Efficient Enumeration of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 9.69102 12.533" width="9.69102pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-101" d="M530 686C535 705 530 712 521 712C504 712 448 684 359 674L358 648H393C437 648 439 646 429 593L400 435C372 447 345 448 332 448C286 448 194 414 144 373C68 311 23 203 23 111C23 26 57 -12 91 -12C120 -12 147 3 188 29C227 54 290 102 341 170H343L322 71C308 6 320 -12 341 -12C373 -12 442 27 501 96L485 120C455 91 422 67 408 67C401 67 401 76 404 91C440 294 479 473 530 686ZM387 375L355 241C326 187 200 53 142 53C126 53 109 73 109 130C109 217 154 337 218 381C240 396 265 404 297 404S372 390 387 375Z"/></g></svg>-</span>Minimal Paths in Reliability Evaluation of Multistate Networks

Xu, Xiu-Zhen; Niu, Yi-Feng; Li, Qing

doi:https://doi.org/10.1155/2019/4561845

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Research Article | Open Access

Volume 2019 | Article ID 4561845 | https://doi.org/10.1155/2019/4561845

Efficient Enumeration of -Minimal Paths in Reliability Evaluation of Multistate Networks

Xiu-Zhen Xu,^1,2Yi-Feng Niu ,¹and Qing Li³

Academic Editor: Dimitri Volchenkov

Received15 Jan 2019

Revised21 Feb 2019

Accepted26 Feb 2019

Published21 Mar 2019

Abstract

A number of real-world complex networks can be modeled as multistate networks for performance analysis. A multistate network consists of multistate components and possesses multiple different performance levels. For such a network, reliability is concerned with the probability of the network capacity level greater than or equal to a predetermined demand level . One major method for multistate network reliability evaluation is using d-minimal paths. This paper proposes an efficient algorithm to find d-minimal paths. First, a new concept of qualified state vector is defined so as to fix a relatively smaller search space of d-minimal paths, and a sufficient and necessary condition for a qualified state vector to be d-minimal path is established. Then, the max-flow algorithm and the enumeration algorithm are integrated to search for d-minimal paths in the determined search space that is recursively divided into subspaces such that the searching efficiency can be increased as much as possible. Both analytical and numerical results show that the proposed algorithm is more efficient in finding all d-minimal paths. In addition, a case study related to power transmission network is performed to demonstrate the implication of network reliability.

1. Introduction

Network models have been extensively applied in analyzing real-world complex systems, such as computer and information networks [1, 2], power transmission networks [3], manufacturing networks [4], logistics networks [5–13], and gene regulatory networks [14–16]. From the viewpoint of network reliability, network model is generally classified as binary state network model and multistate network model [10]. Binary state network model assumes two states for network components: fully working or completely failed [17–20]. For example, in a gene regulatory network, the state of each gene is described by two levels: either active (fully working) or inactive (completely failed) [20], so it can be modeled as a binary state network. However, a number of actual complex systems are often affected by various random parameters due to external or internal uncertainties [21], such that they could exhibit multiple behaviors. In such situation, binary state network model is insufficient in network analysis. For example, consider a power transmission network constructed by a set of nodes (high voltage transmission towers, high voltage substations, or power plants) and a set of arcs (transmission lines). Each transmission line linking two nodes is combined with several physical lines and each physical line may provide a specific capacity or fail with a probability [3]. That is, each transmission line has several capacities due to complete failure, partial failure or maintenance, etc. In this sense, the behavior of the power transmission network is multistate. For practical applications, binary state network model has been extended and replaced by multistate network model to characterize complex behaviors of multistate networks.

A multistate network is composed of multistate components and holds a finite number of different states for performance rate. For such a network, an interesting question is to evaluate its reliability defined as the probability of the network capacity level greater than or equal to a predetermined demand level . Researchers have presented various algorithms to evaluate the reliability of multistate networks, including direct algorithm [22–24] and indirect algorithm [25–33]. Indirect algorithm is often called network-based algorithm established in terms of minimal paths or minimal cuts. One common indirect method for reliability computation of multistate networks is based on minimal path vectors to level d (d-minimal paths for short) [25–33]. And, all of these methods pay attention to the enumeration of d-minimal paths.

