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+\documentclass[stropt]{svjour}
+%\usepackage{times}
+\usepackage{graphicx}
+%
+%Definitions
+%
+\def\be{\begin{equation}}%
+\def\ee{\end{equation}}%
+\def\m{\mathbf{M}}%
+\def\b{\mathbf{B}}%
+\def\y{\mathbf{Y}}%
+\def\x{\mathbf{X}}%
+\def\p{\mathbf{P}}%
+\def\xx{\mathbf{x}}%
+\def\zz{\mathbf{z}}%
+\def\bb{\mathbf{\beta}}%
+\def\kb{\mathbf{b}}%
+%
+\begin{document}
+%
+\title{Using response surface approximations\\ in fuzzy set based
+design optimization\thanks{Presented as paper 98--1776 at the
+39th AIAA\-/ASME/\-ASCE/\-AHS/\-ASC Structures, Structural Dynamics, and Materials
+Conference, Long Beach, California, April 20-23, 1998}}
+\author{G. Venter and R.T. Haftka}
+%
+\institute{Department of Aerospace Engineering, Mechanics and Engineering
+Science, University of Florida, Gainesville, Florida 32611--6250, USA\\
+\email{gventer@ufl.edu} and \email{haftka@ufl.edu}}
+%
+\date{Received: August 24, 1998}
+% The correct dates will be entered by Springer
+%
+\maketitle
+%
+\begin{abstract}
+The paper focuses on modelling uncertainty typical of the aircraft industry.
+The design problem involves maximizing a safety measure of an isotropic
+plate for a given weight.
+Additionally, the dependence of the weight on the level of uncertainty,
+for a specified allowable possibility of failure, is also studied.
+It is assumed that the plate will be built from future materials, with
+little information available on the uncertainty.
+Fuzzy set theory is used to model the uncertainty.
+Response surface approximations that are accurate over the entire design
+space are used throughout the design process, mainly to reduce the
+computational cost associated with designing for uncertainty.
+All of the problem parameters are assumed to be uncertain, and both a
+yield stress and a buckling load constraint are considered.
+The fuzzy set based design is compared to a traditional deterministic
+design that uses a factor of safety to account for the uncertainty.
+It is shown that, for the example problem considered, the fuzzy
+set based design is superior.
+Additionally, the use of response surface approximations results in
+substantial reductions in computational cost, allowing the final results to
+be presented in the form of design charts.
+\end{abstract}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Introduction}
+\label{sec01}
+
+In the aircraft industry, structures are often designed that will be built
+well into the future from materials available then, leading to uncertainty
+in material properties.
+Apart from the uncertain material properties, the manufacturing cost is also
+uncertain.
+However, unlike the uncertainty in the material properties, the designer has
+some control over the manufacturing cost, which is closely linked to the
+required tolerances in geometry.
+For such design problems, little information regarding the uncertainty is
+known, and the uncertainty is typically modeled based on expert opinion and
+assumptions made by the designer.
+Fuzzy set theory can use limited available data and caters for worst case
+scenarios.
+Fuzzy set theory is thus capable of by compensating for the fact that the
+uncertainty is modeled based on subjective opinions and assumptions
+\citep{Maglaras97}.
+In contrast, probabilistic methods require large amount of data and the
+results obtained are, in some cases, very sensitive to both the accuracy of
+this data as well as to the assumptions made during the modeling process
+\citep[e.g.][pp.~11--32]{BenHaim90}.
+
+Fuzzy set theory was introduced by \citet{Zadeh65}
+as a mathematical
+tool for the quantitative modeling of uncertainty, and makes use of fuzzy
+numbers to represent uncertain problem parameters.
+The designer only needs to specify the range of uncertainty and a membership
+function that denotes the possibility of occurrence of an element in the
+specified range to represent an uncertain parameter as a fuzzy number.
+Membership functions are generally constructed subjectively, based on
+expert opinion.
+In recent years, fuzzy set theory has been applied to a wide range of
+structural optimization problems.
+For example, \cite{Liu92}
+performed a fatigue reliability
+analysis of a portal frame, \cite{Jung96}
+considered the
+optimal plastic design of a fixed-fixed beam and a portal frame, and
+\cite{Jensen97}
+minimized the weight of a 25-bar transmission tower.
+Fuzzy set theory has also been used in multidisciplinary optimization by
+\cite{Rao93}
+to design the main rotor of a helicopter as well as by
+\cite{Wu96}
+to optimize the machine room layout of a ship.
+Additionally, \cite{Shih95}
+applied multicriteria
+optimization to various truss examples, considering both weight and
+displacement as objectives.
+
+Unfortunately, designing for uncertainty is computationally intensive and
+typically requires at least an order of magnitude more computational cost as
+compared to a corresponding deterministic design.
+In the present paper, response surface approximations are used to reduce the
+high computational cost associated with designing for uncertainty by using
+approximations that are accurate over the entire design space to replace
+costly finite element analyses.
+Response surface approximations have attracted a lot of interest from the
+structural optimization community in recent years, since they filter out
+numerical noise inherent to most numerical analysis procedures
+\citep[e.g.][]{Giunta94},
+they provide the designer with a global
+perspective of the response over the entire design space
+\citep[e.g.][]{Mistree94},
+and they enable easy integration of
+various software codes \citep[e.g.][]{Kaufman96}.
+
+An isotropic plate with a change in thickness across its width is considered
+as a design problem.
+All of the problem parameters are considered to be uncertain and the
+objective is to maximize a safety measure of the plate for a given weight.
+Both deterministic and fuzzy set based designs are considered and the results
+are compared.
+The safety measure is maximized by maximizing the factor of safety in the
+deterministic design and by minimizing the possibility of failure in the
+fuzzy set based design.
+Finally, the dependence of the weight on the level of uncertainty
+associated with the key geometric parameters is presented in the form of a
+design chart, based on results obtained from a number of optimizations.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Fuzzy set theory}
+\label{sec02}
+
+Fuzzy set theory presents a methodology for the mathematical modeling of
+uncertainty.
+In contrast to classical set theory where a sharp transition exists between
+membership and non-membership, fuzzy set theory makes use of membership
+functions to denote the degree to which an element belongs to a fuzzy set.
