samples/texts/2926839/page_3.md

Fig. 1. Concept of robust optimization

The objective function of the robust design optimization is conventionally formulated as the weighted sum of the mean value and the standard deviation of the objective function, as follows:

$$
\text{Minimize} \quad f_r(x) = E[f(x,z)] + \alpha \sqrt{\mathrm{Var}[f(x,z)]} \quad (1)
$$

where x and z are the design and random variables, respective-
ly, and α is a weighting coefficient. As the linear approxima-
tion, the mean value and variance of the objective function f
can be evaluated according to the following equation [8]:

$$
E[f(\mathbf{x},z)] \approx f(E[\mathbf{x}]) E[z] \quad (2)
$$

$$
\operatorname{Var}[f(\mathbf{x}, z)] \approx \sum_{i=1}^{n_x} \left(\frac{\partial f}{\partial x_i}\right)^2 \operatorname{Var}[x_i] + \sum_{i=1}^{n_z} \left(\frac{\partial f}{\partial z_i}\right)^2 \operatorname{Var}[z_i] \quad (3)
$$

where $n_x$ and $n_z$ are the numbers of the design and random variables, respectively.

The single-objective optimization using Eq. (1) sometimes fails to obtain the desired solution, especially when the Pareto set has a non-convex shape, as shown in Fig. 2. In this study, the robust design optimization is formulated as the following multiobjective optimization problem:

$$
\begin{align}
\text{Minimize: } & f(x,z) = (f_1, f_2) \tag{4} \\
& f_1 = E[f(x,z)] \nonumber \\
& f_2 = \sqrt{\text{Var}[f(x,z)]} \nonumber \\
\text{subject to: } & g_j(x,z) \le 0 \quad (j=1,\dots,m) \nonumber \\
& x_i^L \le x_i \le x_i^U \quad (i=1,\dots,n_x) \nonumber
\end{align}
$$

where $g_j(x, z)$ and $(j = 1, ..., m)$ are constraint conditions, and $x_i^U$ and $x_i^L$ are the upper and lower limits of the design variables, respectively.

**3. Satisficing Trade-off Method (STOM)**

STOM is known to be an interactive optimization method and converts a multiobjective optimization problem into the equivalent single-objective optimization problem by introducing an aspiration level that corresponds to the user's preference for each objective function value. The flow of the STOM is

Fig. 2. Case of non-convex Pareto set

Fig. 3. Computational flow of STOM

summarized in Fig. 3 and briefly described as follows.

Step 1: Set the ideal point $f_i^I$, ($i$ = 1, ..., $k$) of each objective function. The ideal point is usually determined by solving a single-objective optimization problem considering only the corresponding objective function, $f_i(\mathbf{x}, \mathbf{z})$. The ideal point for the mean performance is obtained by solving the deterministic design problem.

Step 2: Set the aspiration level $f_i^A$, ($i$ = 1, ..., $k$) of each objec-
tive function and evaluate the weight coefficient as fol-
lows:

$$
w_i = \frac{1}{f_i^A - f_i^I}, \quad (i = 1, \dots, k) \tag{5}
$$

Step 3: Formulate the multiobjective optimization problem in Eq. (4) into the weighted Tchebyshev norm problem as follows:

$$
\begin{align*}
\text{Minimize:} \quad & \max_{i=1,\cdots,k} w_i (f_i(\mathbf{x}) - f_i^I) && (6) \\
\text{subject to:} \quad & g_j(\mathbf{x},\mathbf{z}) < 0, && (j=1,\cdots,m) \\
& x_i^L \le x_i \le x_i^U && (i=1,\cdots,n_x)
\end{align*}
$$

Step 4: The min-max problem in Eq. (6) is transformed into
the equivalent single-objective problem by introducing a