Two important types of models have been reported to solve d-minimal paths. One is proposed by Lin et al. [25], and the other is proposed by Yeh [26]. All minimal paths are assumed to be known; the model by Lin et al. [25] is constructed via the network structure and the flow conservation law. A path is a sequence of arcs that connect the source node to the sink node, and a minimal path is such a path that removing any arc will make it no longer a path. Afterward, some algorithms have been developed to find d-minimal paths in terms of the model by Lin et al. [25], such as the methods by Lin [27], Yeh [28], Chen and Lin [29], and Xu et al. [13]. It is noted that there are two drawbacks for these methods in [13, 25, 27–29]. The first is that they require all minimal paths as prior knowledge, which is an NP-hard problem, and the second is the extra burden for removing duplicate d-minimal paths.

The model by Yeh [26] requires no minimal paths knowledge, and thus possesses an advantage relative to the model by Lin et al. [25]. But, one remarkable limitation of Yeh's model is that a huge number of state vectors need to be enumerated, which leads to high computational complexity. Recently, Niu et al. [30] have attempted to improve the method of Yeh [26] by means of the concept of lower capacity bound, but the efficiency of their method highly depends on the network structure and the specified demand level . In particular, the method by Niu et al. [30] is even inferior to the one by Yeh when the demand level is low. The method by Xu and Niu [31] is another improvement to the method by Yeh [26] and is proven to be more efficient in solving d-minimal paths.

The enumeration of d-minimal paths is a difficult task because it is an NP-hard problem [23], and developing an efficient algorithm is worthwhile from the viewpoint of reliability evaluation. Therefore, this paper focuses on the efficient enumeration of d-minimal paths in multistate network reliability evaluation. More explicitly, the incremental contributions of this paper are twofold. First, a new concept of qualified state vector is defined to determine the search space of d-minimal paths more accurately, and a sufficient and necessary condition for a qualified state vector to be d-minimal path is established. Second, the max-flow algorithm and the enumeration method are integrated to search for d-minimal paths in the determined search space that is recursively divided into subspaces so that the efficiency of finding d-minimal paths can be increased as much as possible. Both analytical and numerical results indicate that the proposed algorithm compares favorably with the existing algorithm. Finally, a case study related to power transmission network is performed to demonstrate the implications of network reliability.

The rest of this paper is arranged as follows: Section 2 introduces multistate network model and reviews the concept of network reliability. In Section 3, a concept of qualified state vector is introduced, and a sufficient and necessary condition for a qualified state vector to be d-minimal path is proposed. Moreover, the main ideas for searching for d-minimal paths are discussed, based on which an algorithm is developed along with analyses on its time complexity. In Section 4, an illustrative example and a numerical example are provided to validate the effectiveness and efficiency of the proposed algorithm. A practical case is studied in Section 5 to demonstrate the implications of network reliability. The final section presents some concluding remarks.

2. Multistate Network Model and Network Reliability

2.1. Multistate Network Model

Mathematically, a multistate network is modeled as a directed graph , where is the set of nodes, is the set of arcs, and is the universal space of state vectors. The state (capacity) of arc defined by is a nonnegative integer random variable ranging from the smallest value 0 to the largest value with a known probability distribution. A state vector c = (, ,…, ) represents the current states of all arcs in the network. The state (capacity) of a network under state vector is determined by the so-called structure function M(c) denoting the maximum amount of flow that can be sent from the source node to the sink node when the sates of arcs are defined by . For simplicity, the universal space can be denoted by its smallest state vector and largest state vector as . Let be a set of state vectors, then is a subset of .

For example, consider a multistate network shown in Figure 1 with , , and = [(0, 0, 0, 0, 0, 0), (3, 2, 1, 1, 1, 2)] according to Table 1. For a state vector c = (2, 1, 1, 1, 1, 2), the state of the network under is M(c) = 3, that is, the maximum amount of flow that can be sent from the source node to the sink node is 3 when = 2, = 1, = 1, = 1, =1, = 2. Given a set of state vector , its smallest state vector and largest state vector are l^C = (1, 1, 0, 0, 0, 0) and u^C = (3, 2, 1, 0, 0, 2), respectively.

Following the literature in [25, 26, 28–33], this paper considers four assumptions: (1) the state/capacity of each arc is a nonnegative integer random variable following a given probability distribution; (2) the states/capacities of different arcs are statistically independent; (3) each node is perfectly reliable; (4) all flows in the network obey the flow conservation law, i.e., total flows into and from a node (other than the source and sink nodes) are all equal.

