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Reliability evaluation of multi-state systems under cost consideration
Applied Mathematical Modelling 36 (2012) 4261–4270
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Reliability evaluation of multi-state systems under cost consideration Yi-Feng Niu a,⇑, Xiu-Zhen Xu b a b
School of Mathematics & Information Science, Henan Polytechnic University, Jiaozuo 454000, PR China School of Computer Science & Technology, Henan Polytechnic University, Jiaozuo 454000, PR China
a r t i c l e
i n f o
Article history: Received 27 February 2011 Received in revised form 11 November 2011 Accepted 15 November 2011 Available online 23 November 2011 Keywords: Reliability evaluation Multi-state system Cost constraint Lower capacity bound (d, c)-MP/d-MP/MP/d-flow
a b s t r a c t A more practical and desirable performance index of multi-state systems is the twoterminal reliability for level (d, c) (2TRd, c), defined as the probability that d units of flow can be transmitted from the source node to the sink node with the total cost less than or equal to c. In this article, a simple algorithm is developed to calculate 2TRd, c in terms of (d, c)-MPs. Two major advantages of the proposed algorithm include: (1) as of now, it is the only algorithm that searches for (d, c)-MPs without requiring all minimal paths (MPs) and the procedure of transforming feasible solutions; (2) it is more practical and efficient in solving (d, c)-MP problem in contrast to the best-known method. An example is provided to illustrate the generation of (d, c)-MPs by using the presented algorithm, and 2TRd, c is thus evaluated. Furthermore, the computational experiments are conducted to verify the performance of the presented algorithm. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Reliability is usually an important index to evaluate the service quality of many real-life systems such as transportation systems, logistics systems, computer systems, and manufacturing systems. The capacitated-flow network explored by Ford and Fulkerson [1] is the most widely employed model for reliability analysis. A capacitated-flow network consists of multistate edges which have independent, finite and multi-integer-valued random capacities [2], and has a finite number of different states for performance rate. For such a network, the two-terminal reliability for level d (2TRd), defined as the probability that d units of flow can be transmitted from the source node to the sink node through multi-state edges [3,4], is a combination of source-sink connection, edge capacity, and flow demand [5]. To compute the NP-hard 2TRd without considering the cost constraint, a variety of methods [2–13] have been presented. In general, these methods can be classified into two types: One is based on d-minimal paths (d-MPs) [2,3,7,9,10,13], and the other is based on d-minimal cuts (d-MCs) [6,8,11,12]. However, in addition to capacity limitation, each edge is always associated with a cost that must be paid for every unit of flow passing through it. So, it is more practical and suitable to incorporate cost attribute as an integral part of reliability analysis. Taking cost into account, the two-terminal reliability for level (d, c) (2TRd, c), which is defined as the probability that d units of flow can be transmitted from the source node to the sink node under the cost constraint c, is a more desirable index for evaluating the performance of many real-life systems. As 2TRd can be computed in terms of d-MPs, 2TRd, c can be computed in terms of (d, c)-MPs [14–16]. A (d, c)-MP is a special d-MP such that the total capacity cost is not greater than c. Once all (d, c)-MPs are known, 2TRd, c can be calculated by using inclusion-exclusion principle. Hence, searching for all (d, c)-MPs plays an important part in solving the problem of 2TRd, c. Based on MPs, Lin [14] proposed an algorithm to search for all (d, c)-MPs. Lin’s algorithm first uses implicit enumeration algorithm to search for (d, c)-MP candidates. Then, all (d, c)-MP candidates are further verified whether they are (d, c)-MPs by ⇑ Corresponding author. E-mail address: [email protected] (Y.-F. Niu). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.11.055
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comparison process. It is a time-consuming task to check whether (d, c)-MP candidates are (d, c)-MPs by comparison process due to the exponentially growing number of (d, c)-MP candidates [16]. Lin [15] extended the work to the case with unreliable node. The presented method in [15] uses MPs to assign the flow to each component (edge or node). All (d, c)-MPs can be obtained by using implicit enumeration algorithm. Notice that this algorithm also involves the comparison process in checking (d, c)-MP candidates. With improvement on checking (d, c)-MPs candidates, Yeh [16] proposed a new approach to find (d, c)-MPs based on MPs. Yeh’s algorithm also uses implicit enumeration algorithm to search for (d, c)-MP candidates by solving a mathematical programming model imposed by the limitation of max-capacity, flow-conservation law and capacity cost. Then, an efficient cycle-checking method without requiring comparison process is employed to verify (d, c)-MP candidates. Yeh’s algorithm is superior to the algorithms in [14,15] in solving (d, c)-MP problem. However, just like the algorithms in [14,15], Yeh’s algorithm is also subject to the assumption that all MPs must be known in advance. The above introduction shows that the well-known existing algorithms for 2TRd, c [14–16] must require all MPs information in advance. It is a very cumbersome and time-consuming task if all MPs must be computed, since the MP problem is known to be NP-hard [17,18]. In this article, a more efficient algorithm for 2TRd, c. is presented. The proposed algorithm requires neither MPs information, nor the procedure of transforming feasible solutions. Besides, in order to improve the efficiency of seeking (d, c)-MPs, the proposed algorithm introduces a novel concept-the lower capacity bound to reduce the amount of enumerated state vectors. Therefore, all (d, c)-MPs can be efficiently obtained by using implicit enumeration algorithm. The rest of this paper is organized as follows: Acronyms, nomenclature, notations and assumptions are described in Section 2. In Section 3, the network model is introduced, and the best-known method for solving (d, c)-MP problem is briefly reviewed. Meanwhile, some related theorems are discussed. In Section 4, the proposed algorithm is described in detail, and the computational complexity of the algorithm is also analyzed along with a contrast to the best-known method. Section 5 introduces the inclusion-exclusion method for reliability evaluation. In Section 6, an illustrative example is provided to illustrate how to search for all (d, c)-MPs by using the proposed algorithm. Then, the reliability is evaluated in terms of all (d, c)-MPs. Computational experiments are conducted in Section 7 to further test the performance of the proposed algorithm, together with comparisons between the proposed algorithm and the best-known method. The final section presents the concluding remarks. 2. Acronyms, nomenclature, notations and assumptions 2.1. Acronyms MP/MC 2TRd 2TRd, c Max-flow
minimal path/minimal cut two-terminal reliability for level d two-terminal reliability for level (d, c) maximum flow
2.2. Nomenclature Level d d-flow d-MP (d, c)-MP Network reliability for level (d, c)
a non-negative integer-valued flow demand from s to t a flow vector f1d ; f2d ; . . . ; fmd representing the flows through each edge ei when the amount of flow sent from s to t is d a system state vector x = (x1, x2, . . . , xm) is a d-MP if and only if M(x) = d, and M(x 0(ei)) < d for each xi > 0 a (d, c)-MP is a d-MP such that the total capacity cost is not greater than c The probability that d units of flow can be transmitted from the source node to the sink node such that the total capacity cost is less than or equal to c
2.3. Notations G(V, E, U, C) s, t ei
a capacitated-flow network with the set of nodes V = {s, 1, 2, . . . , n, t}, the set of edges E = {e1, e2, . . . , em}, the largest state vector U = (u1, u2, . . . , um), and the cost vector C = (c1, c2, . . . , cm) the source and sink nodes, respectively the ith edge in E
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m, n (v, ) (, v) ci xi ui L(ei) x U(0i) C(x) Pi p M(x) fi fid f Fi 0(ei) R(d,c)
4263
the number of edges, and the number of nodes except for nodes s and t a set of edges emanating from node v a set of edges pointing to node v the unit capacity cost of ei a state of ei representing the amount of flow allowed to be sent through edge ei the largest state of ei the minimal capacity of ei such that the max-flow from s to t is d the system state vector x = (x1, x2, . . . , xm) a special system state vector U(0i) = (u1, . . . , ui1, 0, ui+1, . . . , um), i.e., state level is 0 for edge ei and the largest for other edges. P CðxÞ ¼ m i¼1 xi c i is the total capacity cost of the system state vector x the ith MP in G(V, E, U, C) the number of MPs in G(V, E, U, C) the max-flow of the network under x the flows through edge ei the flows through edge ei when there are d units of flow sent from s to t a flow vector f = (f1, f2, . . . , fm) Fi = min{ui | for each ej 2 Pi} is the minimal capacity on Pi 0(ei) = (0, . . . , 0, 1, 0, . . . , 0), i.e., state level is 1 for edge ei and 0 for other edges the reliability for level (d, c)
2.4. Assumptions The network satisfies the following assumptions [16]: 1. Each node is perfectly reliable. 2. The state/capacity of each edge is a successive random variable which takes integer values from 0 to ui according to a given distribution. 3. The states/capacities of different edges are s-independent. 4. All flows in the network obey the conservation law. 3. Network model and preliminaries 3.1. Network model Let G(V, E, U, C) denote a capacitated-flow network with the unique source node s and the unique sink node t, where V = {s, 1, 2, . . . , n, t} is the set of nodes, E = {e1, e2, . . . , em} is the set of edges, U = (u1, u2,. . . , um) is the largest state vector, and C = (c1, c2, . . . , cm) is the cost vector, where ci is the unit capacity cost of ei. The current state of ei is defined by xi which takes integer values from 0 to ui, where ui denotes the largest state of edge ei. A system state vector x = (x1, x2, . . . , xm) indicates the current state of each edge ei. Let M(x) represent the max-flow of the network under x, then M(x) is always called the structure function of a multi-state network. For example, consider a capacitated-flow network G(V, E, U, C) in Fig. 1, which shows V = {s, 1, 2, t}, E = {e1, e2, e3, e4, e5, e6}, U = (3, 2, 1, 1, 2, 2), and C = (3, 1, 1, 1, 1, 3). For a given system state vector x = (2, 1, 1, 0, 1, 2), which indicates the current states of e1, e2, e3, e4, e5, e6 are 2, 1, 1, 0, 1, and 2, respectively, the total capacity cost of the system state vector x is C(x) = 2 3 + 1 1 + 1 1 + 0 1 + 1 1 + 2 3 = 15, and the max-flow of the network under x is M(x) = 3. 3.2. Preliminaries Before presenting the proposed algorithm, there is need to briefly review the best-known method for solving (d, c)-MP problem. As of now, the method proposed by Yeh [16] is the most efficient approach to search for (d, c)-MPs. The following Lemma [16] is the foundation of Yeh’s method. Lemma 1. Any state vector x = (x1, x2, . . . , xm) is a (d, c)-MP candidate, if and only if its corresponding flow vector f = (f1, f2, . . . , fp) satisfies the following four conditions. p X j¼1
fj ¼ d;
ð1Þ
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Fig. 1. A simple flow network.
