An Improved Method for Multistate Flow Network ... - IEEE Xplore

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 41, NO. 2, MARCH 2011

An Improved Method for Multistate Flow Network Reliability With Unreliable Nodes and a Budget Constraint Based on Path Set Wei-Chang Yeh Abstract—Evaluating multistate flow network reliability and reducing system cost are important tasks when planning and designing systems. Existing methods are based on (d, c)-minimal paths ((d, c)-MP), which are vectors, such that d units of flow transmit between two specified nodes with a total cost that does not exceed c. However, these methods only work for directed networks. This correspondence paper finds all (d, c)-MPs before calculating network reliability under budget constraints using a novel method. The proposed algorithm is easier to understand and implement and is superior to existing algorithms. This correspondence paper analyzes and proves the correctness of the proposed algorithm, using two examples to demonstrate how to generate, verify, and implement all (d, c)-MPs to solve multistate flow network reliabilities under budget constraints using the proposed algorithm. Index Terms—Budget constraint, (d, c)-minimal path (MP)/d-MP/MP, multistate network, reliability.

N OMENCLATURE1 MFN MUFN MP/d-/(d, c)-MP IE G(V, E, W, C)

n, m ei p, π, δ

Xi

Fig. 1.

Directed bridge network.

Fig. 2.

Network corresponding to (2, 2, 0, 0, 1, 1).

Xi (ek ) = xij φi

Multistate flow network that satisfies the conservation law. An MFN with unreliable nodes. Minimal/d-minimal/(d, c)-minimal path. Inclusion–exclusion method. MFN with the set of nodes V = {1, 2, . . . , n}; the set of arcs E = {ei |1 ≤ i ≤ m}; W = (w1 , w2 , . . . , wn ), where wi = W (ei ) denotes the max-capacity of arc ei ; and C = (c1 , c2 , . . . , cn ), where ci = C(ei ) denotes the unit capacity cost of arc ei ; with nodes 1 and n being the source and sink nodes, respectively. For example, Fig. 1 is an MFN with V = {1, 2, 3, 4}, E = {e1 , e2 , e3 , e4 , e5 , e6 }, node 1 is the source node, and node 4 is the sink node. Numbers of nodes in V and arcs in E in G (V, E, W, C). ith arc in E. Numbers of MPs, (d, c)-MP candidates , (p = O (2m−n+1 ) and π = Min{ m+d−1 d mr [W (e ) + 1]}), and real (d, c)-MPs in i i=1 G(V, E, W, C). Xi = (xi1 , xi2 , . . . , xim ) is the ith systemstate vector. For example, X1 = (2, 2, 0, 0, 1, 1) is a system-state vector in Fig. 1.

fij Pi Fi G(V, E, Xi , C)

C(Xi )

W (Xi ) Manuscript received April 12, 2007; revised December 8, 2008; accepted September 23, 2009. Date of publication October 7, 2010; date of current version January 19, 2011. This work was supported in part by the National Science Council (NSC) of Taiwan under Grant NSC 97-2221-E-007-099-MY3. This paper was recommended by Associate Editor H. Pham. The author is with the e-Integration&Collaboration Laboratory, Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu 300, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMCA.2010.2069093 1 The

Capacity level d

singular and plural of an acronym are always spelled the same. 1083-4427/$26.00 © 2010 IEEE