2.2. Network Reliability

Reliability is a fundamental attribute for the safe operation of modern technological networks [34], and generally speaking, it represents the ability of a network to achieve its intended function within a given environment. For a multistate network, the reliability index is defined as the probability that at least units of flow can be sent from the source node to the sink node or the probability that the network capacity level is greater than or equal to a predetermined demand level d, that is . Evidently, to calculate , one direct way is to enumerate every state vector of by checking the condition M(c) ≥ d, and then sum up their probabilities. And yet, it is extremely time-consuming due to the huge number of state vectors in for a multistate network. Instead, most of studies have focused on the d-minimal path method. A d-minimal path is the minimal state vector satisfying M(c) = d. In other words, a state vector is a d-minimal path if and only if M(c) = d, and M(c – 0()) < d for each > 0, where 0() = (0, …, 0, 1, 0, …, 0), i.e., the state of is 1 and the states of other arcs are 0. Given that all d-minimal paths of a network are found, can be calculated by the well-known Inclusion-Exclusion method or Sum of Disjoint Product method. Therefore, finding d-minimal paths is the key for computing . In the following discussions, we focus on how to efficiently search for d-minimal paths in a multistate network.

3. Development of -Minimal Path Algorithm

3.1. Determination of Search Space of -Minimal Paths

Theoretically, the universal space is the whole search space of d-minimal paths, where is the set of states of . It is a cumbersome task to search for d-minimal paths in due to the large number of state vectors. The definition of d-minimal path indicates that if a state vector is a d-minimal path, two conditions must be met: M(c) = d, and M(c – 0()) < d for each > 0. As the source node and the sink node are two special nodes with the unique feature that only has outgoing arcs and only has incoming arcs, when is a d-minimal path, the following conditions must be satisfied:

Now we define a new concept called qualified state vector to determine a smaller search space of d-minimal paths compared to .

Definition 1. For a given demand level d, a state vector is called a qualified state vector if satisfies conditions (1) and (2).

The following theorem points out one basic property of qualified state vectors.

Theorem 2. If is a qualified state vector, then M(c) ≤ d.

Proof. Note that is a minimal cut of the network. Because , M(c) ≤ d follows from the max-flow min-cut theorem of network flows [35, 36].
Obviously, a d-minimal path is a qualified state vector, but a qualified state vector is not necessarily a d-minimal path. In conditions (1) and (2), suppose and . Let δ^j = (, ,…, , , ,…, ) (1 ≤ j ≤ H) represent a combination satisfying conditions (1) and (2), and for (1 ≤ j ≤ H) is a set of state vectors corresponding to δ^j, then every state vector in C^j is a qualified state vector. Thus, composes a new search space that is a subspace of the universal space and includes all qualified state vectors. Because a d-minimal path is a qualified state vector, the search space of d-minimal paths can be limited to instead of .
For illustration, we consider to find 3-minimal paths in Figure 1. Note that , and , then all possible combinations satisfying conditions (1) and (2) are δ¹ = (2, 1, 1, 2), δ² = (2, 1, 2, 1), δ³ = (3, 0, 1, 2), and δ⁴ = (3, 0, 2, 1). Thus, , , , and . That is, all 3-minimal paths are included in search space .

3.2. The Condition for Qualified State Vector to Be -Minimal Path

When the amount of flow sent from to is d, a d-flow is a vector (, ,…, ) consisting of flow through each arc (1 ≤ i ≤ n) [31]. Thus, according to the theory of network flows [35, 36], a state vector is a d-flow if and only if M(c) = d and satisfies the flow conservation law. The following theorem gives the condition for a qualified state vector to be a d-flow.

Theorem 3. A qualified state vector is a d-flow if and only if satisfies the flow conservation law.

Proof. If a qualified state vector satisfies the flow conservation law, then is a feasible flow; meanwhile, and , which means M(c) = d. Hence, c is a d-flow.
A d-flow is not necessarily a d-minimal path, and their relationship is determined by Lemma 4 proposed by Xu and Niu [31].

Lemma 4. A d-flow is a d-minimal path if and only if there is no directed cycle in it.

Thus, we can obtain the following conclusion.

Theorem 5. A qualified state vector is a d-minimal path if and only if satisfies the flow conservation law and there is no directed cycle in it.