fj F j for each j ¼ 1; 2; . . . ; p; xi ¼
X
ð2Þ
fj for each i ¼ 1; 2; . . . ; m;
ð3Þ
ei 2P j m X
xi ci c:
ð4Þ
i¼1
Firstly, Yeh’s method uses implicit enumeration algorithm to solve all feasible solutions to constraints (1) and (2). Then, all feasible solutions are transformed into (d, c)-MP candidates according to constraints (3) and (4). Finally, all (d, c)-MP candidates are verified whether they are (d, c)-MPs by using cycle-checking method. The total number of feasible solutions sat p þ d 1 Qp isfying constraints (1) and (2) is bounded by Min ; j¼1 ð1 þ MinfF j ; dgÞ [16]. It takes O(mp) time to transform d a feasible solution into its corresponding (d, c)-MP candidate. For a (d, c)-MP candidate, Yeh’s algorithm takes O(n) time to verify if it is a (d, c)-MP [16]. Therefore, we have the following statement. p þ d 1 Qp Corollary 1. The time complexity of Yeh’s algorithm is O(mp Min ; j¼1 ð1 þ MinfF j ; dgÞ ) for finding all (d, c)-MPs. d Corollary 1 indicates that the time complexity of Yeh’s algorithm is directly proportional to the number of MPs. Unfortunately, the number of MPs grows exponentially with the number of nodes and edges [9]. Consequently, Yeh’s method is impractical and inefficient in searching for (d, c)-MPs. A more practical and efficient method is thus developed to solve (d, c)-MP problem in the following discussion. The definition of (d, c)-MP shows that a state vector x is a (d, c)-MP if and only if x is a d-MP and C(x) 6 c. Then, the first important step to solving (d, c)-MP problem is to search for all d-MPs. When the amount of flow transmitted from s to t is d, a d-flow which is a flow vector (f1d ; f2d ; . . . ; fmd Þ consists of flows fid through each edge ei for 1 6 i 6 m. Thus, if x is a d-flow, we have M(x) = d and x satisfies flow-conservation law [19,20]. Therefore, the following theorem holds. Theorem 1. x = (x1, x2, . . . , xm) is a d-flow if and only if the following three conditions are satisfied.
X
xi ¼
ei 2ðs;Þ
X ei 2ð;v Þ
X
xj ¼ d;
ð5Þ
ej 2ð;tÞ
xi ¼
X
xj
for ev ery node
v 2 f1; 2; . . . ; ng;
ð6Þ
ej 2ðv ;Þ
0 xi Minfui ; dg for 1 i m:
Proof. Immediately from the definition of d-flow and flow-conservation law.
ð7Þ
h
Any state vector which satisfies conditions (5) and (6) is called a d-MP candidate [9]. Then, any d-flow, in fact, is also a dMP candidate. Each d-MP candidate needs to be verified whether it is a d-MP. The most efficient method for verifying d-MP candidate is the cycle-checking method proposed by Yeh [9,10,16]. Such a cycle-checking method is based on the following lemma [10,16].
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Lemma 2. A d-MP candidate x is a d-MP if and only if there is no directed cycle in it. The following theorem specifies the relationship between d-flow and d-MP. Theorem 2. A d-flow x is a d-MP if and only if there is no directed cycle in it. Proof. Directly from Lemma 2.
h
According to Theorem 1, Theorem 2 and the definition of (d, c)-MP, it is easy to have the following conclusions. Corollary 2. A d-flow x is a (d, c)-MP if and only if C(x) 6 c, and there is no directed cycle in it. Corollary 3. A state vector x = (x1, x2, . . . , xm) is a (d, c)-MP if and only if the following conditions are satisfied.
X
xi ¼
ei 2ðs;Þ
X
X
xj ¼ d;
ð8Þ
ej 2ð;tÞ
xi ¼
ei 2ð;v Þ
X
xj
for every node
v 2 f1; 2; . . . ; ng;
ð9Þ
ej 2ðv ;Þ
0 xi Minfui ; dg for 1 i m;
ð10Þ
CðxÞ c;
ð11Þ
there is no directed cycle in x:
ð12Þ
Corollary 3 provides a simple enumeration method to search for (d, c)-MPs without knowing any MP information. However, given a capacitated-flow network G(V, E, U, C), it is obvious that the total number of state vectors satisfying condition (10) is Q Qm bounded by m i¼1 ðMinfui ; dg þ 1Þ. That is, the amount of state vectors to be enumerated is up to i¼1 ðMinfui ; dg þ 1Þ. So, it is time-consuming to search for (d, c)-MPs by using Corollary 3 in a straightforward manner, and there is a room for reducing the number of enumerated state vectors. The following discussion will introduce a novel concept-the lower capacity bound as a limited condition to reduce the number of enumerated state vectors. The lower capacity bound L(ei) is the minimal capacity of ei such that the max-flow from s to t is equal to d [11]. L(ei) is a restricted condition for minimal state of edge ei, and can be found according to the following Theorem 3. Theorem 3. Given the demand d (0 < d 6 M(U)), let U(0i) denote a special system state vector in which state level is 0 for edge ei and the largest for other edges, then
Lðei Þ ¼
0; if MðUð0i ÞÞ d; d MðUð0i ÞÞ; if MðUð0i ÞÞ < d:
ð13Þ
Proof. If M(U(0i)) P d, it means that even when the capacity of ei is 0, d units of flow can be transmitted from the source node to the sink node. So, L(ei) = 0. If M(U(0i)) < d, since M(U) P d, it means that at least d M(U(0i)) units of flow must be sent through edge ei such that d units of flow can be transmitted from the source node to the sink node. So, L(ei) = d M(U(0i)). h Theorem 3 shows that to find the lower capacity bound L(ei) of ei, there is only need to compute M(U(0i)). Hence, the total time complexity is O(mn3) for finding all lower capacity bounds, where n3 is the time complexity for evaluating the max-flow [19,20]. The role of lower capacity bound lies in the following theorem. Theorem 4. For a state vector x = (x1, x2, . . . , xm), if there exists a state xi satisfying xi < L(ei) (1 6 i 6 m), then x must not be a (d, c)MP. Proof. By the definition of lower capacity bound, we have M(x) < d if xi < L(ei) for some edge ei. Consequently, x is not a (d, c)MP. h By Theorem 4, it is easy to have the following conclusion. Corollary 4. For a state vector x = (x1, x2, . . . , xm), if x is a (d, c)-MP, then L(ei) 6 xi for 1 6 i 6 m. The following theorem which is the foundation of the proposed algorithm provides a practical method to search for all (d, c)-MPs.
