Reliability-Based Optimization With Discrete and Continuous Decision and Random Variables

Engineering design problems frequently involve both discrete and continuous random and design variables, and system reliability may depend on the union or intersection of multiple limit states. Solving reliability-based design optimization (RBDO) problems, where some or all of the decision variables must be integer valued, can be expensive since the computational effort increases exponentially with the number of discrete variables in discrete optimization problems, and the presence of both system and component level reliability makes RBDO more expensive. The presence of discrete random variables in a RBDO problem has usually necessitated the use of Monte Carlo simulation or some other type of enumeration procedure, both of which are computationally expensive. In this paper, the theorem of total probability is used to allow for the use of the first-order reliability method in solving mixed-integer RBDO problems. Single-loop RBDO formulations are developed for three classes of mixed-integer RBDO with both discrete and continuous random variables and component and system-level reliability constraints. These problem formulations can be solved with any appropriate discrete optimization technique. This paper develops, for each of the three problem classes, greedy algorithms to find an approximate solution to the mixed-integer RBDO problem with both component and system reliability constraints and/or objectives. These greedy algorithms are based on the solution of a relaxed formulation and require hardly additional computational expense than that required for the solution of the continuous RBDO problem. The greedy algorithms are verified by branch and bound and genetic algorithms. Also, this paper develops three algorithms, which can allow for calibration of reliability estimation with a more accurate reliability analysis technique. These algorithms are illustrated in the context of a truss optimization problem.