Capacity level of arc ei under the systemstate vector Xi = (xi1 , xi2 , . . . , xim ). φi = (fi1 , fi2 , . . . , fip ) is the flow-state vector corresponding to the system-state vector Xi = (xi1 , xi2 , . . . , xim ) such that fik , for all ej ∈ Pk , j = 1, 2, . . . , xij = m, and k = 1, 2, . . . , p, where fij is the flow through MP Pj and p is the total number of MPs. For example, (f11 , f12 , f13 , f14 ) = (2, 0, 0, 1) is the corresponding flowstate vector of X1 = (2, 2, 0, 0, 1, 1) in Fig. 1, where P1 = {e1 , e2 }, P2 = {e1 , e3 , e6 }, P3 = {e2 , e4 , e5 }, and P4 = {e5 , e6 }. Amount of flow through MP Pj when the flow-state vector is φi = (fi1 , fi2 , . . . , fip ). ith MP in G(V, E, W, C). Fi = min{W (ej )| for each ej ∈ Pi } is the max-capacity on Pi . Subnetwork of G(V, E, W, C), where Xi = (xi1 , xi2 , . . . , xim ) and W (ek ) = xik in G(V, E, X, C) for all ek ∈ E. For example, Fig. 2 is the subnetwork represented by G(V, E, X1 , C) of Fig. 1 if X1 = (2, 2, 0, 0, 1, 1) and mW = (3, 2, 1, 1, 2, 2). C(Xi ) = j=1 cj xij is the total capacity cost for the system-state vector Xi = (xi1 , xi2 , . . . , xim ). For example, the capacity cost for X1 = (2, 2, 0, 0, 1, 1) is C(X1 ) = 2 × 3 +2×1+0×1+0×1+1×1+ 1 × 3 = 12 in Fig. 2 if C = (3, 1, 1, 1, 1, 3). Max-flow from the source node to the sink node under the state vector Xi = (xi1 , xi2 , . . . , xim ) such that the flow of ek is equal to xk in G(V, E, Xi , C). For example, W (X1 ) = 3 if X1 = (2, 2, 0, 0, 1, 1), but W (X1 ) does not exist if X1 = (2, 1, 0, 0, 1, 1) in Fig. 2, where W = (3, 2, 1, 1, 2, 2). Capacity level d is a nonnegative integervalued flow requirement for a given problem. Typically, d is a random variable, through which continuous observation and forecasting determine its distribution.


(Network) Reliability

Implicit enumeration algorithm

MP

d-MP (d, c)-MP candidate (d, c)-MP Reliability for level (d, c) Flow conservation law Acyclic network Endpoint

Probability that the required amount transmits successfully from the source node to the sink node. This algorithm is a special enumeration algorithm that uses special constraints and/or bounds to eliminate redundant feasible combinations in the solution space without needing to assess all alternatives when obtaining the required or best solutions. Further details can be found in [17]. Path/cut set that does not remain as a path/cut set if any arc is removed from this path/cut set. d-MP candidate: A system-state vector X = (x1 , x2 , . . . , xm ) is a d-MP candidate iff W (X) = d for all i. A d-MP candidate X = (x1 , x2 , . . . , xm ) is a d-MP if there is no directed cycle in G(V, E, X, C). A d-MP candidate is a (d, c)-MP candidate if C(X) ≤ c. A (d, c)-MP is a d-MP such that the total capacity cost is less than or equal to c. Probability that d units of flow can transmit from the source node to the sink node such that the total capacity cost of each (d, c)-MP is less than or equal to c. Total flows into and from a node (not source and target nodes) are all equal. A directed network is acyclic if it contains no directed cycle. Endpoints of an arc are the vertices that it joins. A SSUMPTIONS

The MUFN satisfies the following assumptions. 1) The component capacity is an independent discrete random variable and takes a nonnegative integer value according to a given distribution. 2) The failure of a node inhibits the work of all arcs incident from it. 3) The capacities of different components are statistically independent. 4) The network obeys the conservation law. I. I NTRODUCTION Many real-world systems have extensively applied the network reliability theory, such as computer and communication systems, power transmission and distribution systems, transportation systems, and oil/gas production systems. The reliability of such multistate networks [1]–[25] belongs to systems engineering fields to deal with issue formulation, analysis, modeling, decision making, and issue interpretation at any systems engineering life cycle phase associated with defining, developing, and deploying large systems. Network reliability also applies to systems management, systems engineering processes, and a variety of system engineering methods such as optimization, modeling, and simulation. Evaluating network reliability is an NP-hard problem even when a network is binary state and/or without budget constraints. The general method for reliability evaluation limits discussion to two-terminal reliability analysis. For MFN reliability with budget constraints, all existing algorithms are based on (d, c)-MPs, and these algorithms only