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Theorem 5. x = (x1, x2, . . . , xm) is a (d, c)-MP if and only if the following conditions are satisfied.
X
xi ¼
ei 2ðs;Þ
X
X
xj ¼ d;
ð14Þ
ej 2ð;tÞ
xi ¼
ei 2ð;v Þ
X
xj
for every node
v 2 f1; 2; . . . ; ng;
ð15Þ
ej 2ðv ;Þ
Lðei Þ xi Minfui ; dg for 1 i m;
ð16Þ
CðxÞ c;
ð17Þ
there is no directed cycle in x:
ð18Þ
Proof. Directly from Corollary 3 and Corollary 4.
h
By Theorem 5 we have the following conclusion. Corollary 5. For an acyclic network and a state vector x, if x satisfies conditions (14)–(17), then x is a (d, c)-MP. Q It is clear that the total number of state vectors satisfying condition (16) is bounded by m ðMinfui ; dg þ 1 Lðei ÞÞ. That Qm Qmi¼1 is, the amount of state vectors to be enumerated has reduced from i¼1 ðMinfui ; dg þ 1Þ to i¼1 ðMinfui ; dg þ 1 Lðei ÞÞ. Since Qm Qm i¼1 ðMinfui ; dg þ 1 Lðei ÞÞ is always much less than i¼1 ðMinfui ; dg þ 1Þ, searching for (d, c)-MPs by using Theorem 5 is more practical and efficient in contrast to Corollary 3. 4. The algorithm for generating all (d, c)-MPs Through the above discussion, all (d, c)-MPs can be found by the following steps without knowing all MPs in advance. Input: A capacitated-flow network G(V, E, U, C). Output: All (d, c)-MPs. Step 0: Find the lower capacity bound L(ei) of ei by using Theorem 3, where 1 6 i 6 m. Step 1: Use implicit enumeration algorithm to solve all of the feasible solutions, say y1, y2, . . . , yq, to constraints (14)–(16). Step 2: If C(yj) 6 c and there is no directed cycle in yj, yj is a (d, c)-MP; otherwise, yj is not a (d, c)-MP, where 1 6 j 6 q. The detailed solution steps show that in fact, the proposed algorithm is also an enumeration algorithm. To clearly understand the differences between the proposed algorithm and the well-known existing methods, the main characteristics of different algorithms are listed in Table 1. The time complexity will be discussed below. As we mentioned before, Step 0 requires O(mn3) time to find all lower capacity bounds, where n3 is the time complexity for evaluating the max-flow [19,20]. The total number of state vectors satQ isfying constraint (16) is bounded by m ; dg þ 1 Lðei ÞÞ: It takes O(n) time to check if constraints (14) and (15) are i¼1 ðMinfu Qi satisfied for all nodes. Then, Step 1 requires Oðn m i¼1 ðMinfui ; dg þ 1 Lðei ÞÞÞ time in total. For a state vector yj, it only needs O(n) time to find a directed cycle [20], and O(m) time to check if C(yj) 6 c. Hence, Step 2 requires O((m + n)q) time. Since q is Q Qm bounded by m i¼1 ðMinfui ; dg þ 1 Lðei ÞÞ; Step 2 takes O((m + n) i¼1 ðMinfui ; dg þ 1 Lðei ÞÞ) amount of time in the worst case. Therefore, the total time complexity of finding (d, c)-MPs by using the proposed algorithm is
Oðmn3 Þ þ O n
m Y
!
ðMinfui ; dg þ 1 Lðei ÞÞ þ Oððm þ nÞ
i¼1
O ðm þ nÞ
m Y
m Y
ðMinfui ; dg þ 1 Lðei ÞÞÞ
i¼1
! ðMinfui ; dg þ 1 Lðei ÞÞ :
ð19Þ
i¼1
Table 1 The characteristics of different algorithms.
a b
Algorithm
Based on IEA?a
Needs MPs?
Needs transforming procedure?b
Use what method to verify (d, c)-MPs?
The The The The
Yes Yes Yes Yes
Yes Yes Yes No
Yes Yes Yes No
Comparison method Comparison method Cycle-checking method Cycle-checking method
algorithm in [14] algorithm in [15] algorithm in [16] proposed algorithm
Implicit enumeration algorithm. The procedure of transforming feasible solutions into (d, c)-MP candidates.
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Hence, the following statement holds. Theorem 6. The time complexity of the proposed algorithm is O((m + n)
Qm
i¼1 ðMinfui ; dg
þ 1 Lðei ÞÞ) for finding all (d, c)-MPs.
p þ d 1 Qp By Corollary 1, the time complexity of Yeh’s algorithm is OðmpMin ; j¼1 ð1 þ MinfF j ; dgÞÞ. Since the number d of MPs grows exponentially with the number of nodes and edges [9], we have n 6 m p. Meanwhile, we know d 6 M(U) p. Then,
! Y p pþd1 Oððm þ nÞ ðMinfui ; dg þ 1 Lðei ÞÞÞ O mp Min ; ð1 þ MinfF j ; dgÞ : d i¼1 j¼1 m Y
ð20Þ
Through the above detailed analysis, it is understood that from the viewpoint of time complexity, the proposed algorithm is more efficient than the best-known algorithm [16] in solving (d, c)-MP problem. 5. Reliability evaluation Once all (d, c)-MPs are known, the inclusion-exclusion method can be used to calculate 2TRd, c. Assume y1, y2, . . . , yq are all (d, c)-MPs, and let B1 = {x | x P y1}, B2 = {x | x P y2}, . . . , Bq = {x | x P yq}. Then, the two-terminal reliability for level (d, c) by using the inclusion–exclusion method can be evaluated as follows:
Rðd;cÞ ¼ PrðB1 [ B2 [ . . . [ Bq Þ ¼
q X
PrðBi Þ
i¼1
where Pr{x | x P yi} =
Qm
j¼1 Prfxj
q X j1 X j¼2
PrðBi \ Bj Þ þ þ ð1Þq1 PrðB1 \ B2 \ . . . \ Bq Þ
ð21Þ
i¼1
P yij g and yi = (yi1, yi2, . . . , yim).
6. An illustrative example A simple flow network in Fig. 1 which is also discussed in Ref. [16] is chosen to demonstrate how the proposed algorithm works. The probability distribution of each edge is given in Table 2. We would like to know the probability that 3 units of flow can be transmitted from the source node s to the sink node t such that the total capacity cost is less than or equal to 14. Without knowing all MPs in advance, all (3, 14)-MPs (d = 3, c = 14) can be found as follows:
Table 2 States and state probabilities of edges of Fig. 1. Edge
States
e1 e2 e3 e4 e5 e6
0 0 0 0 0 0
State probabilities 1 1 1 1 1 1
2 2 – – 2 2
3 – – – – –
0.05 0.10 0.10 0.10 0.10 0.05
Fig. 2. The network with no directed cycle under y1.
0.10 0.30 0.90 0.90 0.10 0.25
0.25 0.60 – – 0.80 0.70
0.60 – – – – –
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Solve: Step 0: By Theorem 3, it is easy to have L(e1) = 1, L(e2) = 1, L(e3) = L(e4) = L(e5) = 0, and L(e6) = 1. Step 1: Use implicit enumeration algorithm to solve all of the feasible solutions yi = (yi1, yi2, yi3, yi4, yi5, yi6) of the following constraints:
yi1 þ yi5 ¼ yi2 þ yi6 ¼ 3;
ð22Þ
yi1 þ yi4 ¼ yi2 þ yi3 ;
ð23Þ
yi3 þ yi5 ¼ yi4 þ yi6 ;
ð24Þ
1 yi1 3;
ð25Þ
1 yi2 2;
ð26Þ
0 yi3 1;
ð27Þ
0 yi4 1;
ð28Þ
0 yi5 2;
ð29Þ
1 yi6 2;
ð30Þ
As a result, there are seven feasible solutions: y1 = (2, 2, 0, 0, 1, 1), y2 = (2, 1, 1, 0, 1, 2), y3 = (3, 2, 1, 0, 0, 1), y4 = (1, 2, 0, 1, 2, 1), y5 = (1, 1, 0, 0, 2, 2), y6 = (2, 2, 1, 1, 1, 1), y7 = (1, 1, 1, 1, 2, 2).
