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 the 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, the 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.

中文翻译：

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中断后相互依存系统恢复的控制策略

我们研究了一个控制问题，其中系统组件的状态在中断后恶化，如果它们没有被实体修复。组件之间存在一组优先约束形式的依赖关系，由有向无环图 (DAG) 捕获。该实体的目标是在给定时间内最大限度地增加状态恢复到完全修复状态的组件数量。我们证明一般问题是 NP-hard，因此我们为问题的特殊实例描述了接近最优的控制策略。我们表明，当劣化率大于或等于修复率并且优先约束由 DAG 给出时，最好继续修复组件，直到其状态达到完全恢复状态，然后再切换到修复任何其他组件。在上述假设下，当所有组件的劣化和修复率都是同质的时，我们证明了在每个时间步针对最健康组件的控制策略，同时尊重优先级和时间限制，完全修复了至少一半的组件。将通过最佳策略完全修复的组件。最后，我们证明当修复率足够大于恶化率时，优先约束由一组不相交的树给出，每个树最多包含 k 个节点，并且没有时间约束，