351

work for directed networks [11], [17]. An MP is a subset of arcs; the d-MP candidate, i.e., X = (x1 , x2 , . . . , xm ), is a system-state vector such that the max-flow in this vector is d; when no other distinct dMP candidate exists, for example, Y = (y1 , y2 , . . . , ym ), with yi ≤ xi for i = 1, 2, . . . , m, then X is a real d-MP. A (d, c)-MP is a special d-MP such that d units of flow transmit from the source node to the sink node, and the total capacity cost is less than or equal to c [17]. The following displayed list summarizes the primary steps of the best known algorithms for directed MUFN reliability with budget constraint problems [11], [17]. 1) Assume that all MPs are known in advance, where an MP is a path from nodes s to t such that one subpath is not an MP. 2) Construct a mathematical programming model for all MPs. 3) Employ the implicit enumeration algorithm (similar to the branch-and-bound algorithm) to solve this mathematical programming problem to obtain all feasible solutions, referred to as flow-state vectors in this correspondence paper. 4) Transform flow-state vectors into (d, c)-MP candidates defined in the Nomenclature. 5) Determine which (d, c)-MP candidates are real by scanning the corresponding subnetworks. A (d, c)-MP candidate that has no directed cycle in the corresponding subnetwork is a (d, c)-MP; otherwise, it is a not a real (d, c)-MP [17]. 6) Calculate the MUFN reliability in terms of (d, c)-MPs. As the number of MPs and (d, c)-MP candidates grows exponentially as network size increases, steps 1)–4) are NP-hard and are superfluous [11], [17]–[19]. All related algorithms only work for a directed MUFN in which all arcs are directed [11], [17]. Thus, a simple algorithm is needed to evaluate general MUFN reliability. This correspondence paper proposes an efficient and effective method to calculate MUFN reliability under budget constraints. II. M ODEL F ORMULATION AND P RELIMINARY This section describes some useful properties and observation results before introducing the proposed algorithm. The following unproven simple observations are core to the proposed algorithm accuracy and all known algorithms [11], [17]. These are the important relationships between system-state vectors, d-MPs, (d, c)-MP candidates, and (d, c)-MPs. Property 1: If the max-flow in G(V, E, W, C) is less than d, i.e., there are no d-MP candidates and (d, c)-MP candidates. Property 2: X is a (d, c)-MP if and only if X is also a (d, c)-MP candidate without directed cycles. The study [18] proposes and proves the origin of Property 2. The most efficient tool derives from Property 2 to verify whether a d-MP candidate is a real d-MP, and extends Property 2 to verify (d, c)-MP candidates. The following derives directly from Property 2. Property 3: If the network is acyclic, each (d, c)-MP candidate is a (d, c)-MP. If the MFN is acyclic, no directed cycle exists in each (d, c)-MP candidate, i.e., each (d, c)-MP is a real (d, c)-MP by Property 2. Therefore, this correspondence paper focuses only on a general MFN with cycles. Property 4: A system-state vector Xi = (xi1 , xi2 , . . . , xim ) is a (d, c)-MP if and only if the following conditions are satisfied. 1) W (Xi ) = d and C(Xi ) ≤ c. 2) There is no directed cycle in G(V, E, Xi , C). Property 4 is the basis of all known methods for obtaining (d, c)MPs [11], [17]. The following theorem is important for generating (d, c)-MP candidates in the best known algorithm [17] based on Property 4. All (d, c)-MP candidates are located by the proposed

352


implicit enumeration algorithm. The best known algorithm [11], [17] uses all MPs known beforehand, and all (d, c)-MP candidates must transfer from the corresponding flow-state vectors. Theorem 1: Any system-state Xi = (xi1 , xi2 , . . . , xim ) is a (d, c)MP candidate, i.e., W (Xi ) = d and C(Xi ) ≤ c, if and only if its corresponding flow-state vector φi = (fi1 , fi2 , . . . , fip ) satisfies the following four conditions: p

fik = d

(1)

k=1

fik ≤ Fk , xij =

for each k = 1, 2, , . . . , p

(2)

for all ej ∈ Pk ,

fik ,

j = 1, 2, . . . , m, and k = 1, 2, . . . , p

(3)

m

cj xij ≤ c,

for j = 1, 2, . . . , m.

