Efficient probabilistic multi-objective optimization of complex systems using matrix-based Bayesian network

For optimal design and maintenance of complex systems such as civil infrastructure systems or networks, the optimization problem should take into account the system-level performance, multiple objectives, and the uncertainties in various factors such as external hazards and system properties. Influence Diagram (ID), a graphical probabilistic model for decision-making, can facilitate modeling and inference of such complex problems. The optimal decision rule for ID is defined as the probability distributions of decision variables that minimize (or maximize) the sum of the expected values of utility variables. However, in a discrete ID, the interdependency between component events that arises from the definition of the system event, results in the exponential order of complexity in both quantifying and optimizing ID as the number of components increases. In order to address this issue, this paper employs the recently proposed matrix-based Bayesian network (MBN) to quantify ID for large-scale complex systems. To reduce the complexity of optimization to polynomial order, a proxy measure is also introduced for the expected values of utilities. The mathematical condition that makes the optimization problems employing proxy objective functions equivalent to the exact ones is derived so as to promote its applications to a wide class of problems. Moreover, the proposed proxy measure allows the analytical evaluation of a set of non-dominated solutions in which the weighted sum of multiple objective values is optimized. By using the strategies developed to compensate the errors by the approximation as well as the weighted sum formulation, the proposed methodology can identify even a larger set of non-dominated solutions than the exact objective function of weighted sum. Four numerical examples demonstrate the accuracy and efficiency of the proposed methodology. The supporting source code and data are available for download at https:/github.com/jieunbyun/GitHub-MBN-DM-code.