Dynamic Set-Covering for Real-Time Multiple Fault Diagnosis With Delayed Test Outcomes

The set-covering problem is widely used to model many real-world applications. In this paper, we formulate a generalization of set-covering, termed dynamic set-covering (DSC), which involves a series of coupled set-covering problems over time. We motivate the DSC problem from the viewpoint of a dynamic multiple fault diagnosis problem, wherein faults, possibly intermittent, evolve over time; the fault-test dependencies are deterministic (components associated with passed tests cannot be suspected to be faulty and at least one of the components associated with failed tests is faulty), and the test outcomes may be observed with delay. The objective of the DSC problem is to infer the most probable time sequence of a parsimonious set of failure sources that explains the observed test outcomes over time. The DSC problem is NP-hard and intractable due to the fault-test dependency matrix that couples the failed tests and faults via the constraint matrix, and the temporal dependence of failure sources over time. By relaxing the coupling constraints using Lagrange multipliers, the DSC problem can be decoupled into independent subproblems, one for each fault. Each subproblem is solved using the Viterbi decoding algorithm, and a primal feasible solution is constructed by modifying the Viterbi solutions via a heuristic. The Lagrange multipliers are updated using a subgradient method. The proposed Viterbi-Lagrangian relaxation algorithm provides a measure of suboptimality via an approximate duality gap. As a major practical extension of the above problem, we also consider the problem of diagnosing faults with delayed test outcomes, termed delay DSC. A detailed experimental evaluation of the algorithms is provided using real-world problems that exhibit masking faults.