Step 2: Since C(y1) = 2 3 + 2 1 + 0 1 + 0 1 + 1 1 + 1 3 = 12 6 14, and there is no directed cycle in y1 (see Fig. 2), y1 = (3, 2, 1, 0, 0, 1) is a (3, 14)-MP. In the same way, we have that both y4 = (1, 2, 0, 1, 2, 1) and y5 = (1, 1, 0, 0, 2, 2) are also (3, 14)-MPs. But, since C(y2) = C(y3) = 15 > 14, y2 = (2, 1, 1, 0, 1, 2) and y3 = (3, 2, 1, 0, 0, 1) are not (3, 14)-MPs. y6 = (2, 2, 1, 1, 1, 1) and y7 = (1, 1, 1, 1, 2, 2) are also not (3, 14)-MPs due to the fact that there exist directed cycles in both y6 and y7 (see Fig. 3 and Fig. 4).
Fig. 3. The network with a directed cycle 1 ? e3 ? 2 ? e4 ? 1 under y6.
Fig. 4. The network with a directed cycle 1 ? e3 ? 2 ? e4 ? 1 under y7.
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C(yi) 6 14?
A directed cycle in yi?
A (3, 14)-MP?
y1 = (2, 2, 0, 0, 1, 1) y2 = (2, 1, 1, 0, 1, 2) y3 = (3, 2, 1, 0, 0, 1) y4 = (1, 2, 0, 1, 2, 1) y5 = (1, 1, 0, 0, 2, 2) y6 = (2, 2, 1, 1, 1, 1) y7 = (1, 1, 1, 1, 2, 2)
Yes No No Yes Yes Yes Yes
No No No No No Yes Yes
Yes No No Yes Yes No No
Fig. 5. A capacitated-flow network.
Table 4 The largest state and unit capacity cost of each edge in Fig. 5. Edges
e1
e2
e3
e4
e5
e6
e7
e8
e9
e10
e11
e12
e13
ui ci
2 3
4 2
2 1
2 2
1 1
3 2
2 3
3 2
1 1
2 3
2 1
2 2
4 3
Table 5 Comparison of the proposed algorithm with Yeh’s algorithm. (d, c)-MP
(1, 10)-MP (2, 17)-MP (3, 27)-MP (4, 36)-MP (5, 49)-MP (6, 60)-MP
Computational time (s) Yeh’s algorithm
The proposed algorithm
89.047 89.343 89.859 90.468 90.812 92.249
0.721 5.799 9.705 5.752 0.831 0.627
The solution results for all (3, 14)-MPs obtained by the proposed algorithm are listed in Table 3. From the solution results in Table 3, we know y1 = (2, 2, 0, 0, 1, 1), y4 = (1, 2, 0, 1, 2, 1), and y5 = (1, 1, 0, 0, 2, 2) are all (3, 14)-MPs. Set B1 = {x | x P (2, 2, 0, 0, 1, 1)}, B2 = {x | x P (1, 2, 0, 1, 2, 1)}, B3 = {x | x P (1, 1, 0, 0, 2, 2)}, then the reliability for level (3, 14) is obtained as follows:
Rð3;14Þ ¼ PrfB1 [ B2 [ B3 g ¼ PrfB1 g þ PrfB2 g þ PrfB3 g PrfB1 \ B2 g PrfB1 \ B3 g PrfB2 \ B3 g þ PrfB1 \ B2 \ B3 g ¼ Prfxjx ð2; 2; 0; 0; 1; 1Þg þ Prfxjx ðð1; 2; 0; 1; 2; 1Þg þ Prfxjx ð1; 1; 0; 0; 2; 2Þg Prfxjx ð2; 2; 0; 1; 2; 1Þg Prfxjx ð2; 2; 0; 0; 2; 2Þg Prfxjx ð1; 2; 0; 1; 2; 2Þg þ Prfxjx ð2; 2; 0; 1; 2; 2Þg ¼ 0:43605 þ 0:38988 þ 0:4788 0:34884 0:28728 0:2856 þ 0:2856 ¼ 0:66861:
7. Computational experiments In this section, computational experiments are conducted to further test the performance of the presented algorithm, along with comparisons between the proposed algorithm and Yeh’s [16]. Both algorithms are implemented in a MATLAB program with a PC (AMD Athlon (tm) 64 X2 1.71 GHz CPU). A capacitated-flow network in Fig. 5 has 8 nodes, 13 edges. The
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largest state and unit capacity cost of each edge are listed in Table 4. The max-flow of the network under the largest state vector (2, 4, 2, 2, 1, 3, 2, 3, 1, 2, 2, 2, 4) is 6. To clearly embody the efficiencies of the two algorithms, we choose six different combinations of demand level and cost constraint: (1, 10)-MP, (2, 17)-MP, (3, 27)-MP, (4, 36)-MP, (5, 49)-MP, (6, 60)-MP. The computational time of searching for different (d, c)-MPs is listed in Table 5. It is observed from Table 5 that the proposed algorithm takes far less time than Yeh’s [16] no matter what kind of (d, c)-MPs is solved. It is not surprising, since the earlier analysis of time complexity in Section 4 has shown that the proposed algorithm is more efficient in solving (d, c)-MP problem. 8. Conclusion Reliability evaluation is an important issue in performance analysis of multi-state systems. Multi-state reliability under cost consideration is a more practical and desirable performance index of multi-state systems, and can be evaluated in terms of (d, c)-MPs. There is no doubt that several researchers [14–16] have been devoted to developing efficient methods for solving (d, c)-MP problem. To search for (d, c)-MPs, the existing methods [14–16] all require MPs information. It is a very timeconsuming task to compute all MPs, because the MP problem is NP-hard [17,18]. In addition, the existing methods [14–16] all involve a step of transforming feasible solutions into (d, c)-MP candidates. Unfortunately, the number of feasible solutions is always quite enormous [9]. This article presents a practical algorithm to evaluate the multi-state reliability under the cost constraint. The proposed algorithm requires neither MPs information nor the procedure of transforming feasible solutions into (d, c)-MP candidates. Furthermore, a novel concept-the lower capacity bound is introduced as a restricted condition for minimal capacity of each edge to reduce the total number of enumerated state vectors. As a result, all (d, c)-MPs can be efficiently obtained by using implicit enumeration algorithm. Like the existing algorithms [14–16] which contribute to reliability evaluation under the cost constraint, the proposed algorithm provides a more efficient algorithm for solving the problem of reliability evaluation under cost consideration. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
L.R. Ford, D.R. Fulkerson, Flows in Networks, Princeton University Press, Princeton, 1962. J.S. Lin, C.C. Jane, J. Yuan, On reliability evaluation of a capacitated-flow network in terms of minimal pathsets, Networks 25 (1995) 131–138. S. Satitsatian, K.C. Kapur, An algorithm for lower reliability bounds of multi-state two-terminal networks, IEEE Trans. Reliab. 55 (2006) 199–206. C.C. Jane, Y.W. Laih, A practical algorithm for computing multi-state two-terminal reliability, IEEE Trans. Reliab. 57 (2008) 295–302. C.C. Jane, Y.W. Laih, Algorithms to determine the threshold reliability of flow networks, IIE Trans. 36 (2004) 469–479. C.C. Jane, J.S. Lin, J. Yuan, On reliability evaluation of a limited-flow network in terms of minimal cutsets, IEEE Trans. Reliab. 42 (1993) 354–361. Y.K. Lin, A simple algorithm for reliability evaluation of a stochastic-flow network with node failure, Comput. Oper. Res. 28 (2001) 1277–1285. Y.K. Lin, Using minimal cuts to evaluate the system reliability of a stochastic-flow network with failures at nodes and arcs, Reliab. Eng. Syst. Safety 75 (2002) 41–46. W.C. Yeh, A novel method for the network reliability in terms of capacitated minimum-paths without knowing minimum-paths in advance, J. Oper. Res. Soc. 56 (2005) 1235–1240. W.C. Yeh, A simple method to verify all d-minimal path candidates of a limited-flow network and its reliability, The Int. J. Adv. Manuf. Technol. 20 (2002) 77–81. W.C. Yeh, A fast algorithm for searching all multi-state minimal cuts, IEEE Trans. Reliab. 57 (2008) 581–588. Yan. Z, M. Qian, Improving efficiency of solving d-MC problem in stochastic-flow network, Reliab. Eng. Syst. Safety 92 (2007) 30–39. J.E. Ramirez-Marquez, D.W. Coit, M. Tortorella, A generalized multi-state-based path vector approach to multi-state two-terminal reliability, IIE Trans. 38 (2006) 477–488. J.S. Lin, Evaluate the reliability of limited-flow networks under the cost constraint, IIE Trans. 30 (1998) 1175–1180. Y.K. Lin, Reliability of a stochastic-flow network with unreliable branches and nodes under budget constraints, IEEE Trans. Reliab. 53 (2004) 381–387. W.C. Yeh, A new approach to evaluate reliability of multi-state networks under the cost constraint, The Int. J. Manage. Sci. 33 (2005) 203–209. C.J. Colbourn, The Combinatorics of Network Reliability, Oxford University Press, New York, 1987. M.O. Ball, Computational complexity of network reliability analysis an overview, IEEE Trans. Reliab. 35 (1993) 230–239. R.K. Ahuja, M. Kodialam, A.K. Mishra, J.B. Orlin, Computational investigations of maximum flow algorithms, Eur. J. Oper. Res. 97 (1997) 509–542. R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows-Theory, Algorithms, and Applications, Prentice-Hall International, Englewood cliffs, NJ, 1993.
