fuzzy variable with the membership function Π ( y ) is said to satisfy the unity if and only if there exists unique y= d such that Π ( y ) =. This unique value d is called the maximal grade point. A fuzzy variable with the membership function Π ( y ) is convex if and only if the event { y Π ( y) α} is convex α (0,]. An alternative definition of convexity is Π ( λ y+ ( λ) y) min{ Π( y), Π ( y)} y, y and λ [0,]. A fuzzy variable with the membership function Π ( y ) is strongly convex if and only if the event { y Π ( y) α} is strongly convex α (0,]. A fuzzy variable with the membership function Π ( y ) is bounded if and only if the event { y Π ( y) α} is bounded α (0,]. Gert de Cooman [9] provided the possibility integral with respect to the possibility measure. owever, since the summation and multiplication in the possibility theory are maximum and minimum, respectively, the possibility integral cannot represent the variation of the fuzzy variable ust like the variance of the random variable. In the probability theory, the integral for the moment calculation can be interpreted as the weighted average because the probability has the property of summation to one. 4.. Random Variable Corresponding to Fuzzy Variable The investigation of the possibility theory shows that, the unity, convexity, and boundedness properties of membership function are similar to the properties of CDF. Theorem: A unit membership function of a continuous fuzzy variable satisfies the boundedness and convexity properties if and only if the following conditions hold:. lim Π ( y) = lim Π ( y) = 0 ; y y. Π ( y ) is a non-decreasing function for y d, and a non-increasing function for y d. Proof: Suppose the unity membership function Π ( y ) satisfies boundedness and convexity properties. It is to show that, Π ( y ) satisfies the two conditions above. The first part of second condition is, if y < y < d, Π( y) Π ( y). Since unity means Π ( d) =, if Π ( y) >Π ( y), then Π ( y) < min { Π( y), Π ( d) = }, which conflicts the convexity property. Similarly we can show that, if d < y < y, then Π( y) Π ( y). To show the first condition, we need to show lim Π ( y) = 0. Since Π ( y ) is a non-decreasing function for y d, and Π ( y ) 0, y

3 lim Π ( y) = c 0 exists. If c > 0, choosing α = c / y, then we have α -cut is not bounded from left. Thus, lim Π ( y) = c = 0. Similarly we can show that lim Π ( y) = 0. y y Now let s show that, if the membership function satisfies two conditions above, then it also satisfies boundedness and convexity properties. First, if there exists α > 0 such that α -cut is not bounded, then either lim Π ( y) > 0 or y lim Π ( y) > 0, which contradicts first condition. Second, assume the convexity is not satisfied, i.e., there exist y y, y, y3 : y y y3 Π ( y) < min Π( y), Π ( y3). If y < d, then Π ( y) >Π ( y) for y < y < d which conflicts the non-decreasing condition. If y > d, then Π ( y) <Π ( y3) for d < y < y3 which conflicts the non-increasing condition. If y = d, then, Π ( y) =Π ( y) =Π ( y3) = which conflicts the unity property. Thus, the unity membership function that satisfies two conditions above must also satisfy the convexity property. The proof is completed. Based on this theorem, if the continuous fuzzy variable with the membership function Π ( y ) satisfies unity, boundedness, and convexity properties, then the continuous function Π ( x) x d F( x) = () Π ( x) x > d satisfies non-decreasing property, lim F( x) = 0, and lim F( x) =. Thus, it is a CDF of a random variable. < < such that { } x x Definition: A random variable X with CDF in Eq. () is called the random variable corresponding to the fuzzy variable with membership function Π ( y ) Robustness Measure in Possibility Theory The robustness measure of the fuzzy variable can be defined as the variance of the corresponding random variable. Definition: The robustness measure of the fuzzy variable with membership function Π ( y ) is the variance of the random variable X corresponding to the fuzzy variable. If the fuzzy variable has the isosceles triangular membership function on the interval [,], then the corresponding random variable X is uniformly distributed on the interval [,]. According to the above definition, the robustness measure of this fuzzy variable is /3. 