Multi-state flow network reliability evaluation method considering network configuration adjustment

Abstract

The invention provides a multi-state flow network reliability evaluation method considering network configuration adjustment, and belongs to the technical field of computer science and system engineering. The key point of the invention is to improve the efficiency and the accuracy of the MFN reliability evaluation method when network configuration parameters (a topological structure and component state probability distribution) are adjusted, an MFN reliability evaluation framework is provided, and the following methods are provided: an improved state space decomposition (SSD) method, a deep-level state space decomposition (DSSD) method and a state space reconstruction (SSR) method. According to the method, the reliability evaluation method of the multi-state flow network under the condition that the network evaluation configuration changes is modified, so that the reliability evaluation of the multi-state flow network under the condition that the network evaluation configuration changes has better accuracy and higher efficiency; therefore, the requirements of reducing repeated calculation are generated, the calculation cost is reduced, and the evaluation efficiency is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of computer science and systems engineering, and particularly relates to a method for reliability assessment of multi-state flow networks (MFNs). Background Art

[0002] Multi-state flow networks (MFNs) are complex systems that can effectively represent a wide range of real-world scenarios. In these networks, components can operate in various independent, discrete, and finite states, rather than being limited to the binary states of either perfect operation or complete failure.

[0003] The reliability of such a network is defined as the probability that the flow from the source node to the sink node is greater than the demand d. Theoretically, calculating this metric is an NP-hard problem. Therefore, numerous scholars have dedicated themselves to improving the efficiency of MFN reliability assessment. In the known reported literature, there are mainly two direct methods and an indirect method for network reliability assessment.

[0004] Some scholars use the maximum flow algorithm and the state space decomposition (SSD) method to recursively decompose the state space. The reliability is the sum of the probabilities of all disjoint acceptable state sets obtained in each iteration of SSD. Since this method directly takes the network configuration as input, it is called a direct method.

[0005] The indirect method evaluates the reliability through minimal path / cut vectors (i.e., d-MP and d-MC), which are the lower (upper) boundary component state vectors that satisfy the system demand. The first step of this method is to obtain all d-MPs or d-MCs, and the second step is to calculate the probability of the union of these vectors to obtain the reliability of the MAN.

[0006] This study focuses on the reliability research when the network configuration changes. Once the network configuration changes, the reliability assessment method needs to be repeated. For example, reliability assessment plays a crucial role in the design and optimization of systems. Engineers or designers may strive to adjust the network configuration to pursue the best results. Therefore, the design and optimization of MFNs face the challenge of how to efficiently and accurately obtain the network reliability when the network configuration is adjusted.

[0007] Another typical example of network configuration adjustment is the resilience study. The expected performance of the MFN is defined as the expected traffic from the source node to the sink node, which is based on the reliability of all d demands. During the recovery process, the fixed state probability distribution (SPD) of components may shift, and the capacity of components may be lower than the maximum state. Even the topology of the MFN may change. Such changes in network configuration lead to repeated calculations of MFN performance. However, multiple changes in network configuration parameters are involved in both network optimization design and network resilience assessment, which means that multiple repeated calculations of multi-state network reliability are required, becoming the computational bottleneck in the above research.

[0008] The accuracy and effectiveness of existing methods for repeated evaluation of MFN reliability can be further improved. Summary of the Invention

[0009] The focus of the present invention is to improve the efficiency and accuracy of the MFN reliability assessment method when network configuration parameters (topology, component state probability distribution) are adjusted. An MFN reliability assessment framework is proposed, and the following methods are proposed: an improved state space decomposition (SSD) method, a deep state space decomposition (DSSD) method, and a state space reconstruction (SSR) method.

[0010] First, an MFN reliability assessment framework is proposed for different situations of network configuration adjustment. Second, an improved state space decomposition (SSD) method and a deeper SSD method (DSSD) are proposed to obtain an MFN state space set that can guarantee the preset accuracy requirements. Third, a state space reconstruction (SSR) method is developed to update the state space set when the network configuration changes.