+A membership function assigns a grade of membership, ranging between 0 and 1,
+to each element of the universal set as follows
+\begin{equation}
+\m (x):~X\rightarrow [0,1]\, . \label{eq01}
+\end{equation}
+In (\ref{eq01}) $\m$ denotes a membership function that maps the elements of
+the universal set $X$ to the real interval $[0,1]$.
+The same symbol, a bold face capital letter, is used to denote both the fuzzy
+set and its membership function.
+Since each fuzzy set is completely and uniquely defined by only one particular
+membership function, no ambiguity results from the double use of the symbol.
+
+Fuzzy sets are represented numerically by making use of $\alpha$ level cuts.
+An $\alpha$ level cut is defined as the real interval where the membership
+function is larger than a given value, $\alpha$ \citep[p.~19]{Klir95}
+and may be written mathematically for a generic
+fuzzy set $\b$ as follows:
+\begin{equation}
+^{\alpha}B=\left\{ x | \b (x)\geq\alpha\right\}\, . \label{eq02}
+\end{equation}
+Figure~\ref{fig:1} provides a graphical representation of (\ref{eq02}), where
+it is assumed that $\b$ has a triangular and symmetric membership function,
+and shows the end points $^{\alpha}b_1$ and $^{\alpha}b_2$ of the $\alpha$
+level cut.
+
+\begin{figure}
+\vspace{5cm}
+\caption{An $\alpha$ level cut of a triangular and symmetric membership
+function, having support in $(x_L, x_R)$}
+\label{fig:1}
+\end{figure}
+
+A fuzzy number is defined as a fuzzy set that is both normal and convex
+\citep[pp.~97]{Klir95}.
+A normal fuzzy set has a maximum membership function equal to 1, while all
+possible $\alpha$ level cuts are convex for a convex fuzzy set.
+The fuzzy set $\b$
+shown in Fig.~\ref{fig:1} is thus a fuzzy number.
+In fact the triangular and symmetric membership function is most often used to
+represent fuzzy numbers, mainly due to its simplicity, and was used throughout
+the present paper to represent all of the uncertain problem parameters.
+
+A fuzzy function $\y$ is a function of fuzzy variables $\x_i$ and may be
+written as
+\begin{equation}
+\y=\y (\x_1,\x_2,\ldots,\x_n) \label{eq03}
+\end{equation}
+for the case where $n$ fuzzy variables are considered.
+%Klir and Yuan 1995,
+%(pp.~105--109)
+\citet[pp.~105--109]{Klir95} summarized and proved the
+following properties of a fuzzy function.
+
+\begin{enumerate}
+\item When all of the fuzzy variables of a fuzzy function are continuous
+fuzzy numbers, the fuzzy function itself is also a continuous fuzzy number.
+\item When all of the fuzzy variables of a fuzzy function are fuzzy numbers,
+the $\alpha$ level cut of a fuzzy function $^{\alpha}Y$ may be written in
+terms of the $\alpha$ level cuts of its fuzzy variables $^{\alpha}X_i$ as
+follows:
+
+\begin{gather}
+^{\alpha}Y=^{\alpha}Y(^{\alpha}X_1,%^{\alpha}X_2,
+\ldots,^{\alpha}X_n)=\label{eq04}\\
+\left[
+\min_{^{\alpha}R}
+\left[
+Y(^{\alpha}X_1,%^{\alpha}X_2,
+\ldots,^{\alpha}X_n)
+\right],
+\max_{^{\alpha}R}
+\left[
+Y(^{\alpha}X_1,%^{\alpha}X_2,
+\ldots,^{\alpha}X_n)
+\right]
+\right]\, ,\notag
+\end{gather}
+where $^{\alpha}R$ denotes the $n$-dimensional box, formed by the $\alpha$
+level cuts of the $n$ fuzzy numbers.
+\end{enumerate}
+Based on these properties of a fuzzy function,
+\cite{Dong87}
+introduced the vertex method for evaluating the upper and lower bounds of
+$^{\alpha}Y$ when all of the fuzzy variables of $\y$ are fuzzy numbers.
+This method requires the evaluation of the fuzzy function at the $2^n$
+vertices of the $n$-dimensional rectangle, formed by the $\alpha$ level cuts
+of the $n$ fuzzy variables.
+In addition, interior global extreme points need to be checked.
+This method requires a large number of function evaluations and is
+computationally intensive.
+
+For calculating the possibility of failure it is required to compare a crisp
+number with a fuzzy number.
+Note that a fuzzy number may also be considered as the trace of a possibility
+measure $\Pi$ on the singletons (single elements) $x$ of the universal set $X$
+%(Dubois and Prade 1988,
+%pp.~13--17).
+\citep[p.~13--17]{Dubois88}.
+When a possibility measure defined on the unit interval is considered, its
+possibility distribution $\pi$ is then interpreted as the membership function
+of a fuzzy number $\b$ describing the event that $\Pi$ focuses on, as follows:
+\begin{equation}
+\Pi\left( \{ x\}\right)=\pi(x)=\b (x)\, ,\quad \forall x\in X\, . \label{eq05}
+\end{equation}
+The possibility measure of a crisp number being smaller or equal to a fuzzy
+number $\b$ is then defined \cite[pp.~99--101]{Dubois88} as follows:
+%
+\begin{equation}
+\Pi_{\b}\left( [x,+\infty) \right)=\sup_{y\geq x}\b (y)\, ,\quad \forall
+x\, . \label{eq06}
+\end{equation}
+The possibility distribution function $\pi_{\b}$ corresponding to
+the possibility measure of (\ref{eq06}) is shown graphically in
+Fig.~\ref{fig:2} for the general case where $\b$ has a nonlinear membership
+function.
+
+\begin{figure}
+\vspace{5cm}
+\caption{Possibility distribution of $\b \geq x$ for nonlinear $\b(x)$,
+having support in $(x_L, x_R)$}
+\label{fig:2}
+\end{figure}
+
+Based on (\ref{eq05}) and (\ref{eq06}), the possibility distribution of
+failure $\pi_{(\p-\p_f)}$ is obtained from the fuzzy function
+$(\p-\p_f)$ that contains the fuzzy numbers $\p$ (the applied load) and $\p_f$
+(the failure load) as variables.