(4)

m

number of (d, c)-MP candidates π = Min{ The mr

, d [W (e ) + 1]} grows exponentially as the number of arcs/nodes i i=1 increases [17]. Therefore, one does not need to know all MPs beforehand and transform from the flow-state vector to generate (d, c)-MP candidates using Theorem 1. The next theorem proposes an efficient and intuitive method to generate all (d, c)-MP candidates. The (d, c)-MP candidates are special d-MP candidates that satisfy budget constraints. Obtaining d-MP candidates in an efficient manner proves useful for finding (d, c)-MP candidates. Yeh [18] developed the best known algorithm for the d-MP candidate problem, the first algorithm to search for all d-MP candidates without knowing all MPs in advance and without needing to transform flow-state vectors. The following is the best method for searching for d-MP candidates for MFNs. Theorem 2: Any state vector X = (x1 , x2 , . . . , xm ) is a d-MP candidate, i.e., W (X) = d, if and only if the following conditions are satisfied: n

xij =

j=1 n

n

xhi ,

for all xij , xhi ∈ E, i = 1 or n,

(5)

h=1

x1i =

i=2

n−1

xjn = d,

for all x1i , xjn ∈ E

(6)

j=1

0 ≤ xij ≤ Min {d, W (eij )} , for each directed arc eij ∈ E.

0 ≤ xi ,

xij =

i=j

i=1

for all xij , xhi ∈ E, i = 1 or n

xhi ,

(8)

i=h

x1i =

xjn = d,

(10)

for all k

(11)

for all k.

(12)

The major difference between Theorems 2 and 3 is the use of (10), which is the budget constraint. The proposed algorithm employs Theorem 4 to reduce the total time required to find (d, c)-MP candidates. This method derives directly from Theorem 3 and is one of the most important contributions of this correspondence paper. This method extends to search for all (d, c)-MP candidates in the MUFN that contains unreliable nodes. Theorem 4: Any state vector P = (x1 , x2 , . . . , xm , y1 , y2 , . . . , yn ) is a (d, c)-MP candidate in the MUFN if and only if the following conditions are satisfied:

xij =

i=j

i=h

x1i =

i=1 m

xhi = yi ,

for all eij , ehi ∈ E, i ∈ N − {1, n} (13)

xjn = y1 = yn = d,

for all eij ∈ E

(14)

j=n

ck xk +

k=1

n

C(vk )yk ≤ c,

for all k

(15)

k=1

0 < xk ≤ {wk , d},

for all k

0 < yk ≤ {W (vk ), d},

(16)

for all k.

(17)

After searching for all (d, c)-MP candidates in the MFN/MUFN, all (d, c)-MPs are found by verifying all (d, c)-MP candidates using Property 2. Using these (d, c)-MPs obtains the reliability for the required amount of flow d under the budget constraint. Several methods, such as the sum-of-disjoint-products method, the state-space decomposition method, and the IE method, can calculate reliability [15], [19]– [23]. The IE is a fundamental tool for evaluating multistate network reliability and plays an important role in many areas of mathematics such as combinatorics, number theory, probability theory, and statistics [19]. This correspondence paper applies the IE in Examples 1 and 2 to determine reliability after obtaining all corresponding (d, c)-MPs. Theorem 5: Assuming that X1 , X2 , . . . , Xδ are the (d, c)-MPs, the system reliability using the IE is then given by

(7)