Proof. Directly from Theorem 3 and Lemma 4.
Theorem 5 explicitly presents the sufficient and necessary condition for a qualified state vector to be d-minimal path. In the subsequent discussions, Theorem 5 will be utilized to search for d-minimal paths in set C^j (1 ≤ j ≤ H).

3.3. Searching for d-Minimal Paths in Set (1 ≤ j ≤ H)

Recall that every state vector in C^j (1 ≤ j ≤ H) is a qualified state vector, so the enumeration method, by Theorem 5, can be used to search for d-minimal paths in set C^j. However, it is still not practical to enumerate every state vector of C^j (1 ≤ j ≤ H), and there is room for improvement. Next, we will detail how to integrate the max-flow algorithm with the enumeration method to search for d-minimal paths in set C^j (1 ≤ j ≤ H).

For every set C^j (1 ≤ j ≤ H), we consider its largest state vector . If M () < d, then M (c) ≤ M () < d for any state vector , which indicates there is no d-minimal path in C^j by definition, and thus C^j is discarded (i.e., there is no need to enumerate C^j). Consequently, only the case M () = d needs to be discussed. The following theorem reveals that one d-flow can be derived from set C^j when M () = d.

Theorem 6. For set C^j (1 ≤ j ≤ H), if M() = d, then one d-flow (,,…, ) can be derived from C^j.

Proof. Firstly, note that for , and = 0 for . Since M () = d, the maximum amount of flow that can be sent from to is d, and moreover the max-flow of the network under is just a d-flow () according to the theory of network flows [35, 36]. As for , for , then we have for .
In Theorem 6, the max-flow algorithm can be employed to compute M (), and simultaneously determine a d-flow (,,…,) when M () = d. By Lemma 4, if (,,…,) contains no directed cycle, it is a d-minimal path. After finding a d-flow (,,…,), let A^d(C^j) = . Note that for , and for . If A^d(C^j) = , then for . As a result, we have for . That is, when A^d(C^j) = . In such a case, as M() = d and is the smallest state vector in set C^j, it is impossible for any other state vector of C^j to be d-minimal path, and thus C^j is discarded.
If A^d(C^j) ≠ , postulate that A^d(C^j) = , then we have for , and for . Based on the obtained d-flow () and A^d(C^j) = , set C^j can be divided into disjoint subsets. The technique for dividing C^j is originally reported by Jane and Laih [23], and is adapted for use in this paper. Pivoting on arcs one by one, C^j is divided into , ,…, and as follows:
(1) Pivot on arc :(2) Pivot on arc : …
Pivot on arc :(q+1)

According to Jane and Laih [23], subsets , , ,…, and satisfy two properties: (1) C^j = ; (2) , , ,, and are disjoint. Meanwhile, it is obvious that all of the state vectors in (0 ≤ k ≤ q) are still qualified state vectors.

The remaining work is to search for d-minimal paths in these subsets. First, we consider , and note that the d-flow (,,…,) derived from C^j is the smallest state vector in , then all of the other state vectors in are no d-minimal paths by definition. In such a case, there is no need to enumerate the state vectors in , i.e., is discarded.

Now we consider for 2 ≤ k ≤ q. If M () < d, then there exists no d-minimal path in , and is discarded either. But, if M() = d, the existence of d-minimal paths in is uncertain. In this case, because all of the state vectors in (0 ≤ k ≤ q) are still qualified state vectors, by Theorem 5, the enumeration method can be used to search for d-minimal paths in .

Finally, we consider . If M () < d, then there exists no d-minimal path in , and is discarded. Now we discuss the case M () = d. Observe that = which means , then can be coped with in the same way as C^j. That is, a d-flow (,,…,) can be obtained from when M() = d by Theorem 6, and then is similarly decomposed into disjoint subsets that are discussed in the same way as the subsets derived from C^j. This decomposition process will stop when there exists no subset meeting and M() = d. At this moment, all of the d-minimal paths in set C^j have been found.

3.4. The Developed Algorithm

According to the above discussions, we now present an algorithm for finding d-minimal paths in a multistate network . To make the proposed algorithm more understandable, we use two parts, Algorithm A and Algorithm B, to describe it hereunder.

Algorithm A. Finding d-minimal paths in a multistate network .
Input. A multistate network with demand level d.
Output. All d-minimal paths.
A. Step 1. Use the enumeration method to solve all of the combinations satisfying conditions (1) and (2), and then determine the search space .
A. Step 2. For each C^j(1 ≤ j ≤ H), let C = Cⁱ and call Algorithm B to search for d-minimal paths in C.

