Control Policies for Recovery of Interdependent Systems After Disruptions

We examine a control problem where the states of the components of a system deteriorate after a disruption, if they are not being repaired by an entity. There exist a set of dependencies in the form of precedence constraints between the components, captured by a directed acyclic graph (DAG). The objective of the entity is to maximize the number of components, whose states are brought back to the fully repaired state within a given time. We prove that the general problem is NP-hard and, therefore, we characterize near-optimal control policies for special instances of the problem. We show that when the deterioration rates are larger than or equal to the repair rates and the precedence constraints are given by a DAG, it is optimal to continue repairing a component until its state reaches the fully recovered state before switching to repair any other component. Under the aforementioned assumptions and when the deterioration and the repair rates are homogeneous across all the components, we prove that a control policy that targets the healthiest component at each time-step while respecting the precedence and time constraints fully repairs at least half the number of components that would be fully repaired by an optimal policy. Finally, we prove that when the repair rates are sufficiently larger than the deterioration rates, the precedence constraints are given by a set of disjoint trees that each contain at most $k$ nodes, and there is no time constraint, a policy that targets the component with the least value of health minus the deterioration rate at each time-step while respecting the precedence constraints fully repairs at least $1/k$ times the number of components that would be fully repaired by an optimal policy.