+The possibility of failure $(\p-\p_f\geq 0)$ is then defined as
+\begin{equation}
+\Pi_{(\p-\p_f)}\left( [0,+\infty)\right)=\sup_{y\geq 0}(\p-\p_f)(y)\, .
+\label{eq07}
+\end{equation}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Overview of response surface approximations}
+\label{sec03}
+
+A response surface approximation is an approximate relationship between a
+dependent variable $\eta$ (the response) and a vector $\xx$ of $k$
+independent variables (the predictor variables).
+The response is generally obtained from experiments (which may be numerical
+in nature), where $\eta$ denotes the mean or expected response value.
+It is assumed that the true model of the response may be written as a linear
+combination of given functions $\tilde{\zz}$ with some unknown coefficients
+$\tilde{\bb}$.
+The experimentally obtained response $y$ differs from the expected value
+$\eta$ due to random experimental error $\delta$ as follows:
+\begin{equation}
+y(\xx)=\eta(\xx)+\delta=\tilde{\zz}(\xx)^T \tilde{\bb}+\delta\, .
+\label{eq08}
+\end{equation}
+
+Since the exact dependence of $\eta$ is generally unknown, a response surface
+approximation is used to approximate $\eta(\xx)$ as follows:
+\begin{equation}
+y(\xx)=\zz(\xx)^T\bb+\varepsilon\, ,
+\label{eq09}
+\end{equation}
+where $\zz(\xx)$ contains the assumed functions in the response surface
+approximation and $\bb$ the associated coefficients.
+Furthermore, $\varepsilon$ denotes the total error, which is the
+difference between the predicted and measured response values and includes
+both random (variance) and modeling (bias) error.
+Typically low order polynomials are used as a response surface approximation,
+in which case $\zz(\xx)$ consists of monomials.
+
+The coefficients $\bb$ of the response surface approximation are estimated
+from the experimentally obtained response values to minimize the sum of the
+squares of the error terms, a process known as regression.
+The estimated values of $\bb$ is denoted by $\kb$, resulting in the
+following response surface approximation:
+\begin{equation}
+\hat{y}(\xx)=\zz(\xx)^T\kb\, ,
+\label{eq10}
+\end{equation}
+where the caret symbol implies predicted values.
+
+The assumed form of the response surface approximation, (\ref{eq10}),
+usually includes redundant parameters and parameters that are poorly
+characterized by the experiments.
+These parameters may increase the prediction error of the approximation and
+thus decrease its predictive capabilities.
+In the present paper, redundant parameters are eliminated by using mixed,
+backwards, stepwise regression (e.g.
+\citealt{Ott93}, pp.~648--659; \citealt{Myers95}, pp.~642--655).
+Mallow's $Cp$ statistic is used to identify the best reduced response
+surface approximation from the subset of reduced response surface
+approximations provided by the stepwise regression procedure and is
+defined as
+\begin{equation}
+Cp=\frac{SSE_p}{s_{\varepsilon}^2}-(n-2p)\, ,
+\label{eq11}
+\end{equation}
+where $SSE_p$ is the sum of the squares of the $n$ error terms (one for each
+data point used to estimate $\kb$) for an approximation with $p$ parameters
+and $s_{\varepsilon}^2$ is the mean sum of squares of the error terms obtained
+from the response surface approximation with all of the parameters included.
+
+Optimization has the general tendency of exploiting weaknesses in the
+formulation of the response function, and highly accurate response surface
+approximations are thus a requirement in structural optimization applications.
+To ensure highly accurate approximations, it is important to evaluate the
+predictive capabilities of the approximations.
+In the present paper, the coefficient of determination $(R^2)$ statistic,
+the adjusted $R^2$ (Adj-$R^2$) statistic, the percent root mean square
+error (\%RMSE) as well as the percent root mean square error based on the
+predicted sum of squares (PRESS) statistic (\%RMSE$_{{\rm PRESS}}$) are
+calculated \citep[pp.~28--47]{Myers95}.
+
+The $R^2$ statistic denotes the proportion of the variability in the response
+that is accounted for by the response surface approximation and has a value
+between 0 and 1.
+The Adj-$R^2$ statistic is an alternative measure of the explained variability
+that, unlike $R^2$, has the desirable property that its value does not
+necessarily increase when adding (possibly redundant) parameters to a
+response surface approximation.
+The \%RMSE is an estimate of the root mean square error of the approximation
+that is obtained from the data points used to construct the approximation,
+using the following unbiased estimator:
+%
+\[
+{\rm \%RMSE}=\frac{100}{\overline{y}}=\sqrt{\frac{1}{(n-p)}
+\sum\limits_{i=1}^{n}(y_i-\hat{y}_i)^2}\, ,
+\]
+where
+%
+\begin{equation}
+\overline{y}=\frac1n\sum\limits_{i=1}^{n}| y_i |\, .
+\label{eq12}
+\end{equation}
+
+The \%RMSE$_{{\rm PRESS}}$ is an additional measure of the error, based on
+the PRESS statistic.
+The PRESS statistic is calculated by selecting a data point, say data point
+$i$.
+The response surface approximation obtained from the remaining $(n - 1)$ data
+points is used to predict the response at the withheld data point, denoted by
+$\hat{y}_{(i)}$.
+The prediction error at the withheld data point $e_{(i)}$ is then defined as
+\begin{equation}
+e_{(i)}=y_i-\hat{y}_{(i)}\, , \label{eq13}
+\end{equation}
+and is referred to as the $i$-th PRESS residual.
+This procedure is repeated for all of the data points and the resulting PRESS
+residuals are summed to form the PRESS statistic as follows:
+\begin{equation}
+{\rm PRESS}=\sum\limits_{i=1}^{n}e^2_{(i)}
+=\sum\limits_{i=1}^{n}\left\lbrack y_i-\hat{y}_{(i)}\right\rbrack ^2\, .
+\label{eq14}
+\end{equation}
+The %RMSE$_{{\rm PRESS}}$ is then defined as:
+\begin{equation}
+{\rm \%RMSE}_{{\rm PRESS}}=\frac{100}{\overline{y}}
+\sqrt{\frac1n{\rm PRESS}}\, .
+\label{eq15}
+\end{equation}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Plate example}
+\label{sec04}
+
+An isotropic plate with a change in thickness in the form of a linear ramp
+(see Fig.~\ref{fig:3}) is the design problem considered in the present paper.
