Robust Optimization Using the Probabilistic Robustness Index and the Enhanced Single-Loop Approach

Abstract

I N RECENT years, the robust design optimization (RDO) methodology has been developed to consider the uncertainties in conventional design optimization. RDO is a method that minimizes the effect of variation without eliminating the causes [1–9]. It has been known as an effective designmethod to improve the quality of a product. It deals with both the robustness of the objective function and the feasibility of the constraints. In RDO, the robustness of the objective function can be achieved by reducing the change of the objective function with respect to the changes of the design variables. Therefore, robustness indices of the objective function pursue an insensitive design when there are variations on design variables and/or parameters. To accomplish the purpose, various robustness indices on the objective function have been proposed [1–9]. A design formulation with the weighted sum method has beenwidely used due to simple and easy application [3,8]. Robustness of the constraints means that all of the constraints are satisfied within the range of the variations for the design variables and/or parameters. Typical approaches of previous studies can be classified into three types: the worst-case method, the penalty method, and the probabilistic method [3,10–16]. In the probabilistic method, the constraints with uncertainties are replaced by the probabilistic constraints. Generally, it is well known as reliabilitybased design optimization (RBDO). In RBDO, the feasibility of design constraints is calculated exactly by a probability theory in RBDO, whereas it is approximately calculated in the worst-case method and the penalty method. The reliability-based robust design optimization (RBRDO) is known as a method that treats the constraints with uncertainties by using RBDO methods in RDO [17,18]. In this study, the formulation and the numerical method for RDO are addressed. An RDO algorithm is made with the newly proposed robustness index and the enhanced single-loop single vector approach. First, a probabilistic robustness index is proposed to deal with the robustness of the objective function. It is derived from the probabilistic theory. By using the robustness index, the mean value and the variance of the objective function are simultaneously controlled. Second, an effective RBDOmethod is used to achieve the feasibility of the constraints. The feasibility robustness is exactly guaranteed by using the RBDOmethod.Among theRBDOmethods, the single-loop single-vector approach using the conjugate gradient method (SLSVCG), whichwas recently proposed by Jeong and Park, is employed to treat the constraints efficiently [15]. By using the new algorithm, the robustness of the objective function is considered in the optimization process while the feasibility of the constraints is guaranteed with the specified target reliability. Two mathematical problems are defined to show the distinct efficiency of the used methods, and a practical engineering problem is employed for assessment of the methods in the application viewpoint. The results of the proposed method are compared with those from combinations of other methods for RDO.