My method proposed in the published paper entitled ‘A Novel Method for the Network Reliability in Terms of Capacitated-Minimum-Paths without Knowing Minimum-Paths in Advance’ (J Opl Res Soc, 2005, Vol. 56, pp 1235–1240) can be implemented to search for all CMPs in the network of which the state (ie the capacity level) of each arc, for example arc e, is 0,1,2,…,W(e).

If the arc state is defined as in the special cases discussed in Jane's comment, two modifications are needed for the proposed method to find all CMPs as follows:

  1. 1

    The state variable x i (in the definition of X of Section 2 and in Equation (9) of Theorem 2) needs to change from the original definition ‘x i =0,1,…,W(e i )’ to the new definition ‘x i ∈{c ik | where c ik is the kth state of e i ∈E}’.

  2. 2

    Change Step 1 in the proposed algorithm of Section 4:Step 1 (Implement Theorem 2): Construct and use the implicit algorithm to solve all of the feasible solutions (CMP candidates), say p1,p2,…,p π , to Equations (7)–(9). If there is any feasible solution, go to Step 2. If no feasible solution exists and d is less than the max-flow in the network, let d=d+1 and go to Step 1. Otherwise, halt.

After the above two simple modifications, all CMPs can be found using my method without needing any MPs in advance.