In a recent article, Yeh presents a method for computing the multi-state two-terminal reliability in terms of capacitated-minimum paths (CMPs). It is claimed that the proposed algorithm has the following advantages: (1) it is just based on the special property of CMPs of which the max-flow in these paths are all equal to a given capacity (say d) and can be used to search for all capacitated minimum-paths without knowing all minimum-paths in advance; (2) it is simple and more effective in finding CMP candidates than the existing methods and (3) the proposed method is easier to understand and implement. Unfortunately, the proposed algorithm is incorrect as can be shown by the following counterexample.

Referring to Figure 1 and Table 1, we first note that since capacities of all arcs are 0, 3 and 5, any system-state vector that can (resp. cannot) send 1 or 2 unit of demand from s to t is able (respectively unable) to send 3 units of demand from s to t. Thus the sets of all 1-mps, 2-mps and 3-mps are equivalent. Similarly, the set of all 4-mps equals the set of all 5-mps.

Table 1 Capacities and capacity probabilities of arcs of Figure 1
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Figure 1
figure 1

A limited-flow network.

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Case 1: d=1, 2, or 4.

Since no summation of arc capacities equals d (Equation (8) in Theorem 2), Yeh's algorithm returns an empty set of d-mp when d is 1, 2 or 4.

Case 2: d=5

To derive all 5-mps, a summation of arc capacities being equal to 5 (Equation (8) in Theorem 2) restricts arcs to take on capacity values 0 or 5. It is incorrect, since combination of two disjoint 3-mps from s to t (for example, system-state vector (xs1,x1t,x12,xs2,x2t)=(3,3,0,3,3)) is also a 5-mp but is not searched by Yeh.

Consequently, Yeh's algorithm is unavailable for searching all d-mps when d has values 1, 2 and 4, and is incorrect when d=5.

The mistake is due to the statement of advantage 1 ‘it is just based on the special property of CMPs of which the max-flow in these paths are all equal to a given capacity (say d) and can be used to search for all capacitated minimum-paths without knowing all minimum-paths in advance’ which is the fundamental of Theorem 2. In case 1 (d=1, 2 or 4) of the counterexample, it yields an empty set of 1-mp, 2-mp or 4-mp. In case 2 (d=5) of the counterexample, it yields an incorrect set of all 5-mps. It is also note here that the system reliability for level 3 in Example 2 is 0.611415, value 0.64515 in the article is not correct.

Since Yeh's algorithm is incorrect, there is no existing algorithm that can search for all d-mps without knowing all binary-state minimal paths in advance. As a result, the NP-hard multi-state two-terminal reliability will be solved in terms of three NP-hard problems, say searching all binary-state minimal paths, searching all multi-state minimal paths (d-mps), and applying the inclusion-exclusion method to compute the reliability value. The effort of finding an efficient algorithm that can search all d-mps without locating all binary-state minimal paths in advance is worthwhile.