5. Possibility-Based Robust Design Optimization Using the robustness measure in the possibility theory, the possibility-based robustness analysis; sensitivity analysis of the possibility-based robustness analysis; and possibility-based robust design optimization (PBRDO) formulation are proposed in this section. 5.. Possibility-Based Robustness Analysis For structural systems in large-scale engineering applications, the input uncertainties are treated as fuzzy variables when there is lack of sufficient statistical information. Thus, the output performance measures are also treated as fuzzy variables. This section discusses the calculation of the robustness measure of the output fuzzy variable. Assume the system has the input fuzzy vector with non-interactive oint membership function Π ( y ) = min Π ( y ),, Π ( y ) where nf is number of fuzzy variables. Assume the output performance = h ( ) { } n nf is a fuzzy variable with the membership function Π ( h ). The random variable R corresponding to the fuzzy variable is introduced to compute the robustness measure of the output performance. The CDF of the random variable R is: Π ( hr ) hr d F ( h R R ) = (3) Π ( hr ) hr > d For large-scale engineering problems, the output membership function Π ( h ) and thus the CDF F ( h R R ) are computationally expensive to obtain. To compute the variance of the output random variable efficiently, the performance moment integration method (PMI) [8,], the dimension reduction method (DRM) [3,4,], and the percentile difference method (PDM) [9-] were developed. We use PMI in this paper. Thus, the robustness measure of the output performance can be computed as

4 3 3 = R i i i i i= i= (4) σ σ ωξ ωξ where ω = ω3 =, 6 ω = 3, ( ( ) ξ = F Φ 3 R ), ξ = F ( Φ (0)), and ξ ( ) R 3 = F R ( Φ 3 ). ere Φ() i is the standard normal CDF. Considering CDF in Eq. (3), we have ξ = d, and ξ and ξ 3 are two end points of α -cut of, where α = Φ( 3). Transferring all components of the fuzzy vector into the standard fuzzy variable v with isosceles triangular membership function on the interval [,] (see detail in Refs. 3,4), the left end point of α -cut of can be computed by solving the inverse possibility analysis to minimize h( v) subect to α (5) v The right end point of α -cut of can be computed by solving the inverse possibility analysis to maximize h( v) subect to α (6) v Finally, d = h ( v ) v= 0. These inverse possibility analyses are solved using the maximal possibility search (MPS) method [3,4]. Using the above inverse possibility analysis results, the robustness measure σ of the output performance = h( ) of the system with the input fuzzy vector of non-interactive oint membership function Π ( y ) can be computed as 3 3 σ ωi( hi ) ωihi (7) i= i= where h is the performance value at the point y from Eq. (5); h is the performance value at the maximal grade point y = d ; and h 3 is the performance value at the point y 3 from Eq. (6). 5.. Conservative Robustness Measure in Possibility Theory For robust design problems with lack of input uncertainty information, it is desirable to provide a conservative estimation of the design by providing the simulated robustness measure of the output performance larger than the variance of the output performance when sufficient statistical data is available for the input uncertainties. The robustness measure σ in Eq. (4), which is equivalent to Eq. (7), will increase if ξ decreases and/or ξ 3 increases. To show this, consider f ( x) = ωx ( ωx+ c) (8) Then, f '( x) = ω( ω) x ωc (9) Since ξ < ξ < ξ3 and ω+ ω + ω3 =, if we choose x = ξ, ω = ω and c = ωξ + ω3ξ3, then f '( x) = ω ( ω ) x ω ω ξ + ω ξ ( ) ( ) 3 3 < ω( ω) x ω ωx+ ω3x (0) = ω( ω ω ω3) x = 0 If we choose x = ξ3, ω = ω3 and c = ωξ+ ωξ, then f '( x) = ω3( ω3) x ω3( ωξ + ωξ) > ω3( ω3) x ω3( ωx+ ωx) () = ω3( ω3 ω ω) x = 0 Equation (0) and () completes proof of the statement. Additionally, ξ will decrease and ξ 3 will increase if the search domain enlarges, i.e., α decreases. On the other hand, when the fuzzy variables become all random variables, the robustness measure of the output

