An Algebraic Version of the Sum-of-disjoint-products Method for Multi-state System Reliability Analysis

The evaluation of system reliability is an NP-hard problem even in the binary case. There exist several general methodologies to analyze and compute system reliability. The two main ones are the sum-of-disjoint-products (SDP), which expresses the logic function of the system as a union of disjoint terms, and the Improved Inclusion-Exclusion (IIE) formulas. The algebraic approach to system reliability, assigns a monomial ideal to the system and computes its reliability in terms of the Hilbert series of the ideal, providing an algebraic version of the IIE method. In this paper we make use of this monomial ideal framework and present an algebraic version of the SDP method, based on a combinatorial decomposition of the system's ideal. Such a decomposition is obtained from an involutive basis of the ideal. This algebraic version is suitable for binary and multi-state systems. We include computer experiments on the performance of this approach using the C++ computer algebra library CoCoALib and a discussion on which of the algebraic methods can be more efficient depending on the type of system under analysis.