Algorithm B. Finding d-minimal paths in set C.
B. Step 1. Compute M(u^C). If M (u^C) < d, there is no d-minimal path in C, and stop.
B. Step 2. Find a d-flow (,,…,) in C, if it contains no directed cycle, it is a d-minimal path.
B. Step 3. Compute A^d(C) = . If A^d(C) = Φ, then stop; otherwise, suppose A^d(C) = .
B. Step 4. Divide into q+1 nonempty disjoint subsets , , ,…, .
B. Step 5. If q >1, compute M () (2 ≤ k ≤ q). If M () < d, there is no d-minimal path in ; otherwise, for each (2 ≤ k ≤ q), if satisfies the flow conservation law and there is no directed cycle in it, it is a d-minimal path.
B. Step 6. Let C = , and go to B. Step 1.

To help follow the procedure of the proposed algorithm, we briefly explain the role of every step. In Algorithm A, A. Sep 1 is a proprocessing step for determining a smaller search space of d-minimal paths, which includes sets, and A. Step 2 is to search for d-minimal paths of each set C^j by calling Algorithm B. Note that Algorithm B is a recursive algorithm for solving d-minimal paths. B. Step 1 checks whether there may exist d-minimal paths in by computing the maximum flow under the largest state vector. B. Step 2 is to find a d-flow and to check whether it is a d-minimal path. Once a d-flow is obtained, B. Step 3 and B. Step 4 are to divide set into q+1 disjoint subsets. B. Step 5 searches for d-minimal paths in subsets ,…, . Subset is regarded as a new set to be solved from the beginning step, i.e., B. Step 1 (a new iteration), and thus B. Step 6 is to put subset into the next iteration.

It is noteworthy that the major characteristics of the proposed algorithm are twofold. First, a smaller search space of d-minimal paths is determined in A. Step 1. Second, the search space is recursively divided into subspaces such that the traditional max-flow algorithm (find a d-flow in B. Step 2) and the enumeration method (check whether satisfies the flow conservation law in B. Step 5) are integrated to search for d-minimal paths in subsets. Notably, there is no need to enumerate subset , and when M () < d (2 ≤ k ≤ q) holds in B. Step 5, there is also no need to enumerate subset (no d-minimal paths exist in either), which no doubt contributes to the efficiency improvement in searching for d-minimal paths.

Now, we discuss the time complexity of the proposed algorithm. Complexity analysis of Algorithm A: it takes time to solve all of the combinations satisfying conditions (1) and (2). After obtaining the combinations, it then needs O(Hn) time to determine the search space. As a result, A. Step 1 totally requires time. The time complexity of A. Step 2 is obviously determined by the time complexity of Algorithm B. Complexity analysis of Algorithm B: B. Step 1 takes O(mn²) time to compute M(u^C) [37]. Once M(u^C) = d is determined, a d-flow (,,…,) can be obtained in O(n) time, and checking a directed cycle needs O(m) time. Thus, B. Step 2 requires O(m+n) time to check whether d-flow (,,…,) is a d-minimal path or not. B. Step 3 requires O(n) time to compute A^d(C). B. Step 4 takes O(n) time to divide set . B. Step 6 takes O(n) time to implement assignment operation. Therefore, the algorithm totally takes O(mn²) time without consideration of the time complexity of B. Step 5. Since solving all d-minimal paths is an NP-hard problem, the time complexity of the proposed algorithm mainly depends on B. Step 5 that employs the enumeration method to search for d-minimal paths. The number of enumerated state vectors in B. Step 5 is upper bounded by , i.e., , and it takes O(m) time to check whether a qualified state vector satisfies the flow conservation law, and O(m) time to check a directed cycle. Thus, B. Step 5 takes time in the worst case. Thus, the time complexity of Algorithm B is in the worst case. As a result, the time complexity of the proposed algorithm is .

We compare the proposed algorithm with the existing algorithms theoretically in terms of time complexity. Note that the algorithms in [13, 27, 29] are all based on the model proposed by Lin et al. [25], so they have the same time complexity as the algorithm by Lin et al. [25]. The time complexity of the algorithm by Lin et al. [25] is where is the number of minimal paths and O(p) = O(2^m), and the time complexity of both the algorithm by Yeh [26] and the modified algorithm by Xu and Niu [31] is . As is the total number of possible combinations satisfying conditions (1) and (2), it is trivial to have , and thus ≤ . Furthermore, notice that [26]. Therefore, we can obtain ≤ . That is, the proposed algorithm compares favorably with the algorithms of Lin et al. [25], Yeh [26], and Xu and Niu [31] in terms of time complexity.

