Monomial ideals and the Scarf complex for coherent systems in reliability theory

Abstract

A certain type of integer grid, called here an echelon grid, is an object found both in coherent systems whose components have a finite or countable number of levels and in algebraic geometry. If α=(α₁,…,α_d) is an integer vector representing the state of a system, then the corresponding algebraic object is a monomial x₁^α₁⋯x_d^α_d in the indeterminates x₁,…,x_d. The idea is to relate a coherent system to monomial ideals, so that the so-called Scarf complex of the monomial ideal yields an inclusion–exclusion identity for the probability of failure, which uses many fewer terms than the classical identity. Moreover in the “general position” case we obtain via the Scarf complex the tube bounds given by Naiman and Wynn [J. Inequal. Pure Appl. Math. (2001) 2 1–16]. Examples are given for the binary case but the full utility is for general multistate coherent systems and a comprehensive example is given.