Complexity

Volume 2019, Article ID 4561845, 10 pages

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

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

^{1}School of Economics and Management, Chongqing University of Posts and Telecommunications, Chongqing 400065, China^{2}School of Computer Science and Technology, Henan Polytechnic University, Jiaozuo 454000, China^{3}College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Yi-Feng Niu; moc.621@120gnefi

Received 15 January 2019; Revised 21 February 2019; Accepted 26 February 2019; Published 21 March 2019

Academic Editor: Dimitri Volchenkov

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.

#### 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.