A Prime-Logarithmic Method for Optimal Reliability Design

Optimal reliability design (ORD) problem is challenging and fundamental to the study of system reliability. For a system with n components/stages where each of them can be set in m possible reliability levels, state-of-the-art linear reformulation models of ORD problem require O(nm) binary variables, O(mⁿ) continuous variables together with either O(mⁿ) inequality constraints or O(nm) equality constraints. Using the special property of prime factorization and adopting the logarithmic expression technique, in this article, we propose a novel linear reformulation model of the ORD problem requiring O(nm) binary variables, O(mⁿ ⁄n!) continuous variables, and very few linear constraints. This theoretic reduction in variables and constraints can lead to significant savings in computational efforts. Our numerical experiments further confirm the drastic reduction in computational time for solving ORD problems in large size.