5 performance with the input fuzzy variables should become the variance of the output random variable. That means, Eqs. (5) and (6) become to minimize/maximize h( u ) subect to u 3, and so the target probability of failure for this inverse reliability analysis is Φ( 3). Based on the above discussion, a smaller value α is proposed as α =Φ( 3) in Eqs. (5) and (6) for conservative estimation of the robustness measure in possibility theory Sensitivity Analysis of Possibility-Based Robustness Measure This section derives the sensitivity of the possibility-based robustness measure based on the assumption that the point movement can be approximated by the design movement. Denoting hi h hi, = ( yi ) () y y we have h h y = i h, (3) d y d and h h d 3 h3 d y d h( ) = = h d, (4) y = i h3, (5) d Taking derivative of Eq. (7) and using Eqs. (3-5), we obtain the sensitivity of the possibility-based robustness measure as 3 3 ωi( hi ) ωihi d d i= i= (6) ωihi hi, ωihi ωihi, i= i= i= 5.4. Possibility-Based Robust Design Optimization (PBRDO) The possibility-based robust design optimization (PBRDO) can be formulated as to minimize f ( d, σ ) ( G ) subect to Π ( ) > 0 Π, =,, nc L U i i i t, d d d, i =,, nd where σ is the robustness measure of the output performance ; G ( ) > 0 denotes failure; Π t, is the target possibility of failure for the th constraint; and nc and nd are the numbers of the constraints and design variables; respectively. 6. Numerical Examples 6. Robustness Analysis of Mathematical Problem In the mathematical problem, let X and X be normally distributed random variables with the mean of 5 and the standard deviation of 0.3. Consider the performance function h( X ) = ( X 8) ( X 3) (8) In order to imitate a situation of practical engineering applications, assume the design engineer does not have the exact standard deviation of the input random variable X but has limited data. From the limited data, suppose the standard deviation is estimated to be The possibility-based robustness analysis result and reliability-based robustness analysis result are obtained as shown on Table. These results are compared with the robustness analysis result obtained using the correct standard deviation of 0.3 for the random variable X. The third and forth columns are sensitivities of the robustness measure with respect to the two input variables. The last column shows the number of function and sensitivity evaluations (FE) for the (7)

6 robustness analysis. Table. Robustness analysis for mathematical problem σ x FE Possibility-Based Robustness Analysis Result Reliability-Based Robustness Analysis Result Robustness analysis Result using Correct Data As shown in Table, the reliability-based robustness analysis with insufficient input data over-estimated the robustness (i.e., under-estimated the variance σ ). The possibility-based robustness analysis under-estimated the robustness (i.e., over-estimated the variance σ ), which is conservative estimation. 6. Robustness Analysis of Vehicle Side Impact Problem In the vehicle side impact problem [], the performance function (the lower rib deflection) is h( X ) = 9.9X.9XX X3X0 (9) where X, X, X3 ~ N (,0.), X 8 ~ N (0.3,0.006), and X0 ~ N (0,30). Suppose the design engineer does not have sufficient input data for the random variables X 8 and X 0. Using the limited data, the design engineer constructs the statistical information for these two random variables as X 8 being uniformly distributed on the interval [0.948,0.305]; and X 0 being uniformly distributed on the interval [ 8.660,8.660]. The possibility-based robustness analysis result and reliability-based robustness analysis result are obtained as shown on Table. These results are compared with the robustness analysis result obtained using the correct input data. The columns 3-7 are sensitivities of the robustness measure with respect to the input variables X, X 3, and X 8. The last column shows the number of function and sensitivity evaluations (FE) for the robustness analysis. Table. Robustness analysis for vehicle side impact problem σ x x x3 x8 x0 FE Possibility-Based Robustness Analysis Result Reliability-Based Robustness Analysis Result Robustness analysis Result using Correct Data As shown in Table, the probability-based robustness analysis with insufficient input data over-estimated the robustness. The possibility-based robustness analysis under-estimated the robustness, which is conservative estimation. 