+
+\begin{figure}
+\vspace{5cm}
+\caption{Three-dimensional view of the plate with a thickness change.}
+\label{fig:3}
+\end{figure}
+
+Three nondimensional parameters, $\lambda$, $\beta$ and $\gamma$, are used to
+specify the geometry and location of the change in thickness
+(see Fig.~\ref{fig:4}).
+The plate is simply supported on two edges, free on the other two edges, and
+subjected to an uniformly distributed load applied on the two simply supported
+edges.
+
+\begin{figure}
+\vspace{5cm}
+\caption{Cross-section of plate with response variables shown.}
+\label{fig:4}
+\end{figure}
+
+Both a yield stress failure (according to the Von Mises criterion) and a
+buckling load constraint are considered in the design, and the failure load
+$P_f$ of the plate is calculated from
+\begin{equation}
+P_f=\min\left\{
+\begin{array}{l}
+\frac{\strut\displaystyle\sigma_Y\lambda b t_0}{\strut\displaystyle\tilde{\sigma}_x} \\
+\frac{\strut\displaystyle\tilde{N}_{{\rm crit}} \pi^2 Eb(\lambda t_0)^3}{\strut\displaystyle12(1-\nu^2)a^2}
+\end{array}
+\right.\, , \label{eq16}
+\end{equation}
+and failure is defined to occur when:
+\begin{equation}
+P-P_f\geq 0\, . \label{eq17}
+\end{equation}
+In (\ref{eq16}) and (\ref{eq17}), $\sigma_y$ denotes the yield stress, $E$ the
+Young's modulus and $\nu$ the Poisson's ratio of the material considered,
+while $\lambda$, $a$, $b$, $t_0$ and $r$ describe the geometry of the plate as
+shown in Figs.~\ref{fig:3} and \ref{fig:4} and $P$ denotes the applied load.
+Additionally, $\tilde{\sigma}_x$ denotes the nondimensional, $x$-directional
+stress component on the top surface of the thin section of the plate,
+calculated a distance $r$ from the re-entrant corner, and is defined as
+\begin{equation}
+\tilde{\sigma}_x=\frac{\lambda b t_0\sigma_x}{P}\, ,
+\label{eq18}
+\end{equation}
+while $\tilde{N}_{{\rm crit}}$ denotes the nondimensional buckling load of
+the plate, defined as
+\begin{equation}
+\tilde{N}_{{\rm crit}}=\frac{12(1-\nu^2)a^2 N_{{\rm crit}}}{\pi^2 E b
+(\lambda t_0)^3}\, .
+\label{eq19}
+\end{equation}
+
+\leavevmode\citet{Venter97}
+used a large number of numerical experiments to
+study this problem in detail, and showed that the maximum von Mises stress
+always occurs on the top surface of the thin section of the plate, in which
+case $\sigma_x$ is the only nonzero stress component.
+According to the von Mises criterion, failure then occurs when
+\begin{equation}
+\tilde{\sigma}_x\geq \tilde{\sigma}_Y=\frac{\lambda b t_0\sigma_Y}{P}\, .
+\label{eq20}
+\end{equation}
+\cite{Venter97}
+also determined that the problem has both a local
+and a global buckling mode, and defined a simple geometric criterion to
+distinguish between the two buckling modes as follows:
+\begin{equation}
+\mbox{buckling mode}=\left\{
+\begin{array}{ll}
+\mbox{local if} & \frac{(0.5-\beta-\gamma)}{\lambda}\geq 0.6 \\[4pt]
+\mbox{global if} & \frac{(0.5-\beta-\gamma)}{\lambda}\leq 0.6
+\end{array}\, .
+\right. \label{eq21}
+\end{equation}
+
+\leavevmode\cite{Venter97}
+constructed highly accurate response surface
+approximations for both the $x$-directional stress distribution on the top
+surface of the thin section of the plate and for the buckling load of the
+plate, using a total of 752 finite element analyses.
+Numerical experiments in the form of finite element analyses were conducted
+using MSC$\backslash$NASTRAN Version 68.
+A cross-section of the plate was used to model the stress distribution near
+the re-entrant corner, using four-node, isoparametric, plane strain elements.
+All of these models had a uniform mesh, with roughly 1,800 elements in the
+$x$-direction and 9 elements in the $z$-direction, but the number of elements
+varied slightly from model to model.
+A schematic representation of the finite element model used is shown in
+Fig.~\ref{fig:5}.
+
+\begin{figure}
+\vspace{5cm}
+\caption{Finite element model used for stress distribution about the
+re-entrant corner}
+\label{fig:5}
+\end{figure}
+
+For the buckling load response surface approximations, four-node,
+isoparametric, plate bending elements were used to construct a two-dimensional
+finite element model similar to a plan view of Fig.~\ref{fig:3}.
+Twenty elements were used in each of the $x$- and $y$-directions respectively.
+The eccentricity of the mid-plane was found to have an insignificant impact on
+the buckling load value (note that the sides of the plate are free) and was
+ignored in the analysis.
+
+The stress distribution response surface approximation
+\citep[see][]{Venter97}
+%(see
+%\cite{Venter97})
+may be written in functional form as
+\begin{equation}
+\tilde{\sigma}_x=\tilde{\sigma}_x\left(\lambda,\beta,\gamma,
+\tilde{r}^{\zeta-1}\right)\, ,
+\label{eq22}
+\end{equation}
+where $\zeta$ is a constant that describes the radial stress distribution near
+the re-entrant corner and depends on $\lambda$, $\gamma$ and $a/t_0$ through
+the angle $\Theta$.
+Additionally, $\tilde{r}$ is the nondimensional distance measured from the
+re-entrant corner, defined as:
+\begin{equation}
+\tilde{r}=r/t_0\, .
+\label{eq23}
+\end{equation}
+Additionally, two response surface approximations, corresponding to the local
+and global buckling modes were constructed, which may be written in functional
+form as:
+\begin{equation}
+\tilde{N}_{{\rm loc}}=\tilde{N}_{{\rm loc}}(\lambda,\beta,\gamma)\, ,
+\quad
+\tilde{N}_{{\rm glob}}=\tilde{N}_{{\rm glob}}(\lambda,\beta,\gamma)\, .