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Reliability evaluation of multi-state systems under cost consideration Yi-Feng Niu a,⇑, Xiu-Zhen Xu b a b
School of Mathematics & Information Science, Henan Polytechnic University, Jiaozuo 454000, PR China School of Computer Science & Technology, Henan Polytechnic University, Jiaozuo 454000, PR China
a r t i c l e
i n f o
Article history: Received 27 February 2011 Received in revised form 11 November 2011 Accepted 15 November 2011 Available online 23 November 2011 Keywords: Reliability evaluation Multi-state system Cost constraint Lower capacity bound (d, c)-MP/d-MP/MP/d-flow
a b s t r a c t A more practical and desirable performance index of multi-state systems is the twoterminal reliability for level (d, c) (2TRd, c), defined as the probability that d units of flow can be transmitted from the source node to the sink node with the total cost less than or equal to c. In this article, a simple algorithm is developed to calculate 2TRd, c in terms of (d, c)-MPs. Two major advantages of the proposed algorithm include: (1) as of now, it is the only algorithm that searches for (d, c)-MPs without requiring all minimal paths (MPs) and the procedure of transforming feasible solutions; (2) it is more practical and efficient in solving (d, c)-MP problem in contrast to the best-known method. An example is provided to illustrate the generation of (d, c)-MPs by using the presented algorithm, and 2TRd, c is thus evaluated. Furthermore, the computational experiments are conducted to verify the performance of the presented algorithm. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Reliability is usually an important index to evaluate the service quality of many real-life systems such as transportation systems, logistics systems, computer systems, and manufacturing systems. The capacitated-flow network explored by Ford and Fulkerson [1] is the most widely employed model for reliability analysis. A capacitated-flow network consists of multistate edges which have independent, finite and multi-integer-valued random capacities [2], and has a finite number of different states for performance rate. For such a network, the two-terminal reliability for level d (2TRd), defined as the probability that d units of flow can be transmitted from the source node to the sink node through multi-state edges [3,4], is a combination of source-sink connection, edge capacity, and flow demand [5]. To compute the NP-hard 2TRd without considering the cost constraint, a variety of methods [2–13] have been presented. In general, these methods can be classified into two types: One is based on d-minimal paths (d-MPs) [2,3,7,9,10,13], and the other is based on d-minimal cuts (d-MCs) [6,8,11,12]. However, in addition to capacity limitation, each edge is always associated with a cost that must be paid for every unit of flow passing through it. So, it is more practical and suitable to incorporate cost attribute as an integral part of reliability analysis. Taking cost into account, the two-terminal reliability for level (d, c) (2TRd, c), which is defined as the probability that d units of flow can be transmitted from the source node to the sink node under the cost constraint c, is a more desirable index for evaluating the performance of many real-life systems. As 2TRd can be computed in terms of d-MPs, 2TRd, c can be computed in terms of (d, c)-MPs [14–16]. A (d, c)-MP is a special d-MP such that the total capacity cost is not greater than c. Once all (d, c)-MPs are known, 2TRd, c can be calculated by using inclusion-exclusion principle. Hence, searching for all (d, c)-MPs plays an important part in solving the problem of 2TRd, c. Based on MPs, Lin [14] proposed an algorithm to search for all (d, c)-MPs. Lin’s algorithm first uses implicit enumeration algorithm to search for (d, c)-MP candidates. Then, all (d, c)-MP candidates are further verified whether they are (d, c)-MPs by ⇑ Corresponding author. E-mail address: [email protected] (Y.-F. Niu). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.11.055
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comparison process. It is a time-consuming task to check whether (d, c)-MP candidates are (d, c)-MPs by comparison process due to the exponentially growing number of (d, c)-MP candidates [16]. Lin [15] extended the work to the case with unreliable node. The presented method in [15] uses MPs to assign the flow to each component (edge or node). All (d, c)-MPs can be obtained by using implicit enumeration algorithm. Notice that this algorithm also involves the comparison process in checking (d, c)-MP candidates. With improvement on checking (d, c)-MPs candidates, Yeh [16] proposed a new approach to find (d, c)-MPs based on MPs. Yeh’s algorithm also uses implicit enumeration algorithm to search for (d, c)-MP candidates by solving a mathematical programming model imposed by the limitation of max-capacity, flow-conservation law and capacity cost. Then, an efficient cycle-checking method without requiring comparison process is employed to verify (d, c)-MP candidates. Yeh’s algorithm is superior to the algorithms in [14,15] in solving (d, c)-MP problem. However, just like the algorithms in [14,15], Yeh’s algorithm is also subject to the assumption that all MPs must be known in advance. The above introduction shows that the well-known existing algorithms for 2TRd, c [14–16] must require all MPs information in advance. It is a very cumbersome and time-consuming task if all MPs must be computed, since the MP problem is known to be NP-hard [17,18]. In this article, a more efficient algorithm for 2TRd, c. is presented. The proposed algorithm requires neither MPs information, nor the procedure of transforming feasible solutions. Besides, in order to improve the efficiency of seeking (d, c)-MPs, the proposed algorithm introduces a novel concept-the lower capacity bound to reduce the amount of enumerated state vectors. Therefore, all (d, c)-MPs can be efficiently obtained by using implicit enumeration algorithm. The rest of this paper is organized as follows: Acronyms, nomenclature, notations and assumptions are described in Section 2. In Section 3, the network model is introduced, and the best-known method for solving (d, c)-MP problem is briefly reviewed. Meanwhile, some related theorems are discussed. In Section 4, the proposed algorithm is described in detail, and the computational complexity of the algorithm is also analyzed along with a contrast to the best-known method. Section 5 introduces the inclusion-exclusion method for reliability evaluation. In Section 6, an illustrative example is provided to illustrate how to search for all (d, c)-MPs by using the proposed algorithm. Then, the reliability is evaluated in terms of all (d, c)-MPs. Computational experiments are conducted in Section 7 to further test the performance of the proposed algorithm, together with comparisons between the proposed algorithm and the best-known method. The final section presents the concluding remarks. 2. Acronyms, nomenclature, notations and assumptions 2.1. Acronyms MP/MC 2TRd 2TRd, c Max-flow
minimal path/minimal cut two-terminal reliability for level d two-terminal reliability for level (d, c) maximum flow
2.2. Nomenclature Level d d-flow d-MP (d, c)-MP Network reliability for level (d, c)
