A nested hierarchy of second order upper bounds on system failure probability

For a coherent, binary system made up of elements, the exact failure probability requires knowledge statistical dependence all orders among minimal cut sets. Since sets beyond second order is generally difficult to obtain, bounds on have practical value. The upper bound conservative by definition and can be adopted in reliability based decision making. In this paper we propose new hierarchy m-level bounds, Bm : well-known Kounias-Vanmarcke-Hunter-Ditlevsen (KVHD) - current standard for using joint probabilities turns out weakest member family (m = 1). We prove that non-increasing with level m every ordering sets, derive conditions under which Bm+1 strictly less than any ordering. also optimal + 1 bound, show improvement asymptotically achieves as long are only constrained pair corresponding first probabilities. Numerical examples our yield tighter values previously achieved case exhibit considerable scatter across entire n! orderings compared KVHD bounds. Our results therefore may lead more efficient identification when coupled existing linear programming tree search approaches.