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