a non-negative integer-valued flow demand from s to t a flow vector f1d ; f2d ; . . . ; fmd representing the flows through each edge ei when the amount of flow sent from s to t is d a system state vector x = (x1, x2, . . . , xm) is a d-MP if and only if M(x) = d, and M(x 0(ei)) < d for each xi > 0 a (d, c)-MP is a d-MP such that the total capacity cost is not greater than c The probability that d units of flow can be transmitted from the source node to the sink node such that the total capacity cost is less than or equal to c
2.3. Notations G(V, E, U, C) s, t ei
a capacitated-flow network with the set of nodes V = {s, 1, 2, . . . , n, t}, the set of edges E = {e1, e2, . . . , em}, the largest state vector U = (u1, u2, . . . , um), and the cost vector C = (c1, c2, . . . , cm) the source and sink nodes, respectively the ith edge in E
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m, n (v, ) (, v) ci xi ui L(ei) x U(0i) C(x) Pi p M(x) fi fid f Fi 0(ei) R(d,c)
4263
the number of edges, and the number of nodes except for nodes s and t a set of edges emanating from node v a set of edges pointing to node v the unit capacity cost of ei a state of ei representing the amount of flow allowed to be sent through edge ei the largest state of ei the minimal capacity of ei such that the max-flow from s to t is d the system state vector x = (x1, x2, . . . , xm) a special system state vector U(0i) = (u1, . . . , ui1, 0, ui+1, . . . , um), i.e., state level is 0 for edge ei and the largest for other edges. P CðxÞ ¼ m i¼1 xi c i is the total capacity cost of the system state vector x the ith MP in G(V, E, U, C) the number of MPs in G(V, E, U, C) the max-flow of the network under x the flows through edge ei the flows through edge ei when there are d units of flow sent from s to t a flow vector f = (f1, f2, . . . , fm) Fi = min{ui | for each ej 2 Pi} is the minimal capacity on Pi 0(ei) = (0, . . . , 0, 1, 0, . . . , 0), i.e., state level is 1 for edge ei and 0 for other edges the reliability for level (d, c)
2.4. Assumptions The network satisfies the following assumptions [16]: 1. Each node is perfectly reliable. 2. The state/capacity of each edge is a successive random variable which takes integer values from 0 to ui according to a given distribution. 3. The states/capacities of different edges are s-independent. 4. All flows in the network obey the conservation law. 3. Network model and preliminaries 3.1. Network model Let G(V, E, U, C) denote a capacitated-flow network with the unique source node s and the unique sink node t, where V = {s, 1, 2, . . . , n, t} is the set of nodes, E = {e1, e2, . . . , em} is the set of edges, U = (u1, u2,. . . , um) is the largest state vector, and C = (c1, c2, . . . , cm) is the cost vector, where ci is the unit capacity cost of ei. The current state of ei is defined by xi which takes integer values from 0 to ui, where ui denotes the largest state of edge ei. A system state vector x = (x1, x2, . . . , xm) indicates the current state of each edge ei. Let M(x) represent the max-flow of the network under x, then M(x) is always called the structure function of a multi-state network. For example, consider a capacitated-flow network G(V, E, U, C) in Fig. 1, which shows V = {s, 1, 2, t}, E = {e1, e2, e3, e4, e5, e6}, U = (3, 2, 1, 1, 2, 2), and C = (3, 1, 1, 1, 1, 3). For a given system state vector x = (2, 1, 1, 0, 1, 2), which indicates the current states of e1, e2, e3, e4, e5, e6 are 2, 1, 1, 0, 1, and 2, respectively, the total capacity cost of the system state vector x is C(x) = 2 3 + 1 1 + 1 1 + 0 1 + 1 1 + 2 3 = 15, and the max-flow of the network under x is M(x) = 3. 3.2. Preliminaries Before presenting the proposed algorithm, there is need to briefly review the best-known method for solving (d, c)-MP problem. As of now, the method proposed by Yeh [16] is the most efficient approach to search for (d, c)-MPs. The following Lemma [16] is the foundation of Yeh’s method. Lemma 1. Any state vector x = (x1, x2, . . . , xm) is a (d, c)-MP candidate, if and only if its corresponding flow vector f = (f1, f2, . . . , fp) satisfies the following four conditions. p X j¼1
fj ¼ d;
ð1Þ
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Fig. 1. A simple flow network.
fj F j for each j ¼ 1; 2; . . . ; p; xi ¼
X
ð2Þ
fj for each i ¼ 1; 2; . . . ; m;
ð3Þ
ei 2P j m X
xi ci c:
ð4Þ
i¼1
Firstly, Yeh’s method uses implicit enumeration algorithm to solve all feasible solutions to constraints (1) and (2). Then, all feasible solutions are transformed into (d, c)-MP candidates according to constraints (3) and (4). Finally, all (d, c)-MP candidates are verified whether they are (d, c)-MPs by using cycle-checking method. The total number of feasible solutions sat p þ d 1 Qp isfying constraints (1) and (2) is bounded by Min ; j¼1 ð1 þ MinfF j ; dgÞ [16]. It takes O(mp) time to transform d a feasible solution into its corresponding (d, c)-MP candidate. For a (d, c)-MP candidate, Yeh’s algorithm takes O(n) time to verify if it is a (d, c)-MP [16]. Therefore, we have the following statement. p þ d 1 Qp Corollary 1. The time complexity of Yeh’s algorithm is O(mp Min ; j¼1 ð1 þ MinfF j ; dgÞ ) for finding all (d, c)-MPs. d Corollary 1 indicates that the time complexity of Yeh’s algorithm is directly proportional to the number of MPs. Unfortunately, the number of MPs grows exponentially with the number of nodes and edges [9]. Consequently, Yeh’s method is impractical and inefficient in searching for (d, c)-MPs. A more practical and efficient method is thus developed to solve (d, c)-MP problem in the following discussion. The definition of (d, c)-MP shows that a state vector x is a (d, c)-MP if and only if x is a d-MP and C(x) 6 c. Then, the first important step to solving (d, c)-MP problem is to search for all d-MPs. When the amount of flow transmitted from s to t is d, a d-flow which is a flow vector (f1d ; f2d ; . . . ; fmd Þ consists of flows fid through each edge ei for 1 6 i 6 m. Thus, if x is a d-flow, we have M(x) = d and x satisfies flow-conservation law [19,20]. Therefore, the following theorem holds. Theorem 1. x = (x1, x2, . . . , xm) is a d-flow if and only if the following three conditions are satisfied.
X
xi ¼
ei 2ðs;Þ
X ei 2ð;v Þ
X
xj ¼ d;
ð5Þ
ej 2ð;tÞ
xi ¼
X
xj
for ev ery node
v 2 f1; 2; . . . ; ng;
ð6Þ
ej 2ðv ;Þ
0 xi Minfui ; dg for 1 i m:
Proof. Immediately from the definition of d-flow and flow-conservation law.
ð7Þ
h
Any state vector which satisfies conditions (5) and (6) is called a d-MP candidate [9]. Then, any d-flow, in fact, is also a dMP candidate. Each d-MP candidate needs to be verified whether it is a d-MP. The most efficient method for verifying d-MP candidate is the cycle-checking method proposed by Yeh [9,10,16]. Such a cycle-checking method is based on the following lemma [10,16].
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Lemma 2. A d-MP candidate x is a d-MP if and only if there is no directed cycle in it. The following theorem specifies the relationship between d-flow and d-MP. Theorem 2. A d-flow x is a d-MP if and only if there is no directed cycle in it. Proof. Directly from Lemma 2.
h
According to Theorem 1, Theorem 2 and the definition of (d, c)-MP, it is easy to have the following conclusions. Corollary 2. A d-flow x is a (d, c)-MP if and only if C(x) 6 c, and there is no directed cycle in it. Corollary 3. A state vector x = (x1, x2, . . . , xm) is a (d, c)-MP if and only if the following conditions are satisfied.