+\label{eq24}
+\end{equation}
+
+The design space used for constructing the stress distribution and buckling
+load response surface approximations is summarized in Table~\ref{tab:1}.
+The upper limit on $\tilde{r}$ limits the radius of the yield zone about the
+re-entrant corner to be no greater than 80\% of the thickness of the thin
+section of the plate, while the upper bound on $\gamma$ is dictated by the
+geometry of the transition region.
+
+\begin{table}[htbp]
+\caption{Design space for constructing the response surface approximation
+approximations\hsize=164pt}
+\label{tab:1}
+\tabcolsep=10pt
+\begin{tabular}{@{}ll@{}}
+\hline\noalign{\smallskip}
+Response variable & Range \\
+\noalign{\smallskip}\hline\hline\noalign{\smallskip}
+$\lambda$ & $0.2\leq \alpha\leq 1.0$ \\
+$\beta$ & $-0.475\leq \beta\leq 0.475$ \\
+$\gamma$ & $0\leq \gamma\leq 0.475-\beta$ \\
+$\tilde{r}^{\zeta-1}$ & $0\leq \tilde{r}\leq 0.8\alpha$ \\
+\noalign{\smallskip}\hline
+\end{tabular}
+\end{table}
+
+The stress distribution response surface approximation was constructed from
+288 plate configurations (corresponding to 288 finite element analyses).
+Each plate configuration included a number of data points with different
+$\tilde{r}^{\zeta-1}$ values (corresponding to different finite elements),
+yielding a total of 2,124 data points.
+The buckling load response surface approximations were constructed from an
+additional 288 finite element analyses.
+Using the geometric criterion of (\ref{eq21}), these 288 finite element
+analyses were divided into two groups corresponding to the two buckling modes.
+This process identified 126 data points for constructing the local buckling
+load approximation and 162 data points for constructing the global buckling
+load approximation.
+A quartic polynomial was used as initial response surface approximation for
+both the stress distribution and the global buckling load response surface
+approximations, while a cubic polynomial was used for the local buckling load
+response surface approximation.
+These initial response surface approximations were reduced, using the mixed
+stepwise regression procedure and the $Cp$ statistic.
+The process of constructing the response surface approximations is discussed
+in more detail by \cite{Venter97}.
+The predictive capabilities of the reduced response surface approximations are
+summarized in Table~\ref{tab:2}.
+
+\begin{table}[htbp]
+\caption{Predictive capabilities of stress distribution and buckling load
+response surface approximations\hsize=178pt}
+\label{tab:2}
+\tabcolsep5pt
+\begin{tabular}{@{}lcccc@{}}
+\hline\noalign{\smallskip}
+{Model} & ${\mathbf{R^2}}$ & {Adj-}$\mathbf{R^2}$ &{RMSE}&{PRESS}\\
+&&&{[\%]}&{[\%]} \\
+\noalign{\smallskip}\hline\noalign{\smallskip}
+{Stress} & \multicolumn{4}{c}{{4-th order model (2,124 data
+ points)}} \\
+\noalign{\smallskip}\hline\noalign{\smallskip}
+Reduced\\
+43 terms& 0.9983 & 0.9982 & 3.2964 & 3.3886 \\
+{Local}\\
+{buckling}&\multicolumn{4}{c}{{3-rd order model (126 data points)}} \\
+\noalign{\smallskip}\hline\noalign{\smallskip}
+Reduced\\
+19 terms& 0.9999 & 0.9998 & 0.5550 & 0.6920 \\
+{Global}\\
+ {buckling}& \multicolumn{4}{c}{{4-th order model (162 data points)}} \\
+\noalign{\smallskip}\hline\noalign{\smallskip}
+Reduced\\
+ 25 terms& 0.9910 & 0.9895 & 2.4888
+ & 3.0202 \\
+\noalign{\smallskip}\hline
+\end{tabular}
+\end{table}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Design problem formulation}
+\label{sec05}
+
+The design problem has two objectives.
+The first objective is to maximize a safety measure of the plate for a given
+weight.
+The results obtained from a traditional deterministic approach, using a factor
+of safety to account for the uncertainty, were compared to those obtained from
+a fuzzy set based approach.
+The safety measure of the plate was maximized, by maximizing the factor of
+safety for the deterministic approach and by minimizing the possibility of
+failure for the fuzzy set based approach.
+Note that there exist fundamental differences between the deterministic and
+fuzzy set based approaches for maximizing the safety measure of the plate for
+a given weight.
+The deterministic approach tends to equalize the failure load of each failure
+criterion, while the fuzzy set based design tends to equalize the possibility
+of failure of each failure criterion.
+
+The second objective is to study the dependence of the weight of the final
+design on the level of uncertainty associated with the design variables
+$\lambda$, $\beta$ and $\gamma$.
+In this case, the weight was minimized for a specified allowable possibility
+of failure and different levels of uncertainty associated with the design
+variables.
+The results are presented in the form of a design chart.
+Different levels of uncertainty for the design variables were considered,
+since these geometric variables have the largest influence on the
+manufacturing cost of the plate.
+If the tolerances of these variables can be relaxed without a large penalty
+in terms of weight, substantial cost savings can be achieved in manufacturing
+the plate.
+The problem parameters and associated levels of uncertainty used are
+summarized in Table~\ref{tab:3}.
+Although Table~\ref{tab:3} has a total of 11 uncertain problem parameters,
+only 8 uncertain parameters are associated with each of the two failure
+criteria [see (\ref{eq16}), (\ref{eq22}) and (\ref{eq24})].