6.3 Robust Design Optimization of Mathematical Problem In this mathematical problem, let X and X be normally distributed random variables with the mean of 5 and the standard deviation of 0.3. The obective function is the robustness measure σ of the performance function in Eq. (8). Three design constraints are G(X) = X X / 0 G( X) = ( X+ X 5) / 30 ( X X ) /0 (0) G3( X) = 80/( X + 8X + 5) In order to imitate a situation of practical engineering applications, suppose the input variables are all normally distributed random variables with standard deviation of 0.3, but, due to lack of data, the design engineer does not have the exact standard deviations of X. From the limited, the standard deviation is estimated to be We treat X and X as fuzzy variables, and using the probability-possibility consistency principle and the least conservative principle [3,4], the membership functions are generated from the temporary normally distributed probability density function with the standard deviation 0.3 and 0.59 respectively. The PBRDO formulation is to

7 minimize σ ( G ) subect to Π ( X ) > 0 α, =,, 3 () 0. d 0.0, 0. d 0.0 where α = is the target possibility of failure, which is consistent with -σ design. Using the proposed PBRDO method and SQP optimizer [], the optimal design is obtained as shown in Table 3 and Figure. Iter. Table 3. PBRDO for mathematical problem σ d d G G G 3 FE (3,/0/) (7,/0/) (6,/0/) Opt Ina. Act. Act. (6,3/0/3) Figure. PBRDO history for mathematical problem In Table 3, the first column indicates design iteration. The second column shows the cost, which is the robustness measure of the performance in Eq. (8). The third and fourth columns shows design movement. The columns 5-7 show the possibilistic constraint values. The last column shows the number of function and sensitivity evaluations (FE) for PBRDO design iteration. The number in parentheses shows the number of FE for the robustness analysis and the search for each constraint, respectively. The last row shows the optimal design. The PBRDO and RBRDO results are shown on Table 4. For comparison, RBRDO result with the correct input data is shown on third row. The second column is the robustness measures at the optimal designs. The third and fourth columns are optimal designs. The fifth column shows the number of function and sensitivity evaluations (FE) for design optimization. The sixth column provides the probability-based robustness analysis results of the optimal designs using the correct input data to compare with the second column. The last two columns show the probability of failure for two active constraints using Monte Carlo simulation with sample size of 0,000. Table 4. Robustness design results for mathematical problem σ d d FE Check P G P G3 PBRDO RBRDO Exact Data

8 The table shows that, RBRDO under-estimates the robustness measure of the optimal design, and the optimal design does not satisfy target reliability (0.075). PBRDO provides a conservative robustness measure and conservative reliability. 6.4 Robust Design Optimization of Vehicle Side Impact Problem The vehicle side impact problem [] in Figure is to minimize the robustness measure of the lower rib deflection while enhancing side impact crash performance. The input uncertainties are listed in Table 5. Figure. Vehicle side impact problem Table 5. Vehicle side impact random variable properties Random Variable Std. Distr. Lower Initial Upper Dev. Type Bound Design Bound (B-pillar inner) 0.00 Normal (B-pillar reinforce) 0.00 Normal (Floor-side inner) 0.00 Normal (Cross member) 0.00 Normal (Door beam) 0.00 Normal (Door belt line) 0.00 Normal (Roof rail) 0.00 Normal (Mat. floor inner) Normal (Mat. floor side) Normal (Barrier height) 0.0 Normal 0 th and th variables (Barrier hitting) 0.0 Normal are not design variables Assume that the distribution types and standard deviations for the 8- th random variables are unknown to the design engineer. From limited experimental data, suppose the design engineer constructed