Journal of Mechanical Science and Technology 00 (2013) 0000–0000
www.springerlink.com/content/1738-494x

# Robust multiobjective optimization method using satisfying trade-off method

Masahiro Toyoda¹ and Nozomu Kogiso¹,*

¹Department of Aerospace Engineering, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan

(Manuscript Received 000 0, 2013; Revised 000 0, 2013; Accepted 000 0, 2013)

## Abstract

This study proposes a robust multiobjective optimization approach using the satisfying trade-off method (STOM). STOM is a multi-objective optimization method that obtains a highly accurate single Pareto solution. Conventionally, a robust design is formulated as a single-objective optimization problem, where the objective function is defined as the weighted sum of the mean and standard deviation of the performance index. In this study, the mean and standard deviation are formulated as individual objective function. The effect of uncertainty can be investigated through Pareto surface. In STOM, the multiobjective optimization problem is transformed into the equivalent single objective problem by introducing an aspiration level. As the obtained single Pareto solution corresponds to the aspiration level that implies the ratio of the designer's desired objective function, the designer can investigate only the desired space in detail through setting the aspiration without obtaining full Pareto surfaces. The validity of using STOM for a robust multiobjective optimization problem is discussed using numerical examples.

Keywords: Multiobjective optimization, Robust design, Satisficing trade-off method, Uncertainty

## 1. Introduction

Robust optimal design is widely applied to engineering design problems that consider the uncertainties of design variables and parameters such as the material constants and applied load conditions [1, 2]. Robust design optimization is conventionally defined as a single-objective optimization method, where the objective function is defined as the weighted sum of the expected value and the standard deviation of the performance index. Such a weighted sum approach is one of the methods for transforming a multiobjective optimization problem into the equivalent single-objective optimization problem. It is also known that this approach sometimes fails to obtain the desired optimum solution, especially when the Pareto set has a concave shape.

This paper proposes a method for directly formulating a robust design problem as a multiobjective optimization problem [3], where both the expected value and standard deviation of the performance index are adopted as objective functions. Then, the robust design problem is solved using the satisfying trade-off method (STOM) [4]. STOM is known to be an interactive optimization method and has been applied to many different kinds of multiobjective optimization problems [5, 6]. STOM 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. In addition, the user can interactively repeat the optimization until the desired Pareto solution is found by updating the aspiration level. The automatic trade-off analysis method [7] is one of the methods used to reasonably update the aspiration level.

The validity of using STOM for a robust multiobjective optimization problem is discussed through numerical examples. In particular, the accuracy of the Pareto set obtained by parametrically changing the aspiration level is demonstrated. Then, the effect of the random variables on the Pareto set is investigated in the small region of the Pareto set.

## 2. Robust Multiobjective Optimal Design

The robust design optimization is formulated to obtain the design with the smallest deterioration in performance under a variety of uncertain design parameters, as well as a reasonable higher performance, as illustrated in Fig. 1, where $z_0$ and $\Delta z$ denote the mean and variation of the random variables, respectively. The deterministic optimal solution $x_{opt}$ may typically show larger variation in the objective function $\Delta f_{opt}$ under variation in the random variables $z$. In contrast, the robust optimal solution will yield a smaller corresponding variation $\Delta f_{robust}$ even if the objective function of the robust design $f(x_{robust})$ is worse than that of the deterministic optimal design $f(x_{opt})$.

*Corresponding author. Tel.: +81-72-254-9245, Fax.: +81-72-254-9906
E-mail address: kogiso@aero.osakafu-u.ac.jp

¹Recommended by Editor 000 000
© KSME & Springer 2013