X
xi ¼
ei 2ðs;Þ
X
X
xj ¼ d;
ð8Þ
ej 2ð;tÞ
xi ¼
ei 2ð;v Þ
X
xj
for every node
v 2 f1; 2; . . . ; ng;
ð9Þ
ej 2ðv ;Þ
0 xi Minfui ; dg for 1 i m;
ð10Þ
CðxÞ c;
ð11Þ
there is no directed cycle in x:
ð12Þ
Corollary 3 provides a simple enumeration method to search for (d, c)-MPs without knowing any MP information. However, given a capacitated-flow network G(V, E, U, C), it is obvious that the total number of state vectors satisfying condition (10) is Q Qm bounded by m i¼1 ðMinfui ; dg þ 1Þ. That is, the amount of state vectors to be enumerated is up to i¼1 ðMinfui ; dg þ 1Þ. So, it is time-consuming to search for (d, c)-MPs by using Corollary 3 in a straightforward manner, and there is a room for reducing the number of enumerated state vectors. The following discussion will introduce a novel concept-the lower capacity bound as a limited condition to reduce the number of enumerated state vectors. The lower capacity bound L(ei) is the minimal capacity of ei such that the max-flow from s to t is equal to d [11]. L(ei) is a restricted condition for minimal state of edge ei, and can be found according to the following Theorem 3. Theorem 3. Given the demand d (0 < d 6 M(U)), let U(0i) denote a special system state vector in which state level is 0 for edge ei and the largest for other edges, then
Lðei Þ ¼
0; if MðUð0i ÞÞ d; d MðUð0i ÞÞ; if MðUð0i ÞÞ < d:
ð13Þ
Proof. If M(U(0i)) P d, it means that even when the capacity of ei is 0, d units of flow can be transmitted from the source node to the sink node. So, L(ei) = 0. If M(U(0i)) < d, since M(U) P d, it means that at least d M(U(0i)) units of flow must be sent through edge ei such that d units of flow can be transmitted from the source node to the sink node. So, L(ei) = d M(U(0i)). h Theorem 3 shows that to find the lower capacity bound L(ei) of ei, there is only need to compute M(U(0i)). Hence, the total time complexity is O(mn3) for finding all lower capacity bounds, where n3 is the time complexity for evaluating the max-flow [19,20]. The role of lower capacity bound lies in the following theorem. Theorem 4. For a state vector x = (x1, x2, . . . , xm), if there exists a state xi satisfying xi < L(ei) (1 6 i 6 m), then x must not be a (d, c)MP. Proof. By the definition of lower capacity bound, we have M(x) < d if xi < L(ei) for some edge ei. Consequently, x is not a (d, c)MP. h By Theorem 4, it is easy to have the following conclusion. Corollary 4. For a state vector x = (x1, x2, . . . , xm), if x is a (d, c)-MP, then L(ei) 6 xi for 1 6 i 6 m. The following theorem which is the foundation of the proposed algorithm provides a practical method to search for all (d, c)-MPs.
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Theorem 5. x = (x1, x2, . . . , xm) is a (d, c)-MP if and only if the following conditions are satisfied.
X
xi ¼
ei 2ðs;Þ
X
X
xj ¼ d;
ð14Þ
ej 2ð;tÞ
xi ¼
ei 2ð;v Þ
X
xj
for every node
v 2 f1; 2; . . . ; ng;
ð15Þ
ej 2ðv ;Þ
Lðei Þ xi Minfui ; dg for 1 i m;
ð16Þ
CðxÞ c;
ð17Þ
there is no directed cycle in x:
ð18Þ
Proof. Directly from Corollary 3 and Corollary 4.
h
By Theorem 5 we have the following conclusion. Corollary 5. For an acyclic network and a state vector x, if x satisfies conditions (14)–(17), then x is a (d, c)-MP. Q It is clear that the total number of state vectors satisfying condition (16) is bounded by m ðMinfui ; dg þ 1 Lðei ÞÞ. That Qm Qmi¼1 is, the amount of state vectors to be enumerated has reduced from i¼1 ðMinfui ; dg þ 1Þ to i¼1 ðMinfui ; dg þ 1 Lðei ÞÞ. Since Qm Qm i¼1 ðMinfui ; dg þ 1 Lðei ÞÞ is always much less than i¼1 ðMinfui ; dg þ 1Þ, searching for (d, c)-MPs by using Theorem 5 is more practical and efficient in contrast to Corollary 3. 4. The algorithm for generating all (d, c)-MPs Through the above discussion, all (d, c)-MPs can be found by the following steps without knowing all MPs in advance. Input: A capacitated-flow network G(V, E, U, C). Output: All (d, c)-MPs. Step 0: Find the lower capacity bound L(ei) of ei by using Theorem 3, where 1 6 i 6 m. Step 1: Use implicit enumeration algorithm to solve all of the feasible solutions, say y1, y2, . . . , yq, to constraints (14)–(16). Step 2: If C(yj) 6 c and there is no directed cycle in yj, yj is a (d, c)-MP; otherwise, yj is not a (d, c)-MP, where 1 6 j 6 q. The detailed solution steps show that in fact, the proposed algorithm is also an enumeration algorithm. To clearly understand the differences between the proposed algorithm and the well-known existing methods, the main characteristics of different algorithms are listed in Table 1. The time complexity will be discussed below. As we mentioned before, Step 0 requires O(mn3) time to find all lower capacity bounds, where n3 is the time complexity for evaluating the max-flow [19,20]. The total number of state vectors satQ isfying constraint (16) is bounded by m ; dg þ 1 Lðei ÞÞ: It takes O(n) time to check if constraints (14) and (15) are i¼1 ðMinfu Qi satisfied for all nodes. Then, Step 1 requires Oðn m i¼1 ðMinfui ; dg þ 1 Lðei ÞÞÞ time in total. For a state vector yj, it only needs O(n) time to find a directed cycle [20], and O(m) time to check if C(yj) 6 c. Hence, Step 2 requires O((m + n)q) time. Since q is Q Qm bounded by m i¼1 ðMinfui ; dg þ 1 Lðei ÞÞ; Step 2 takes O((m + n) i¼1 ðMinfui ; dg þ 1 Lðei ÞÞ) amount of time in the worst case. Therefore, the total time complexity of finding (d, c)-MPs by using the proposed algorithm is
Oðmn3 Þ þ O n
m Y
!
ðMinfui ; dg þ 1 Lðei ÞÞ þ Oððm þ nÞ
i¼1
O ðm þ nÞ
m Y
m Y
ðMinfui ; dg þ 1 Lðei ÞÞÞ
i¼1
! ðMinfui ; dg þ 1 Lðei ÞÞ :
ð19Þ
i¼1
Table 1 The characteristics of different algorithms.
a b
Algorithm
Based on IEA?a
Needs MPs?
Needs transforming procedure?b
Use what method to verify (d, c)-MPs?
The The The The
Yes Yes Yes Yes
Yes Yes Yes No
Yes Yes Yes No
Comparison method Comparison method Cycle-checking method Cycle-checking method
algorithm in [14] algorithm in [15] algorithm in [16] proposed algorithm
Implicit enumeration algorithm. The procedure of transforming feasible solutions into (d, c)-MP candidates.
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Hence, the following statement holds. Theorem 6. The time complexity of the proposed algorithm is O((m + n)
Qm
i¼1 ðMinfui ; dg
þ 1 Lðei ÞÞ) for finding all (d, c)-MPs.
p þ d 1 Qp By Corollary 1, the time complexity of Yeh’s algorithm is OðmpMin ; j¼1 ð1 þ MinfF j ; dgÞÞ. Since the number d of MPs grows exponentially with the number of nodes and edges [9], we have n 6 m p. Meanwhile, we know d 6 M(U) p. Then,
! Y p pþd1 Oððm þ nÞ ðMinfui ; dg þ 1 Lðei ÞÞÞ O mp Min ; ð1 þ MinfF j ; dgÞ : d i¼1 j¼1 m Y
ð20Þ
Through the above detailed analysis, it is understood that from the viewpoint of time complexity, the proposed algorithm is more efficient than the best-known algorithm [16] in solving (d, c)-MP problem. 5. Reliability evaluation Once all (d, c)-MPs are known, the inclusion-exclusion method can be used to calculate 2TRd, c. Assume y1, y2, . . . , yq are all (d, c)-MPs, and let B1 = {x | x P y1}, B2 = {x | x P y2}, . . . , Bq = {x | x P yq}. Then, the two-terminal reliability for level (d, c) by using the inclusion–exclusion method can be evaluated as follows:
Rðd;cÞ ¼ PrðB1 [ B2 [ . . . [ Bq Þ ¼
q X
PrðBi Þ
i¼1
where Pr{x | x P yi} =
Qm
j¼1 Prfxj
q X j1 X j¼2
PrðBi \ Bj Þ þ þ ð1Þq1 PrðB1 \ B2 \ . . . \ Bq Þ
ð21Þ
i¼1
P yij g and yi = (yi1, yi2, . . . , yim).
