۱٫ Introduction

۲٫ Quantification of influence diagram (ID) for complex systems using matrix-based Bayesian network (MBN)

۳٫ Proxy objective function for optimizing ID that has multiple strategically relevant decision variables

۴٫ Multi-objective optimization using proxy objective function

۵٫ Numerical examples

۶٫ Conclusions

Acknowledgement

References

Abstract

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.

Introduction

In various efforts to construct, operate, and maintain real-world complex systems such as civil infrastructures and their networks, it is crucial to identify optimal decision-making strategies especially when budgets are limited. When dealing with such systems, the optimization should be able to take into account not only the performance of individual components but also the system-level performance. However, mathematical formulations for the system-level optimization are not straightforward in general. In addition, for large-scale systems, multiple random variables (r.v.’s) are introduced to represent both external factors (e.g. natural or man-made hazards and deterioration-inducing environment) and internal factors (e.g. material or geometric properties of components and system), for which a high-dimensional joint probability distribution needs to be constructed. Probabilistic graphical models (PGMs) can be employed to this end, which can translate realworld causal relationships into mathematical representations [1,2].

Bayesian network (BN) is one of the most widely used PGMs, which facilitates the modeling of causal relationships between r.v.’s as a joint probability distribution by use of nodes and directed arrows (Fig. 1(a)). On the other hand, Influence Diagram (ID) is an extension of BN for the purpose of decision-making in which two additional types of variables are introduced, namely, decision and utility variables (Fig. 1(b)). As the terms imply, decision variables represent the decision alternatives while utility variables quantify the utilities of each instance of interest. The optimal decision rule for an ID is defined as the probability mass functions (PMFs) of decision variables that minimize (or maximize) the sum of the expectations of utility variables. Discrete BN and ID in which all r.v.’s are discrete, allow us to develop inference algorithms that are flexible and widely applicable.