Reliability-based optimization in engineering using decomposition techniques and FORMS

In this paper a review of some decomposition techniques previously given by the authors to solve bi-level problems is presented within a unified formulation, and a new variant is investigated. Different reliability-based optimization problems in engineering works are formulated and solved: (a) the failure-probability safety-factor problem that makes the compatibility of the classical approach, based on safety-factors; and the modern probability-based approach possible; (b) a modern reliability-based approach where the design is based on minimizing initial/construction costs subject to failure-probability and safety-factor bounds for all failure modes; (c) minimizing the expected total cost of a structure, including maintenance and construction, which depend on the failure probabilities; and (d) a mixed model minimizing the expected total cost adding failure-probability and safety-factor bounds for all failure modes. In these four problems, the objective consists of selecting the values of the design variables that minimize the corresponding cost functions subject to some reliability conditions together with geometric and code constraints. The solution becomes complex because the evaluation of failure probabilities using first-order reliability methods (FORM) involves one optimization problem per failure mode, so that decomposition methods are used to solve the problem. The proposed methods use standard optimization frameworks to obtain the reliability indices and to solve the global problem within a decomposition scheme. An advantage of these approaches is that the optimization procedure and the reliability calculations are decoupled. In addition, a sensitivity analysis is performed using a method that consists of transforming the data parameters into artificial variables, and using the dual associated variables. To illustrate the methods, a breakwater design example is used.