Theorem 2 is based on the flow conversation law [see (5)]; the maximal flow for each d-MP candidate, which is d by definition [see (6)]; and the capacity limitation of each arc [see (7)]. This correspondence paper adopts Theorem 2 and extends it to networks with a budget constraint to form an efficient and intuitive method for locating all (d, c)-MP candidates. The following theorem discusses the details of finding (d, c)-MP candidates. Theorem 3: Any state vector P = (x1 , x2 , . . . , xm ) is a (d, c)-MP candidate in the MFN if and only if the following conditions are satisfied:

for all k

xk ≤ {wk , d},

i=1

m+d−1

ck xk ≤ c,

k=1

δ

P r(Xi ) −

i=1

+

j−1 δ

P r(Xi ∩ Xj )

j=2 i=1 j−1 i−1 δ

P r(Xi ∩ Xj ∩ Xk ) + · · ·

j=3 i=2 k=1

+ (−1)δ+1 P r(X1 ∩ . . . ∩ Xδ )

(18)

where P r{Xij = (xi1 , xi2 , . . . , xim )}

= P r X = (x∗i1 , x∗i2 , . . . , x∗im )|xij ≤ x∗ij , for j = 1, 2, . . . , m

(19) for all x1i ,

j=n

xjn ∈ E, i = 1, and j = n

m

= (9)

j=1

P r xij ≤ x∗ij .

(20)


353

III. P ROPOSED A LGORITHM The following discussion describes the implementation of the proposed algorithm in detail to search for all (d, c)-MPs. This discussion highlights the efficiency and simplicity of the proposed algorithm, which does not need to have all MPs in advance. STEP 0) Replace each directed arc by two opposite directed arcs with the original capacity probability and cost. STEP 1) Construct a mathematical programming model based on (8)– (12) for the MFN or (13)–(18) for the MUFN. STEP 2) Solve the feasible solutions (i.e., (d, c)-MP candidates), for example, X1 , X2 , . . . , Xπ , of the model listed in STEP 1) using the implicit enumeration algorithm. STEP 3) Verify each (d, c)-MP candidate based on Property 4 [2)]. STEP 4) Calculate R(d,c) using the (d, c)-MPs obtained in STEP 3).

Fig. 3. Undirected bridge network. TABLE I P ROBABILITY D ISTRIBUTIONS OF THE A RC C APACITIES OF F IG . 3

The proposed algorithm, compared with the algorithms in [11] and [17], is an efficient method for generating all possible (d, c)-MPs via a simple examination of these (d, c)-MP candidates. The correctness of the proposed procedure follows the definition of the (d, c)-MP (candidate), Property 3, and Theorem 4. The major time complexity of the proposed algorithm is from STEP 2), implemented by the implicit enumeration algorithm to obtain all feasible nonnegative integer solutions. The following two properties are needed to analyze time complexity. Property 5: The time complexity to search for all (d, c)-MP candi(1 + Min{W (eij ), d})) dates from STEP 2) is O((m + n) e ∈E

ij

(1 + Min{W (eij ), d})) units of with at most O((m + n) eij ∈E storage space to save all d-MP candidates. Proof: STEP 2), based on Theorem 4, is used to search for all (d, c)-MP candidates. The nonnegative integer solutions that satisfy (1 + Min{W (eij ), d}). (16) and (17) are bounded by eij ∈E∪V Equations (13) and (14) utilize O(n) when employing the flow conservation law to determine whether these two equations are satisfied for all nodes. Equation (15) needs O(m + n) for verification. In addition, (m + n) components (arcs and nodes) exist in each d-MP candidate, i.e., (m + n) units of storage space are needed to save each d-MP candidate. Therefore, Property 5 is true. Property 6: The time complexity to verify all (d, c)-MP candidates from STEP 3) is O(n). Proof: Based on Property 3, STEP 3) verifies each (d, c)-MP candidate by checking whether a directed cycle is included. The time complexity to find any directed cycle is only O(n) in any network with n nodes. This establishes Property 6. Theorem 5: The proposed algorithm finds and verifies all (d, c)-MP candidates in O(nπ) time, where π is the number of total (d, c)-MP candidates. Proof: The proposed algorithm is based only on the definition of (d, c)-MP and some simple concepts discussed in Section II, implying that the proposed algorithm is correct. We assume that the total number of (d, c)-MP candidates generated in STEPs 1) and 2) of the proposed algorithm is π, i.e., π ((m + n) e ∈E (1 + Min{W (eij ), d})) ij from Property 5. The time complexity for finding a directed cycle is only O(n) for each (d, c)-MP candidate [15] in STEP 3). Thus, Theorem 5 is constructively proven. From [18], the algorithm not needing all MPs and transforming from flow-state vectors is more efficient than algorithms that require all MPs in advance in both time and space complexities. Each (d, c)-MP candidate is a d-MP candidate that satisfies the budget constraint. Therefore, there are fewer (d, c)-MP candidates than d-MP candidates. Hence, the proposed algorithm is more efficient than existing algorithms in both time and space complexities [11], [17].

