The exact calculation of all-terminal reliability is not feasible in large networks. Hence estimation techniques and lower and upper bounds for all-terminal reliability have been utilized. Here, we propose using an ordered subset of the mincuts and an ordered subset of the minpaths to calculate an all-terminal reliability upper and lower bound, respectively. The advantage of the proposed new approach results from the fact that it does not require the enumeration of all mincuts or all minpaths as required by other bounds. The proposed algorithm uses maximally disjoint minpaths, prior to their sequential generation, and also uses a binary decision diagram for the calculation of their union probability. The numerical results show that the proposed approach is computationally feasible, reasonably accurate and much faster than the previous version of the algorithm. This allows one to obtain tight bounds when it not possible to enumerate all mincuts or all minpaths as revealed by extensive tests on real-world networks. © 2015 Wiley Periodicals, Inc. NETWORKS, 2015