the statistical information of these random variables as X 8 and X 9 being uniformly distributed on the interval with length 0.004; and X 0 and X being uniformly distributed on the interval [ 8.660, 8.660]. We treat all input variables as fuzzy variables, and using the probability-possibility consistency principle and the least conservative principle [3,4], the membership functions are generated from the temporary normally distributed probability density function with standard deviation 0.00 for X ~ X 7 ; and the temporary uniformly distributed probability density function with the interval length of and 7.3 for X 8 and X 9, X 0 and X ; respectively. The PBRDO formulation is to minimize σ () subect to Π G ( X ) > 0 α, =,,0 ( ) where α = is the target possibility of failure for the -σ design. Using the proposed PBRDO method and SQP optimizer [], the optimal design is obtained as shown in Table 6 and Table 7. Iter. Table 6. PBRDO cost and design history for vehicle side impact problem σ d d d 3 d 4 d 5 d 6 d 7 d 8 d

9 Opt Table 7. PBRDO constraint history and No. of function evaluations for vehicle side impact problem Iter. G G G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 0 FE Opt. Act. Ina. Ina. Ina. Ina. Ina. Act. Ina. Ina. Ina. 6 The RBRDO and PBRDO optimum design results are obtained as shown in Table 8. For comparison, the RBRDO result with the correct input data is listed on third row. The second column is the robustness measures at the optimal design. The columns 3- are optimal designs. The th column shows the number of function and sensitivity evaluations (FE) for design optimization. The last column provides the probability-based robustness analysis results of the optimal designs using the correct input data to compare with the second column. Table 9 shows the Monte Carlo simulation results for all optimal design. The total sample size is 0,000. Only first and seventh constraints are active. Table 8. Robust Design Results for Side Impact Problem σ d d d 3 d 4 d 5 d 6 d 7 d 8 d 9 FE Check PBRDO RBRDO Exact Data Table 9. Monte Carlo simulation of optimal designs G G G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 0 PBRDO RBRDO Exact Data Tables 8 and 9 show that, RBRDO with insufficient data provides unreliable design and underestimates the variance. PBRDO provides reliable design and overestimates the variance because of luck of data. Thus PBRDO provides conservative reliable design and conservative estimation of output variance. 6. Conclusion A robust design concept in the possibility theory is proposed in this paper by using the variance of the random variable corresponding to the fuzzy variable. The random variable corresponding to the fuzzy variable is obtained using boundedness and convexity of the unit membership function. A method to provide conservative estimation of the robustness for problems with lack of input uncertainty information is developed. Based on these proposed methods, sensitivity analysis of the robustness measure is carried out; and PBRDO is formulated and solved. Several numerical examples illustrate the robust design concept, the PBRDO formulation, and the conservative measure at the optimum designs. 7. Acknowledgment This research is supported by the Automotive Research Center that is sponsored by the U.S. Army TARDEC. 8. References. Nikolaidis, E., Cudney,.., Chen, S., aftka, R. T., and Rosca, R., Comparison of Probability and Possibility for Design Against Catastrophic Failure Under Uncertainty, Journal of Mechanical Design, Vol. 6, No. 3, 004, pp Mourelatos Z. P., and Zhou, J., Reliability Estimation and Design with Insufficient Data Based on Possibility Theory, AIAA Journal, Vol. 43, No. 8, 005, pp Du L., Choi K. K., oun B. D., and Gorsich D., Possibility-Based Design Optimization Method For Design Problems With Both Statistical And Fuzzy Input Data, Journal of Mechanical Design, Vol. 8, No. 4, , Du L., Choi K. K., and oun B. D., Inverse Possibility Analysis Method for Possibility-Based Design Optimization, AIAAJ, Vol. 44, No., , Du, L., Choi, K.K., An Inverse Analysis Method for Design Optimization With Both Statistical and Fuzzy

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