+
+\begin{table}[htbp]
+{\hsize=156pt
+\caption{Problem parameters and associated uncertainty}}
+\label{tab:3}
+\tabcolsep5pt
+\begin{tabular}{@{}lcc@{}}
+\hline\noalign{\smallskip}
+Variable & \begin{tabular}{c}Nominal \\ values \end{tabular} &
+ \begin{tabular}{c}Level of \\ uncertainty, $\mathbf{u}$ \end{tabular} \\
+\noalign{\smallskip}\hline\hline\noalign{\smallskip}
+$\lambda^{\dagger}$ & [0.2 -- 1.0] & [$\pm$ 2 -- $\pm$ 20]\% \\
+$\beta^{\dagger}$ & [-0.4 -- 0.4] & [$\pm$ 2 -- $\pm$ 20]\% \\
+$\gamma^{\dagger}$ & [0 -- 0.8] & [$\pm$ 2 -- $\pm$ 20]\% \\
+$a$ & 228.6~cm & $\pm$5\% \\
+$b$ & 127.0~cm & $\pm$5\% \\
+$t_0$ & 7.620~cm & $\pm$5\% \\
+$E$ & 206.84~GPa & $\pm$5\% \\
+$\nu$ & 0.29 & $\pm$5\% \\
+$\sigma_y$ & 197.26~MPa & $\pm$10\% \\
+$r$ & $5\alpha t_0$ & $\pm$10\% \\
+$P$ & 3,224.96~kN & $\pm$10\% \\
+\noalign{\smallskip}\hline
+\end{tabular}
+
+\noindent
+{\footnotesize $^{\dagger}$ Design variables}
+\end{table}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\subsection{Deterministic design}
+\label{sec05.01}
+
+The objective of the deterministic design is to maximize the factor of safety
+for a given weight.
+However, since it is difficult to specify a meaningful weight, it was decided
+to minimize the weight for a given factor of safety.
+The resulting minimum weight was then used as the given weight for the fuzzy
+set based design.
+A factor of safety of 1.5 was assumed and the level of uncertainty associated
+with the design variables was considered to be constant, equal to $\pm$5\%.
+The nondimensional cross-sectional area of the plate $\tilde{A}$ was used as
+a representative value of the weight and the resulting optimization problem
+may be written as\\[6pt]
+minimize:
+\[
+\tilde{A}=\frac{A}{\lambda t_0}=\frac12 (1+2\beta+\gamma)+\frac{\lambda}{2}
+(1-2\beta-\gamma)\, ,
+\]
+subject to
+\begin{gather}
+\frac{\beta}{0.4}+1\geq 0\, ,\quad
+1-\frac{\beta}{0.4}\geq 0\, ,\quad
+\gamma\geq 0\, ,\notag\\
+1-\frac{\gamma+\beta}{0.4}\geq 0\, ,\quad
+\frac{P_f}{P}-1.5\geq0\, .
+\label{eq25}
+\end{gather}
+The constraints involving $\beta$ and $\gamma$ are geometric constraints and
+$P_f$ is calculated from (\ref{eq16}), using the nominal values of the design
+variables.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\subsection{Fuzzy set based design}
+\label{sec05.02}
+
+The fuzzy set based design problem minimizes the possibility of failure, using
+the optimum nondimensional cross-sectional area obtained from (\ref{eq25}) as
+an upper limit of the weight.
+The resulting optimization problem may be written as\\[6pt]
+Minimize:
+\[
+\Pi_{(\p-\p_f)}=\Pi_{(\p-\p_f)}({\vec{\lambda}}, {\vec{\beta}}, {\vec{\gamma}} )\, ,
+\]
+subject to
+\begin{gather}
+\frac{\beta}{0.4}+1\geq 0\, ,\quad
+1-\frac{\beta}{0.4}\geq 0\, ,\quad
+\gamma\geq 0\, ,\notag\\
+1-\frac{\gamma+\beta}{0.4}\geq 0\, ,\quad
+\frac{\tilde{A}(\lambda,\beta,\gamma)}{\tilde{A}^*}-1=0\, .
+\label{eq26}
+\end{gather}
+where bold face Greek symbols denote fuzzy numbers while regular font symbols
+denote nominal values.
+Additionally,\linebreak[4]$\Pi_{(\p-\p_f)}$ denotes the possibility of
+failure and
+$\tilde{A}^*$ denotes the optimum nondimensional cross-sectional area
+obtained from the deterministic design of (\ref{eq25}).
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\subsection{Implementation of the fuzzy set based design}
+\label{sec05.03}
+
+In the present work response approximations form an integral part of the fuzzy
+set based design and two levels of response surface approximations are
+employed during the different stages of the design process.
+On the first level, the stress distribution and buckling load response
+surface approximations (Section~\ref{sec04}) are used to replace
+computationally expensive finite element analysis in evaluating the
+possibility of failure.
+The possibility of failure is calculated from (\ref{eq16}), using the vertex
+method.
+When considering all of the problem parameters as uncertain, the evaluation of
+the possibility of failure for a single $\alpha$ level cut value requires
+$2\times 2^8=512$ (recall that each failure criterion has a total of 8
+uncertain problem parameters) finite element analyses when no response surface
+approximations are used.
+In terms of a single optimization, an estimate of the required number of
+finite element analyses required when not using response surface
+approximations, is obtained from the product of four numbers as follows:
+
+\bigskip\noindent
+\begin{tabular}{@{}lc@{}}
+\tabcolsep5pt
+Average number of design & \\
+optimization iterations: & 5 \\
+\\
+Average number of $\Pi_{(\p-\p_f)}$ & \\
+evaluations per iteration: & 6 \\
+\\
+Average number of $\alpha$ level cut & \\
+evaluations per $\Pi_{(\p-\p_f)}$ evaluation: & 5 \\
+\\
+Number of finite element analyses & \\
+per $\alpha$ level cut evaluation of $\Pi_{(\p-\p_f)}$: & 512 \\
+\cline{2-2}
+\\
+Total number of finite element analyses & \\
+required per optimization: & 76,800
+\end{tabular}
+
+\bigskip
+In contrast, the stress distribution and buckling load response surface
+approximations were constructed from a total of only 752 finite element
+analyses.
+Additionally, these response surface approximations can be used in multiple
+optimizations without the need of performing additional finite element
+analyses.
+
+On the second level, a response surface approximation of the possibility of
+failure as a function of the nominal values of the design variables and the
+level of uncertainty associated with these variables was constructed.
+This second level approximation was constructed to simplify the integration of
+the analysis code with the optimization algorithm as well as to eliminate
+noise in the response function, thus allowing the use of a derivative based
+optimization algorithm.
+In the present paper, the generalized reduced gradient algorithm provided
+with Microsoft Excel Version 7.0 was used.
+
+The $\lambda$, $\beta$ and $\gamma$ design space of Table~\ref{tab:1} was used
+to construct the possibility of failure response surface approximation, with
+numerical experiments conducted at an evenly spaced grid consisting of 11 data
+points in each of the $\lambda$, $\beta$ and $\gamma$ directions.
