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

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 d. 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, themax-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.


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][6][7][8][9][10][11][12][13], and gene regulatory networks [14][15][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][18][19][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][23][24] and indirect algorithm [25][26][27][28][29][30][31][32][33].Indirect algorithm is often called 2 Complexity 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][26][27][28][29][30][31][32][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 dminimal 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][28][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 dminimal 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 dminimal 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.For example, consider a multistate network (, , Ω) shown in Figure 1 with  = {, 1, 2, },  = { 1 ,  2 ,  3 ,  4 ,  5 ,  6 }, and Ω = [(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
Following the literature in [25,26,[28][29][30][31][32][33], this paper considers four assumptions: (1) the state/capacity of each arc is a nonnegative integer random variable following a )) < 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.

Development of 𝑑-Minimal Path Algorithm
Now we define a new concept called qualified state vector to determine a smaller search space of d-minimal paths compared to Ω. 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 𝑐 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].

The Condition for Qualified State Vector to Be 𝑑-Minimal
Path.When the amount of flow sent from  to  is d, a d-flow is a vector (  1 ,   2 ,. ..,    ) consisting of flow    through each arc   (1 ≤ i ≤ n) [31].Thus, according to the theory of network flows [35,36] 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 𝑐 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 𝑐 is a d-minimal path if and only if 𝑐 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).

Searching for d-Minimal Paths in Set 𝐶
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 For every set C j (1 ≤ j ≤ H), we consider its largest state vector 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 (2) Pivot on arc   2 : 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.

The Developed Algorithm.
According to the above discussions, we now present an algorithm for finding dminimal paths in a multistate network (, , Ω).To make the proposed algorithm more understandable, we use two parts, Algorithm A and Algorithm B, to describe it hereunder.

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  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.

B. Step 3. Compute
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 dminimal paths.Now, we discuss the time complexity of the proposed algorithm.Complexity analysis of Algorithm A: it takes ((∏   ∈(,•) (1 + min{  , }) + ∏   ∈(•,) (1 + min{  , }))) 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   Step 5 takes (∏   ∈(\((,•)∪(•,))) (1 + min{  , })) time in the worst case.Thus, the time complexity of Algorithm B is (∏   ∈(\((,•)∪(•,))) (1 + min{  , })) 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 (( + )∏   ∈ (1 + min{  , })).As  is the total number of possible combinations satisfying conditions (1) and (2), it is trivial to have  < ∏   ∈(,•)∪(•,) (1 + min{  , }), and thus (∏ 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.

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  as the computational time of the method in [31] divided by the computational time of the proposed algorithm.Then, the ratio  represents the relative efficiency of two algorithms.
From both Figures 3 and 4, it can be seen that the ratio  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  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.

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  6 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  6 = 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.

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 maxflow 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.The set of states of arc   with respect to  ():

Notations
Maximum amount of flow that can be sent from  to  when the network is under state vector  : Demand level   : Network reliability at demand level  0(  ): 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   : Th ec a p a c i t yo fth minimal path    : The flow through arc   when  unit of flow is sent from  to  | • |: Th en u m b e ro fe l e m e n t si n•.

3. 1 .
Determination of Search Space of -Minimal Paths.Theoretically, the universal space Ω = { |  = ( 1 ,  2 , . . .,   ),   ∈   for 1 ≤  ≤ } is the whole search space of d-minimal paths, where   = {0, 1, . . .,   } is the set of states of   .It is a cumbersome task to search for d-minimal paths in Ω 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: 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  2 ,. ..,   .Subset  1 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  1 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 Complexity for d-minimal paths in subsets.Notably, there is no need to enumerate subset  0 , and when M ) 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 | 1 ∪  2 ∪ . . .∪   |, i.e., ∏   ∈(\((,•)∪(•,))) (1 + min{  , }), 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.

Figure 2 :
Figure 2: A benchmark network for numerical experiments.

Figure 5 :
Figure 5: A power transmission network.

Table 1 :
The states of arcs in Figure1.
M(c) = d, and M(c -0( , 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, ∑   ∈(,•) 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 (   For set C j (1 ≤ j ≤ H), if M(   ) = d, then one d-flow (  1 ,  2 ,. ..,    ) 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 (  1 ,   2 , . . .,    ) according to the theory of network flows [35, 36].As    =     =     for   ∈ ((,•) ∪ (•, )),     = 0 ≤    ≤     for   ∈  \ ((,•) ∪ (•, )), then we have     ≤    ≤     for   ∈ .In Theorem 6, the max-flow algorithm can be employed to compute M (   ), and simultaneously determine a dflow (  1 ,  2 ,. ..,   ) when M (   ) = d.By Lemma 4, if (  1 ,  2 ,. ..,   ) contains no directed cycle, it is a d-minimal path.After finding a d-flow (  1 ,  2 ,. .., ) = 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.

Table 2 :
The obtained results using the proposed algorithm.

Table 3 :
Data of each arc (transmission line) of the power transmission network.
(, , Ω): A multistate network with a set of nodes  = {, 1, 2, . . ., , }, a set of arcs  = {  | 1 ≤  ≤ }, and the universal space of state vectors Ω = { |  = ( 1 ,  2 , . . .,   ),   ∈   for 1 ≤  ≤ } / : The source node/the sink node / : The number of nodes except  and /the number of arcs   : th arc in    : As t a t eo fa r c  : A state vector : A set of state vector   : The set of states of arc   , i.e.,   = {0, 1, . . .,   }   /  : 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     :