6. An illustrative example A simple flow network in Fig. 1 which is also discussed in Ref. [16] is chosen to demonstrate how the proposed algorithm works. The probability distribution of each edge is given in Table 2. We would like to know the probability that 3 units of flow can be transmitted from the source node s to the sink node t such that the total capacity cost is less than or equal to 14. Without knowing all MPs in advance, all (3, 14)-MPs (d = 3, c = 14) can be found as follows:
Table 2 States and state probabilities of edges of Fig. 1. Edge
States
e1 e2 e3 e4 e5 e6
0 0 0 0 0 0
State probabilities 1 1 1 1 1 1
2 2 – – 2 2
3 – – – – –
0.05 0.10 0.10 0.10 0.10 0.05
Fig. 2. The network with no directed cycle under y1.
0.10 0.30 0.90 0.90 0.10 0.25
0.25 0.60 – – 0.80 0.70
0.60 – – – – –
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Solve: Step 0: By Theorem 3, it is easy to have L(e1) = 1, L(e2) = 1, L(e3) = L(e4) = L(e5) = 0, and L(e6) = 1. Step 1: Use implicit enumeration algorithm to solve all of the feasible solutions yi = (yi1, yi2, yi3, yi4, yi5, yi6) of the following constraints:
yi1 þ yi5 ¼ yi2 þ yi6 ¼ 3;
ð22Þ
yi1 þ yi4 ¼ yi2 þ yi3 ;
ð23Þ
yi3 þ yi5 ¼ yi4 þ yi6 ;
ð24Þ
1 yi1 3;
ð25Þ
1 yi2 2;
ð26Þ
0 yi3 1;
ð27Þ
0 yi4 1;
ð28Þ
0 yi5 2;
ð29Þ
1 yi6 2;
ð30Þ
As a result, there are seven feasible solutions: y1 = (2, 2, 0, 0, 1, 1), y2 = (2, 1, 1, 0, 1, 2), y3 = (3, 2, 1, 0, 0, 1), y4 = (1, 2, 0, 1, 2, 1), y5 = (1, 1, 0, 0, 2, 2), y6 = (2, 2, 1, 1, 1, 1), y7 = (1, 1, 1, 1, 2, 2).
Step 2: Since C(y1) = 2 3 + 2 1 + 0 1 + 0 1 + 1 1 + 1 3 = 12 6 14, and there is no directed cycle in y1 (see Fig. 2), y1 = (3, 2, 1, 0, 0, 1) is a (3, 14)-MP. In the same way, we have that both y4 = (1, 2, 0, 1, 2, 1) and y5 = (1, 1, 0, 0, 2, 2) are also (3, 14)-MPs. But, since C(y2) = C(y3) = 15 > 14, y2 = (2, 1, 1, 0, 1, 2) and y3 = (3, 2, 1, 0, 0, 1) are not (3, 14)-MPs. y6 = (2, 2, 1, 1, 1, 1) and y7 = (1, 1, 1, 1, 2, 2) are also not (3, 14)-MPs due to the fact that there exist directed cycles in both y6 and y7 (see Fig. 3 and Fig. 4).
Fig. 3. The network with a directed cycle 1 ? e3 ? 2 ? e4 ? 1 under y6.
Fig. 4. The network with a directed cycle 1 ? e3 ? 2 ? e4 ? 1 under y7.
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C(yi) 6 14?
A directed cycle in yi?
A (3, 14)-MP?
y1 = (2, 2, 0, 0, 1, 1) y2 = (2, 1, 1, 0, 1, 2) y3 = (3, 2, 1, 0, 0, 1) y4 = (1, 2, 0, 1, 2, 1) y5 = (1, 1, 0, 0, 2, 2) y6 = (2, 2, 1, 1, 1, 1) y7 = (1, 1, 1, 1, 2, 2)
Yes No No Yes Yes Yes Yes
No No No No No Yes Yes
Yes No No Yes Yes No No
Fig. 5. A capacitated-flow network.
Table 4 The largest state and unit capacity cost of each edge in Fig. 5. Edges
e1
e2
e3
e4
e5
e6
e7
e8
e9
e10
e11
e12
e13
ui ci
2 3
4 2
2 1
2 2
1 1
3 2
2 3
3 2
1 1
2 3
2 1
2 2
4 3
Table 5 Comparison of the proposed algorithm with Yeh’s algorithm. (d, c)-MP
(1, 10)-MP (2, 17)-MP (3, 27)-MP (4, 36)-MP (5, 49)-MP (6, 60)-MP
Computational time (s) Yeh’s algorithm
The proposed algorithm
89.047 89.343 89.859 90.468 90.812 92.249
0.721 5.799 9.705 5.752 0.831 0.627
The solution results for all (3, 14)-MPs obtained by the proposed algorithm are listed in Table 3. From the solution results in Table 3, we know y1 = (2, 2, 0, 0, 1, 1), y4 = (1, 2, 0, 1, 2, 1), and y5 = (1, 1, 0, 0, 2, 2) are all (3, 14)-MPs. Set B1 = {x | x P (2, 2, 0, 0, 1, 1)}, B2 = {x | x P (1, 2, 0, 1, 2, 1)}, B3 = {x | x P (1, 1, 0, 0, 2, 2)}, then the reliability for level (3, 14) is obtained as follows:
Rð3;14Þ ¼ PrfB1 [ B2 [ B3 g ¼ PrfB1 g þ PrfB2 g þ PrfB3 g PrfB1 \ B2 g PrfB1 \ B3 g PrfB2 \ B3 g þ PrfB1 \ B2 \ B3 g ¼ Prfxjx ð2; 2; 0; 0; 1; 1Þg þ Prfxjx ðð1; 2; 0; 1; 2; 1Þg þ Prfxjx ð1; 1; 0; 0; 2; 2Þg Prfxjx ð2; 2; 0; 1; 2; 1Þg Prfxjx ð2; 2; 0; 0; 2; 2Þg Prfxjx ð1; 2; 0; 1; 2; 2Þg þ Prfxjx ð2; 2; 0; 1; 2; 2Þg ¼ 0:43605 þ 0:38988 þ 0:4788 0:34884 0:28728 0:2856 þ 0:2856 ¼ 0:66861:
7. Computational experiments In this section, computational experiments are conducted to further test the performance of the presented algorithm, along with comparisons between the proposed algorithm and Yeh’s [16]. Both algorithms are implemented in a MATLAB program with a PC (AMD Athlon (tm) 64 X2 1.71 GHz CPU). A capacitated-flow network in Fig. 5 has 8 nodes, 13 edges. The
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largest state and unit capacity cost of each edge are listed in Table 4. The max-flow of the network under the largest state vector (2, 4, 2, 2, 1, 3, 2, 3, 1, 2, 2, 2, 4) is 6. To clearly embody the efficiencies of the two algorithms, we choose six different combinations of demand level and cost constraint: (1, 10)-MP, (2, 17)-MP, (3, 27)-MP, (4, 36)-MP, (5, 49)-MP, (6, 60)-MP. The computational time of searching for different (d, c)-MPs is listed in Table 5. It is observed from Table 5 that the proposed algorithm takes far less time than Yeh’s [16] no matter what kind of (d, c)-MPs is solved. It is not surprising, since the earlier analysis of time complexity in Section 4 has shown that the proposed algorithm is more efficient in solving (d, c)-MP problem. 8. Conclusion Reliability evaluation is an important issue in performance analysis of multi-state systems. Multi-state reliability under cost consideration is a more practical and desirable performance index of multi-state systems, and can be evaluated in terms of (d, c)-MPs. There is no doubt that several researchers [14–16] have been devoted to developing efficient methods for solving (d, c)-MP problem. To search for (d, c)-MPs, the existing methods [14–16] all require MPs information. It is a very timeconsuming task to compute all MPs, because the MP problem is NP-hard [17,18]. In addition, the existing methods [14–16] all involve a step of transforming feasible solutions into (d, c)-MP candidates. Unfortunately, the number of feasible solutions is always quite enormous [9]. This article presents a practical algorithm to evaluate the multi-state reliability under the cost constraint. The proposed algorithm requires neither MPs information nor the procedure of transforming feasible solutions into (d, c)-MP candidates. Furthermore, a novel concept-the lower capacity bound is introduced as a restricted condition for minimal capacity of each edge to reduce the total number of enumerated state vectors. As a result, all (d, c)-MPs can be efficiently obtained by using implicit enumeration algorithm. Like the existing algorithms [14–16] which contribute to reliability evaluation under the cost constraint, the proposed algorithm provides a more efficient algorithm for solving the problem of reliability evaluation under cost consideration. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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