IV. I LLUSTRATIVE E XAMPLES The following example best illustrates the general procedure of the proposed algorithm. The computational difficulty of modeling network reliability grows exponentially as network size increases. This difficulty increases when multistates exist in networks. Instead of using large network systems, a benchmark network (Fig. 3) demonstrates the proposed algorithm in existing algorithms [11], [17]. This benchmark network is the most frequently cited example in [11], [17], which transforms into a directed network to demonstrate the procedure for determining MFN reliability (R(3,14) ) without considering unreliable nodes in Example 1. The network considers Example 2 (Fig. 3) an MUFN and derives the reliability for level (3, 20) using the proposed algorithm. Example 1: Table I lists the probability distributions of arc capacities of Fig. 3. The STEPs 0)–3) in the proposed algorithm are used to find and verify all (3, 14)-MPs (d = 3, c = 14). The last implemented step derives the reliability of Fig. 3 using all obtained (3, 14)-MPs in STEP 3). Solution: STEP 0) Since either the source node or the sink node is one endpoint of e1 , e2 , e5 , and e6 , replace these undirected arcs with a directed arc joining the source node or the sink node. Also, replace the undirected arc e∗3 with two directed arcs e3 and e4 , and let C(e3 ) = C(e4 ) = C(e∗3 ) and W (e3 ) = W (e4 ) = W (e∗3 ) (see Fig. 1). STEP 1) Let X = (x1 , x2 , x3 , x4 , x5 , x6 ), and construct the mathematical model using (8)–(12) as follows: x1 + x4 = x2 + x3

(21)

x3 + x5 = x4 + x6

(22)

x1 + x5 = x2 + x6 = d = 3

(23)

354


TABLE II P ROBABILITY D ISTRIBUTIONS OF THE N ODE C APACITIES OF F IG . 3

STEP 3) Since G(V, E, X1 , C), G(V, E, X3 , C), G(V, E, X4 , C), G(V, E, X6 , C), and G(V, E, X7 , C) contain no cycle, X1 , X3 , X4 , X6 , and X7 are all real (3, 14)-MPs. On the other hand, there is a directed cycle in G(V, E, X2 , C) and G(V, E, X5 , C) from node 1 to node 2 to node 1. Hence, only X2 and X5 are not real (3, 20)-MPs. STEP 4) The system reliability using (18)–(20) in terms of (3, 20)MPs is R(3,20) = 0.5427. V. C ONCLUSION AND F UTURE W ORK

3x1 + x2 + x3 + x4 + x5 + 3x6 ≤ c = 14

(24)

x1 ≤ Min{w1 , d} = 3

(25)

x2 ≤ Min{w2 , d} = 2

(26)

x3 ≤ Min{w3 , d} = 1

(27)

x4 ≤ Min{w4 , d} = 1

(28)

x5 ≤ Min{w5 , d} = 2

(29)

x6 ≤ Min{w6 , d} = 2

(30)

