Dynamic Set-Covering for Real-Time Multiple Fault Diagnosis

Abstract

The set-covering problem has been widely used to model many real world applications. In this paper, we formulate a time-dependent set covering problem, viz., 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 fault diagnosis problem, wherein multiple faults may evolve over time and the fault-test dependencies are deterministic. The objective of the DSC problem is to evaluate the most likely evolution of the minimal set of columns (component fault states) covering the rows (failed tests) of the DSC constraint matrix at a minimum cost or maximum reward. The DSC problem is an NP-hard and intractable due to the coupling among the rows and columns via the constraint matrix, and the temporal dependence of columns over time. By relaxing the constraints using Lagrange multipliers, the DSC problem can be decoupled into several subproblems; each corresponding to a column of the constraint matrix. 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 the subgradient method. The proposed primal-dual optimization framework provides a measure of suboptimality via approximate duality gap. Capabilities of the proposed algorithm are demonstrated by way of its application to the dynamic versions of the set-covering problems from the Beasley's OR-library and to real-world diagnostic models.