This paper considers the component system with stair-type consecutive minimal cuts. The system consists of n components and the set of minimal cuts can be linearly ordered. The proposed system generalizes the typical consecutive-k-out-of-n: F systems. By using integer linear programming, this paper shows that such a system can be converted into the consecutive-k-out-of-n: F systems with the insertion of artificial "broken-down" components. Then the system reliability can be obtained by the product form of component reliability matrices and the limit behavior of system could be easily analyzed. Additionally, we show that the integer constraints of the linear programming can be relaxed due to the total unimodularity. Thus, a general linear programming can be used to solve the problem. Numerical examples show the simple and effective new approach.
A characterization of circle graphs in terms of total unimodularity