+Additionally, seven levels of uncertainty evenly spaced between $\pm$2\% and
+$\pm$20\% were considered, yielding a total of 2,629 data points in the design
+space.
+At each data point the possibility of failure according to each of the two
+failure criteria was evaluated.
+Two response surface approximations (one for each failure mode) were
+constructed using all of the data points with possibility of failure not
+equal to either 0 or 1.
+This process resulted in 499 data points for constructing the yield stress
+failure criterion response surface approximation and 573 data points for the
+buckling load constraint failure criterion response surface approximation.
+The resulting predicted possibility of failure is then obtained from
+\begin{equation}
+\hat{\Pi}_{(\p_-\p_f)}=\min\left(\hat{\Pi}_{{\rm Yield Stress}},
+\hat{\Pi}_{{\rm Buckling}} \right)\, .\label{eq27}
+\end{equation}
+
+It was found that a general fourth-order polynomial (70 parameters) gave
+accurate approximations for both failure modes.
+These general response surface approximations were reduced using the mixed
+stepwise regression procedure and the $Cp$ statistic, with the predictive
+capabilities of the response surface approximations summarized in
+Table~\ref{tab:4}.
+
+\begin{table}[htbp]
+\caption{Predictive capabilities of the possibility of failure
+response surface approximations\hsize=197pt}
+\label{tab:4}
+\tabcolsep5pt
+\begin{tabular}{@{}lcccc@{}}
+\hline\noalign{\smallskip}
+{Model} & $\mathbf{R^2}$ &{Adj-}$\mathbf{R^2}$ &
+ \begin{tabular}{c} {RMSE} \\ {[\%]} \end{tabular} &
+ \begin{tabular}{c} {PRESS} \\ {[\%]} \end{tabular} \\
+\noalign{\smallskip}\hline\noalign{\smallskip}
+{Stress} & \multicolumn{4}{c}{{4-th order model (499 data
+ points)}} \\
+\noalign{\smallskip}\hline\noalign{\smallskip}
+Full\\
+ 70 terms& 0.9988 & 0.9986 & 2.2525
+ & 2.6205 \\
+Reduced\\
+ 59 terms& 0.9988 & 0.9986 & 2.2371
+ & 2.5205 \\
+{Buckling} & \multicolumn{4}{c}{{4-th order model
+ (573 data points)}} \\
+\noalign{\smallskip}\hline\noalign{\smallskip}
+Full\\
+70 terms& 0.9982 & 0.9980 & 2.7342
+ & 3.0991 \\
+Reduced\\
+57 terms& 0.9982 & 0.9980 & 2.7118
+ & 3.0211 \\
+\noalign{\smallskip}\hline
+\end{tabular}
+\end{table}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Dependence of the weight on the level of uncertainty}
+\label{sec05.04}
+
+In order to study the dependence of the weight of the plate on the level of
+uncertainty associated with the design variables, different levels of
+uncertainty between $\pm$2\% and $\pm$20\% were considered.
+For each of these levels, the nondimensional cross-sectional area of the
+plate was minimized for an allowable possibility of failure.
+The allowable possibility of failure (allowable was assumed to be equal to the
+optimum value obtained from the fuzzy set based design problem of
+(\ref{eq26}).
+The resulting optimization problem may be written as\\[6pt]
+minimize:
+\[
+\tilde{A}=\frac{A}{\lambda t_0}=\frac12 (1+2\beta+\gamma)+\frac{\lambda}{2}
+(1-2\beta-\gamma)\, ,
+\]
+subject to
+\begin{gather}
+\frac{\beta}{0.4}+1\geq 0\, ,\quad
+1-\frac{\beta}{0.4}\geq 0\, ,\quad
+\gamma\geq 0\, ,\notag\\
+1-\frac{\gamma+\beta}{0.4}\geq 0\, ,\quad
+\frac{\hat{\Pi}_{(\p-\p_f)}(\lambda,\beta,\gamma,u)}{\Pi_{{\rm allowable}}}
+-1\geq0\, , \label{eq28}
+\end{gather}
+where $u$ denotes the level of uncertainty associated with the design
+variables and $\hat{\Pi}_{(\p-\p_f)}$ denotes the predicted possibility of
+failure, obtained from (\ref{eq27}).
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Results}
+\label{sec06}
+
+In order to obtain an upper limit of the weight for the fuzzy set based
+design, the deterministic design was evaluated first.
+The nondimensional cross-sectional area of the plate was minimized for a
+factor of safety equal to 1.5, making use of the formulation of (\ref{eq25}).
+The corresponding optimum design is summarized in Table~\ref{tab:5},
+where the values in parentheses are the possibility of failure values obtained
+from the reduced possibility of failure response surface approximations.
+
+\begin{table}[htbp]
+\caption{Deterministic optimum (uncertainty of the design variables equal
+to $\pm$5\%)\hsize=109pt}
+\tabcolsep5pt
+\label{tab:5}
+\begin{tabular}{@{}lc@{}}
+\hline\noalign{\smallskip}
+Variable & Value \\
+\noalign{\smallskip}\hline\hline\noalign{\smallskip}
+$\lambda$ & 0.6287 \\
+$\beta$ & -0.4000 \\
+$\gamma$ & 0.0447 \\
+$\tilde{A}^*$ & 0.6741 \\
+Factor of safety & 1.5 \\
+$\Pi_{{\rm Yield stress}}$ & \begin{tabular}{c} 0.0977 \\ (0.1181)\end{tabular}\\
+$\Pi_{{\rm Buckling}}$ & \begin{tabular}{c} 0.3411 \\ (0.3331)\end{tabular}\\
+\noalign{\smallskip}\hline
+\end{tabular}
+\end{table}
+
+For the deterministic optimum design, both failure criteria are active.
+The optimum design corresponds to a plate with a change in thickness that
+starts at the minimum allowable distance from the left endpoint of the plate
+(see Fig.~\ref{fig:4}) with a very short transition zone (small $\gamma$
+value).
+Even though both failure criteria are active for the optimum design, a large
+difference exists between the possibility of failure for the two failure
+criteria, with the buckling load constraint being critical.
+Both the possibility of failure values obtained from the vertex method and
+the values obtained from the reduced possibility of failure response surface
+approximation are shown.