4. Computational Examples

4.1. Illustrative Example

The multistate network in Figure 1 is adopted to elucidate the proposed algorithm. Given the demand level d = 3, the following steps illustrate how to search for all of 3-minimal paths using the proposed algorithm (For simplicity, a set of state vectors is denoted by its smallest state vector l^C and largest state vector u^C as [l^C, u^C] ).

Algorithm A.
A. Step 1. Use the enumeration method to solve all of the combinations satisfying conditions (1) and (2) as follows: + =3, where 0 ≤ ≤ 3 and 0 ≤ ≤ 1 + =3, where 0 ≤ ≤ 2 and 0 ≤ ≤ 2 And the obtained combinations are (2, 1, 1, 2), (2, 1, 2, 1), (3, 0, 1, 2), and (3, 0, 2, 1), Thus, the search spaces are = [(2, 1, 0, 0, 1, 2), (2, 1, 1, 1, 1, 2)], = [(2, 2, 0, 0, 1, 1), (2, 2, 1, 1, 1, 1)], = [(3, 1, 0, 0, 0, 2), (3, 1, 1, 1, 0, 2)] and = [(3, 2, 0, 0, 0, 1), (3, 2, 1, 1, 0, 1)].
A. Step 2. Let C = = [(2, 1, 0, 0, 1, 2), (2, 1, 1, 1, 1, 2)] and search for 3-minimal paths by calling Algorithm B as follows:
B. Step 1. M (u^C) = 3.
B. Step 2. (2, 1, 1, 0, 1, 2) is a 3-flow in C, and it contains no directed cycle, then (2, 1, 1, 0, 1, 2) is a 3-minimal path.
B. Step 3. (C) = ≠ Φ, q = 1.
B. Step 4. C is divided into two subsets = [(2, 1, 1, 0, 1, 2), (2, 1, 1, 1, 1, 2)] and = [(2, 1, 0, 0, 1, 2), (2, 1, 0, 1, 1, 2)].
B. Step 5. q = 1.
B. Step 6. Let C = =[(2, 1, 0, 0, 1, 2), (2, 1, 0, 1, 1, 2)], and go to B. Step 1. next iteration
B. Step 1. M (u^C) = 2 < 3, then there is no 3-minimal path in C, and stop.
A. Step 2. Let C = = [(2, 2, 0, 0, 1, 1), (2, 2, 1, 1, 1, 1)] and search for 3-minimal paths by calling Algorithm B as follows:
B. Step 1. M (u^C) = 3.
…

Finally, there are totally obtained three 3-minimal paths, as shown in Table 2. Let , , , then can be evaluated via the Inclusion-Exclusion method as follows: = = = + + – – – + = + + – – – + = 0.685104.

4.2. Numerical Example for Efficiency Investigation

In this section, we focus on the computational efficiency of the proposed algorithm in finding all d-minimal paths. As mentioned before, the method in [31] has been proved to be more efficient than the algorithms of Lin et al. [25], and Yeh [26] in terms of finding d-minimal paths, so we compare the proposed algorithm with it through numerical computations. Both algorithms are coded in a MATLAB program, and run on the same personal computer with Intel(R) Core (TM) i5-3210M 2.50 GHz CPU.

A medium-sized network shown in Figure 2 is used as benchmark network to perform numerical experiments. For comparison, we consider two scenarios. Scenario 1: the largest states of all arcs are equal to 4, i.e., = 4 (1 ≤ i ≤ 13), and Scenario 2: the largest states of all arcs are equal to 5, i.e., = 5 (1 ≤ i ≤ 13). For each scenario, the demand level ranges from 1 to the largest value M(u), i.e., all of the d-minimal paths corresponding to multiple demand levels are solved. We focus on the computational time required to search for all d-minimal paths of each demand level. Meanwhile, to make intuitive comparisons, we define ratio λ as the computational time of the method in [31] divided by the computational time of the proposed algorithm. Then, the ratio λ represents the relative efficiency of two algorithms.