0 ≤ xi ,

(31)

for all i

STEP 2) Use the implicit enumeration algorithm to find all (3, 14)-MP candidates from (21)–(31): X1 = (2, 2, 0, 0, 1, 1), X2 = (2, 2, 1, 1, 1, 1), X3 = (1, 2, 0, 1, 2, 1), X4 = (1, 1, 0, 0, 2, 2), and X5 = (1, 1, 1, 1, 2, 2). STEP 3) Since G(V, E, X1 , C), G(V, E, X3 , C), and G(V, E, X4 , C) contain no cycle, X1 , X3 , and X4 are all real (3, 14)MPs. However, there is a directed cycle in G(V, E, X2 , C) and G(V, E, X5 , C) from node 1 to node 2 to node 1. Hence, X2 and X5 are not real (3, 14)-MPs. STEP 4) The system reliability for level (3, 14) using (18)–(20) is

Evaluating network reliability with budget (capacity cost) constraints is highly practical. This correspondence paper has formulated all known algorithms for this problem in terms of (d, c)-MPs [11], [17]. Thus, they need to identify all MPs in advance, find all flowstate vectors based on MPs, transform flow-state vectors into (d, c)MP candidates, and locate all (d, c)-MPs by verifying (d, c)-MP candidates. However, computational difficulty grows exponentially as network size increases when searching for all MPs. A novel algorithm has been developed by modifying and extending the algorithm proposed in [17] for the d-MP problem to search for all (d, c)-MPs before evaluating MUFN reliability. The proposed algorithm overcomes serious obstacles when searching for all MPs, and it finds and transforms from flow-state vectors. The proposed algorithm is more effective, practical, and easy to implement than other algorithms. According to the time and space complexities required by the proposed algorithm, our analytical results compare favorably with those obtained using methods in literature. Future development of the proposed algorithm depends on realscenario simulations of the proposed method for a large network (10–20 nodes) and experimental comparisons with existing methods. The proposed algorithm needs practical implementation or real-world applications in the future to strengthen the claim that it is more effective, practical, and easy to implement than other works.

R(3,14) = P r(X1 ∪ X3 ∪ X4 ) ACKNOWLEDGMENT = [P r(X1 ) + P r(X3 ) + P r(X4 )] − [P r(X1 ∩ X3 ) + P r(X1 ∩ X4 ) + P r(X3 ∩ X4 )] + P r(X1 ∩ X3 ∩ X4 ) = 0.64005.

The author would like to thank the Chief Editor and referees for their constructive comments and recommendations, which have significantly improved the presentation of this paper. R EFERENCES

(32)

Example 2: Consider the MUFN (Fig. 3). Tables I and II list the probability distributions of the arc and node capacities/costs. In this illustrative network, each edge represents a transmission line and consists of several physical transmission lines, e.g., T3 and E1 cables and optical fiber. Each node represents a computer center and consists of several switches [14]. The proposed algorithm is utilized to find the corresponding reliability using (3, 20)-MPs in building the related network system. Solution: STEP 0) Transfer the MUFN into a directed MUFN analog to STEP 1) in Example 1. STEP 1) Let X = (x1 , x2 , x3 , x4 , x5 , x6 , y1 , y2 , y3 , y4 ) and construct the mathematical model using (13)–(17). STEP 2) Use the implicit enumeration algorithm to find all (3, 20)-MP candidates from STEP 1): X1 = (2, 2, 0, 0, 1, 1, 3, 2, 1, 3), X2 = (2, 2, 1, 1, 1, 1, 3, 3, 2, 3), X3 = (1, 2, 0, 1, 2, 1, 3, 2, 2, 3), X4 = (1, 1, 0, 0, 2, 2, 3, 2, 3, 3), and X5 = (1, 1, 1, 1, 2, 2, 3, 1, 2, 3), X6 = (3, 2, 1, 0, 0, 1, 3, 3, 1, 3), and X7 = (2, 1, 1, 0, 1, 2, 3, 2, 2, 3).

