Theorem specifying how all minimal cut (path) vectors to level j can be obtained from all minimal path (cut) vectors to level j in any multistate monotone system is proven. Characterizations of binary type multistate monotone systems and binary type multistate strongly coherent systems by minimal cut and path vectors and corresponding to them, binary type cut and path sets are also demonstrated.
SinnamonRMAndrewsJD. New approaches to evaluating fault trees. Reliab Eng Syst Safe1997; 58: 89–96.
2.
VatnJ. Finding minimal cut sets in a fault tree. Reliab Eng Syst Safe1992; 36: 59–62.
3.
RauzyA. New algorithms for fault trees analysis. Reliab Eng Syst Safe1993; 40: 203–211.
4.
JungWSHanSHHaJ. A fast BDD algorithm for large coherent fault trees analysis. Reliab Eng Syst Safe2004; 83: 369–374.
5.
BolligBWegenerI. Improving the variable ordering of OBDDs is NP-complete. IEEE T Comput1996; 45: 993–1002.
6.
RochdiZDrissBMohamedT. Industrial systems maintenance modelling using Petri nets. Reliab Eng Syst Safe1999; 65: 119–124.
7.
LiuTSChiouSB. The application of Petri nets to failure analysis. Reliab Eng Syst Safe1997; 57: 129–142.
8.
AhmadSH. Simple enumeration of minimal cutsets of acyclic directed graph. IEEE T Reliab1988; 37: 484–487.
9.
SinghB. Enumeration of node cutsets for an s-t network. Microelectron Reliab1994; 34: 559–561.
10.
YanLTahaHALandersTL. A recursive approach for enumerating minimal cutsets in a network. IEEE T Reliab1994; 43: 383–388.
11.
AbelUBickerR. Determination of all minimal cut-sets between a vertex pair in an undirected graph. IEEE T Reliab1982; R-31(2): 167–171.
12.
FardNSLeeT-H. Cutset enumeration of network systems with link and node failures. Reliab Eng Syst Safe1999; 65: 141–146.
13.
YehWC. Search for all MCs in networks with unreliable nodes and arcs. Reliab Eng Syst Safe2003; 79: 95–101.
14.
EmadiAAfrakhteH. A novel and fast algorithm for locating minimal cuts up to second order of undirected graphs with multiple sources and sinks. Int J Elec Power2014; 62: 95–102.
15.
RebaiaiaMLAit-KadiD. New technique for generating minimal cut sets in nontrivial network. AASRI Procedia2013; 5: 67–76.
16.
ShenY. A new simple algorithm for enumerating all minimal paths and cuts of a graph. Microelectron Reliab1995; 35: 973–976.
17.
JaneCCLinJSYuanJ. On reliability evaluation of a limited-flow network in terms of minimal cutsets. IEEE Trans. Rel1993; 42(3): 354–361.
18.
YanZQianM. Improving efficiency of solving d-MC problem in stochastic-flow network. Reliab Eng Syst Saf2007; 92(1): 30–39.
19.
YehWC. A fast algorithm for searching all multi-state minimal cuts. IEEE Trans Rel2008; 57(4): 581–588.
20.
Forghani-elahabadMMahdavi-AmiriN. A new efficient approach to search for all multi-state minimal cuts. IEEE Trans Rel2014; 63(1): 154–166.
21.
NiuYFGaoZYLamWHK. A new efficient algorithm for finding d-minimal cuts in multi-state networks. Reliab Eng Syst Saf2017; 166: 151–163.
22.
YehWC. A new approach to the d-MC problem. Reliab Eng Syst Saf2002; 77(2): 201–206.
23.
YehWC. A simple MC-based algorithm for evaluating reliability of stochastic-flow network with unreliable nodes. Reliab Eng Syst Saf2004; 83(1): 47–55.
24.
YehWCBaeCHuangCL. A new cut-based algorithm for the multi-state flow network reliability problem. Reliab Eng Syst Saf2015; 136: 1–7.
25.
LinK. Using minimal cuts to evaluate the system reliability of a stochastic-flow network with failures at nodes and arcs. Reliab Eng Syst Saf2002; 75(1): 41–46.
26.
NiuYFXuXZ. A new solution algorithm for the multistate minimal cut problem. IEEE Trans Rel. Epub ahead of print 2 September 2019. Doi: 101109; /TR201929356302019.
27.
NiuYFGaoZYLamWHK. Evaluating the reliability of a stochastic distribution network in terms of minimal cuts. Transp Res Part E2017; 100: 75–97.
28.
YehWC. Fast algorithm for searching d-MPs for all possible d. IEEE Trans Rel2018; 67(1): 308–315.
29.
HudsonJCKapurKC. Reliability analysis for multistate systems with multistate components. IIE Trans1983; 15(2): 127–135.
30.
ZuoMTianZGHuangHZ. An efficient method for reliability evaluation of multistate networks given all minimal path vectors. IIE Trans2007; 39(8): 811–817.
31.
BaiGHZuoMTianZG. Ordering heuristics for reliability evaluation of multistate networks. IEEE Trans Rel2015; 64(3): 1015–1023.
32.
YehWC. An improved sum-of-disjoint-products technique for symbolic multi-state flow network reliability. IEEE Trans Rel2015; 64(4): 1185–1193.
33.
AlexopoulosC. Note on state-space decomposition methods for analyzing stochastic flow networks. IEEE Trans Rel1995; 44(2): 354–357.
34.
BaiGHTianZGZuoM. Reliability evaluation of multistate networks: an improved algorithm using state-space decomposition and experimental comparison. IISE Trans2018; 50(5): 407–418.
35.
ProvanJSBallMO. The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J Comput1983; 12: 777–788.
36.
TanZ. Minimal cut sets of ST networks with k-out-of-n nodes. Reliab Eng Syst Saf2003; 82: 49–54.
37.
YehWC. A simple algorithm to search for all MCs in networks. Europ J Oper Res2006; 174: 1694–1705.
38.
Forghani-elahabadMMahdavi-AmiriN. A note on ‘A simple algorithm to search for all MCs in networks’. Europe J Oper Res2014; 174: 1694.
39.
EiterTMakinoKGottlobG. Computational aspects of monotone dualization: a brief survey. Discrete Appl Math2008; 156: 2035–2049.
40.
EiterTGottlobG. Identifying the minimal transversals of a hypergraph and related problems. SIAM J Comput1995; 24: 1278–1304.
41.
ElbassioniKM. On the complexity of monotone dualization and generating minimal hypergraph transversals. Discrete Appl Math2008; 156: 2109–2123.
42.
KhachiyanLBorosEElbassioniK, et al. An efficient implementation of a quasi-polynomial algorithm for generating hypergraph transversals and its application in joint generation. Discrete Appl Math2006; 154: 2350–2372.
43.
RauzyA. Mathematical foundations of minimal cutsets. IEEE T Reliab2001; 50: 389–396.
44.
GurvichVKhachiyanL. On generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions. Discrete Appl Math1999; 96–97: 363–373.
45.
MihovaMPopeskaZ. Estimation of minimal path vectors of multi-state systems from failure data. Adv Intel Soft Comput2011; 2011: 199–206.
46.
CrowderMJKimberACSmithRL, et al. Data. London: Chapman and Hall, 1991.
47.
NatvigB. Multi-state reliability theory. Statistical research report, Department of Mathematics, University of Oslo, Oslo, 2007.
48.
NatvigB. Multistate systems reliability theory with applications. Hoboken, NJ: John Wiley & Sons, 2011.
49.
MihovaMPopeskaZ. Minimal path and cut vectors of binary type multi-state monotone systems. Math Maced2006; 4: 75–80.