From both Figures 3 and 4, it can be seen that the ratio λ is always above one for every demand level under both scenarios, implying the proposed algorithm is more efficient than the method in [31]. Meanwhile, note that as the demand level increases, the ratio λ increases, indicating the efficiency advantage of the proposed algorithms becomes much stronger. For example, the proposed algorithm is more than 10 times faster than the algorithm in [31] when solving d-minimal paths with demand level above 5 under both Scenario 1 and Scenario 2, and the proposed algorithm is more than 20 times faster than the algorithm in [31] when solving d-minimal paths with demand level above 6 under Scenario 1 and with demand level above 7 under Scenario 2, respectively.

5. Case Study of Power Transmission Network

In modern society, electric power is the necessity to our daily life. Thus, a reliable power transmission network is critical to the stable operation of the whole society. As exemplified in Section 1, power transmission networks can be regarded as typical multistate networks. Therefore, we take a power transmission network as case study to demonstrate the utility of the proposed algorithm and the related implications of network reliability.

A power transmission network with sixteen transmission lines is shown in Figure 5. The capacity and capacity distribution of each transmission line of the power transmission network are presented in Table 3. Suppose that the supervisor would like to assess the capability of the power transmission network to successfully transmit 6 units (MW) of electric power from the power plant (node s) to the targeted high voltage substation (node t). Then, the network reliability can be computed in terms of 6-minimal paths. Using the proposed algorithm, a total of 143 6-minimal paths are obtained. As a result, it can be calculated that = 0.928546 by the Inclusion-Exclusion method. That is, the probability of the power transmission network being able to successfully transmit 6 units (MW) of electric power from the power plant to the targeted high voltage substation is 0.928546. This value 0.928546 provides the supervisor with intuitive information regarding the capability of the network to ensure the transmission of 6 units (MW) of electric power from the power plant to the targeted high voltage substation. If 0.928546 is above the supervisor's expectation, it means the operational state of the power transmission network is desirable. However, if 0.928546 is below the supervisor's expectation, it means the operational state of the power transmission network is undesirable, and then some improvement measures are needed to enhance the power transmission network.

6. Conclusion

In this paper, an efficient algorithm is developed to search for d-minimal paths in reliability evaluation of multistate networks. Unlike the existing algorithms that enumerate d-minimal paths in a larger search space, the proposed algorithm first determines a relatively smaller search space of d-minimal paths by employing a concept of qualified state vector. A sufficient and necessary condition for a qualified state vector to be d-minimal path is derived. And, the search space is recursively divided into subspaces, so that the max-flow algorithm and the enumeration algorithm are integrated to search for d-minimal paths with the purpose of increasing the efficiency of enumeration. Furthermore, it is theoretically proven that the proposed algorithm compares favorably with the existing algorithms in terms of time complexity. Finally, numerical examples have shown the effectiveness and efficiency of the proposed algorithm.

Notations

:	A multistate network with a set of nodes , a set of arcs , and the universal space of state vectors
	The source node/the sink node
	The number of nodes except and /the number of arcs
:	th arc in
:	A state of arc
:	A state vector
:	A set of state vector
:	The set of states of arc , i.e.,
/:	The smallest/largest state of arc
:	The smallest/largest state vector
:	The smallest/largest state vector with respect to set
/:	The smallest/largest state of arc with respect to set
:	The set of states of arc with respect to
:	Maximum amount of flow that can be sent from to when the network is under state vector
:	Demand level
:	Network reliability at demand level
:	A special state vector with the state of being 1 and the states of other arcs being 0, i.e., 0() = (0, …, 0, 1, 0, …, 0)
:	The set of outgoing arcs of
:	The set of incoming arcs of
:	th combination satisfying conditions (1) and (2)
:	th set of state vectors corresponding to
:	th subset derived from the decomposition of set
:	The total number of combination satisfying conditions (1) and (2)
:	The number of minimal paths
:	The capacity of th minimal path
:	The flow through arc when unit of flow is sent from to
:	The number of elements in .

Data Availability

All data are provided in full in the results section of this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is mainly supported by the National Natural Science Foundation of China (Projects nos. 71601072 and 61872126), the Research Project from the Science and Technology Department of Henan Province (Project no. 172102310677), and the Fundamental Research Funds for the Universities of Henan Province (Project no. NSFRF170914).

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Copyright © 2019 Xiu-Zhen Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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