[1] R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network Flows—Theory, Algorithms, and Applications. Englewood Cliffs, NJ: Prentice-Hall, 1993. [2] C. J. Colbourn, The Combinatorics of Network Reliability. New York: Oxford Univ. Press, 1987. [3] H. Pham, “Special issue on critical reliability challenges and practices,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 37, no. 2, pp. 141– 142, Mar. 2007. [4] A. G. Sutcliffe and A. Gregoriades, “Automating scenario analysis of human and system reliability,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 37, no. 2, pp. 249–261, Mar. 2007. [5] Y.-S. Dai, M. Xie, and X. Wang, “A heuristic algorithm for reliability modeling and analysis of grid systems,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 37, no. 2, pp. 189–200, Mar. 2007. [6] W. Kuo and R. Wan, “Recent advances in optimal reliability allocation,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 37, no. 2, pp. 143–156, Mar. 2007. [7] W. J. Ke and S. D. Wang, “Reliability evaluation for distributed computing networks with imperfect nodes,” IEEE Trans. Rel., vol. 46, no. 3, pp. 342–349, Sep. 1997. [8] G. Levitin, Universal Generating Function in Reliability Analysis and Optimization. Berlin, Germany: Springer-Verlag, 2005. [9] A. Lisnianski and G. Levitin, Multi-State System Reliability. Assessment, Optimization and Applications. Singapore: World Scientific, 2003. [10] S. Hwang and H. Pham, “Quasi-renewal time-delay fault-removal consideration in software reliability modeling,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 39, no. 1, pp. 200–209, Jan. 2009.


[11] Y. K. Lin, “Reliability evaluation for an information network with node failure under cost constraint,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 37, no. 2, pp. 180–188, Mar. 2007. [12] M. A. Samad, “An efficient algorithm for simultaneously deducing MPs as well as cuts of a communication network,” Microelectron. Reliab., vol. 27, no. 3, pp. 437–441, 1987. [13] J. Xue, “New approach for multistate systems analysis,” Reliab. Eng., vol. 10, no. 4, pp. 245–256, 1985. [14] W.-C. Yeh, “A simple heuristic algorithm for generating all minimal paths,” IEEE Trans. Rel., vol. 56, no. 3, pp. 488–494, Sep. 2007. [15] W.-C. Yeh, “A greedy branch-and-bound inclusion–exclusion algorithm for calculating the exact multi-state network reliability,” IEEE Trans. Rel., vol. 57, no. 1, pp. 88–93, Mar. 2008. [16] L. Xing, “An efficient binary-decision-diagram-based approach for network reliability and sensitivity analysis,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 38, no. 1, pp. 105–115, Jan. 2008. [17] W.-C. Yeh, “A new approach to evaluate the reliability of multistate networks under the cost constraint,” Omega, vol. 33, no. 3, pp. 203–209, Jun. 2005. [18] W.-C. Yeh, “A novel method for the network reliability in terms of capacitated-minimal-paths without knowing minimal-paths in advance,” J. Oper. Res. Soc., vol. 56, no. 10, pp. 1235–1240, Oct. 2005.

355

[19] W.-C. Yeh, “A simple universal generating function method to search for all MPs in networks,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 39, no. 6, pp. 1247–1254, Nov. 2009. [20] A. M. Al-Ghanim, “A heuristic technique for generating minimal paths and cutsets of a general network,” Comput. Ind. Eng., vol. 36, no. 1, pp. 45–55, Jan. 1999. [21] R. Robidoux, H. Xu, L. Xing, and M. C. Zhou, “Automated modeling of dynamic reliability block diagrams using colored Petri nets,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 40, no. 2, pp. 337–351, Mar. 2010. [22] G. Levitin, “Redundancy optimization for multi-state system with fixed resource-requirements and unreliable sources,” IEEE Trans. Rel., vol. 50, no. 1, pp. 52–58, Mar. 2000. [23] C. Y. Chang, J. P. Sheu, Y. C. Chen, and S. W. Chang, “An obstacle-free and power-efficient deployment algorithm for wireless sensor networks,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 39, no. 4, pp. 795– 806, Jul. 2009. [24] Y. K. Lin, “Optimal pair of minimal paths under both time and budget constraints,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 39, no. 3, pp. 619–625, May 2009. [25] A. G. Sutcliffe and A. Gregoriades, “Automating scenario analysis of human and system reliability,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 37, no. 2, pp. 249–261, Mar. 2007.