+The accuracy of the reduced possibility of failure response surface
+approximation is demonstrated since the difference between the critical
+predicted and calculated possibility of failure values at the optimum design
+is only 2.3\%.
+
+The equivalent fuzzy set based design, using the $\tilde{A}^*$ value
+of Table~\ref{tab:5} as an upper limit of the weight are summarized in
+Table~\ref{tab:6}.
+Again, the values in parentheses are the possibility of failure values
+obtained from the reduced possibility of failure response surface
+approximations.
+The fuzzy set based optimum design corresponds to a plate where the change in
+thickness starts at the minimum allowable distance from the left endpoint of
+the plate with no transition zone ($\gamma$ value equal to 0).
+The fuzzy set based design eliminates the weight of the ramp and uses it to
+thicken the thin section of the plate.
+The result is an increase in the stress concentration and an improvement in
+the buckling load of the plate.
+The fuzzy set based design thus attempts to equalize the possibility of
+failure of the two failure criteria by making the yield stress failure
+criterion more critical and the buckling load constraint less critical.
+However, for the present example problem, the design variable limits kept
+the possibility of failure values from becoming equal at the optimum design.
+
+\begin{table}[htbp]
+\caption{Fuzzy optimum (uncertainty of the design variables equal
+to $\pm$5\%)\hsize=109pt}
+\label{tab:6}
+\tabcolsep5pt
+\begin{tabular}{@{}lc@{}}
+\hline\noalign{\smallskip}
+Variable & Value \\
+\noalign{\smallskip}\hline\hline\noalign{\smallskip}
+$\lambda$ & 0.6379 \\
+$\beta$ & -0.4000 \\
+$\gamma$ & 0.0000 \\
+$\tilde{A}^*$ & 0.6741 \\
+Factor of safety & 1.4898 \\
+$\Pi_{{\rm Yield stress}}$ & \begin{tabular}{c} 0.1319 \\ (0.1262)\end{tabular}\\
+$\Pi_{{\rm Buckling}}$ & \begin{tabular}{c} 0.2788 \\ (0.2721)\end{tabular}\\
+\noalign{\smallskip}\hline
+\end{tabular}
+\end{table}
+
+For the fuzzy set based design, the factor of safety is not much different
+from that of the deterministic design (only 0.7\% lower), however, there is a
+large difference in the possibility of failure between the two designs.
+The possibility of failure for the fuzzy set based design is 22.3\% lower than
+that of the deterministic design.
+As before, both the predicted and calculated possibility of failure values
+are shown in Table~\ref{tab:6}, with the difference between the critical
+values equal to only 2.4\%.
+
+The possibility distributions of failure for each failure mode of the optimum
+designs obtained from the two methods are shown graphically in
+Fig.~\ref{fig:6}.
+The possibility distributions of Fig.~\ref{fig:6} clearly illustrate the
+differences in the way each method maximizes the safety measure for a given
+weight as discussed in Section~\ref{sec05}.
+
+\begin{figure}
+\vspace{5cm}
+\caption{Possibility distributions of failure for the deterministic and fuzzy
+set based optimum designs}
+\label{fig:6}
+\end{figure}
+
+An important tool for determining the tolerances to which a structure will be
+manufactured, is to know the dependence of the weight on the uncertainty
+associated with the geometry of the structure.
+The dependence of the weight of the structure on the level of uncertainty
+associated with the design variables was thus also studied.
+For this study, the possibility of failure was kept constant at the optimum
+value obtained from the fuzzy set based design (i.e., 0.2788 as summarized in
+Table~\ref{tab:6}), while the level of uncertainty associated with the design
+variables $\lambda$, $\beta$ and $\gamma$ was varied between $\pm$2\% and
+$\pm$20\%.
+Seven levels of uncertainty, evenly distributed between $\pm$2\% and
+$\pm$20\%, were considered.
+For each of these levels, the reduced possibility of failure response surface
+approximations and the Microsoft Excel solver was used to minimize the
+nondimensional cross-sectional area for the specified possibility of failure.
+As expected, the nondimensional cross-sectional area of the plate increased
+with an increase in the level of uncertainty and the results are shown
+graphically in Fig.~\ref{fig:7}.
+
+\begin{figure}
+\vspace{5cm}
+\caption{Nondimensional cross-sectional area associated with different level
+of uncertainty in the design variables}
+\label{fig:7}
+\end{figure}
+
+Figure~\ref{fig:7} indicates that the increase in weight is almost linearly
+proportional to the increase in the uncertainty associated with the design
+variables.
+The nondimensional cross-sectional area increased by 10.7\% with an 18\%
+increase in the uncertainty associated with the design variables.
+Using Fig.~\ref{fig:7} and the dependence of the manufacturing cost on the
+tolerance of $\lambda$, $\beta$ and $\gamma$, the designer may determine what
+tolerance to use in manufacturing the plate.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Concluding remarks}
+\label{sec07}
+
+It is shown that response surface approximations provide an effective
+approach for reducing the computational cost associated with performing a
+fuzzy set based design for uncertainty.
+The large number of computationally expensive finite element analyses required
+to perform the fuzzy set based design is replaced by response surface
+approximations that are inexpensive to evaluate.
+By using response surface approximations, the computational burden shifts
+from the optimization problem to the problem of constructing the response
+surface approximations.
+Due to the iterative nature of the design process, the fact that response
+surface approximations allow multiple optimizations at minimal cost should be
+an attractive feature to any designer.
+The present paper also made use of response surface approximations to simplify
+the integration of the analysis code and the optimization algorithm.
+
+It was shown that for the same upper limit of the weight, the fuzzy set based
+design resulted in an optimum design with a possibility of failure 22.3\%
+lower than the corresponding deterministic design.
+Additionally, the factor of safety of the fuzzy set based design is only 0.7\%
+smaller than that of the deterministic design and for this example problem the
+fuzzy set based design is thus clearly superior.
+Finally, the dependence of the structural weight on the uncertainty of some
+key geometric parameters is presented in the form of a design chart and may be
+used, together with the manufacturing cost, to determine the tolerances that
+when manufacturing the plate.
+This design chart would have been very time consuming to construct if response
+surface approximations were not used to reduce the computational cost.
+
+\begin{acknowledgement}
+This work was supported by NASA grants NAG1-1669 and NAG1-2000.
+\end{acknowledgement}
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