[0011] To solve the above technical problems, the specific technical solution of the multi-state flow network reliability assessment method considering network configuration adjustment of the present invention is as follows:

[0012] Step 1: Initialize parameters; take all minimal path sets d-MP in the existing network configuration and the initial probability distribution matrix P as inputs;

[0013] Step 2: If there is an available state space matrix, go to Step 4; otherwise, go to Step 3;

[0014] Step 3: Based on all d-MP in the current network configuration, use the improved state space decomposition method to obtain the initial state space set and obtain the lower reliability boundary LRB with the preset accuracy requirement γ d ; The initial state space set specifically includes a state space matrix for storing key vectors and limit points for each group of acceptable and unacceptable states and a cell for storing information about the remaining set of unspecified states; finally, go to Step 7;

[0015] Step 4: Network Configuration Adjustment If the network topology is changed by introducing new components or the maximum state of a component is adjusted to be greater than its original value, search for the d-MP of all new network configurations, then go to Step 3 to update the output LRB d , the state space matrix, and the current set of cells with unspecified states; otherwise, go to Step 5;

[0016] Step 5: Calculate the LRB using the reliability assessment method based on state space reconstruction based on the current state space matrix d , if the LRB d meets the accuracy requirement, go to Step 7; otherwise, go to Step 6;

[0017] Step 6: If the obtained reliability accuracy is greater than the accuracy requirement, combine the output of the SSR method and cell U and use the proposed deep SSD method to obtain an LRB that meets the accuracy requirement d ; at the same time, update the state space matrix and cell U, and finally go to Step 7;

[0018] Step 7: If the network needs to adjust the configuration, go to Step 2; otherwise, terminate the evaluation.

[0019] The specific steps of the improved state space decomposition method are as follows:

[0020] Step 3.1: Set the initialization parameters, the lower reliability boundary LRB d = 0, the upper reliability boundary URB d = 1; at this time, the loop count k = 1, and the stop flag st = 0; create an initialization state space set to store the initial temporary parameters;

[0021] Step 3.2: Set the number of the set of unspecified states in the current generation to uns, uns = k, and set the initial index of the set of unspecified states to in = 1; set the index τ for storing the state space to τ = 1;

[0022] Step 3.3: Start looping k times, for nu = 1:uns, for is the loop instruction, and nu is the loop variable whose value loops from 1 to uns times; perform the following operations:

[0023] Step 3.3.1: Update the limit vector, b 0 = B 0 (nu), b = B(nu), w = 0, and w stores the number of the current set of qualified d-MPs;

[0024] Step 3.3.2: Based on the upper limit vector b, the d-MP 0Decompose into qualified d-MPs and unqualified d-MPs, save the index e of the current set of qualified d-MPs, save the number of the current set of qualified d-MPs, and update w;

[0025] Step 3.3.3: Obtain all qualified d-MPs in the current set, and denote the l-th d-MP in the current multi-state network as z l , compare the lower limit vector with z l , and obtain the critical vector v of the unacceptable state set;

[0026] Select z l such that z l minimizes the following formula:

[0027]

[0028] b i is the i-th element of the lower limit vector. If there are multiple z l that simultaneously minimize H2(z l ), then select the one that maximizes the following formula from them:

[0029]

[0030] where, x i is the i-th element of the state vector, and b 0 i is the i-th element of the upper limit vector. If there are still multiple options, then arbitrarily select one from them;

[0031] Compare the selected z l with the lower limit vector to obtain the critical vector v of the acceptable state set 0 ;

[0032] Update the state space matrix, let V(τ,:) = v, V0(τ,:) = v 0 , Z(τ,:) = b and Z 0 (τ,:) = b 0 ;

[0033] Step 3.3.4: If st = 0, execute the following steps, otherwise jump to Step 3.3.5:

[0034] Step 3.3.4.a: Set the partial bound LT = 0; Calculate the probability from to

[0035]

[0036] LRB d = LRB​d +LT

[0037] p i is the probability that the i-th side meets the reliability requirement;

[0038] Step 3.3.4.b: Set part of the boundary UT = 0, and update UT and URB d :

[0039]

[0040] URB d = URB d -UT (13)

[0041] Step 3.3.4.c: Set the accuracy of reliability DR = URB d -LRB d ; if DR < γ, then st = 1;

[0042] Step 3.3.5: Let the set of i that satisfies be a q , q = 1, 2,..., s, i ∈ {1, 2,..., n};

[0043] If there is no such i, then set s = 0; if for q = 1, 2,..., s, i = 1, 2,..., n, there exists s ≥ 1, then update the parameters of the initial state space set, update the setting k ← k - 1 + s, in ← in + s, τ ← τ + 1; if nu = nus, end the for loop;

[0044] Step 3.4: If k = 0, stop and output LRB d and URB d ; otherwise:

[0045] If st = 1, output LRB d and URB d , then store the cell U{B 0 (1:k) = t_B 0 (1:k), B(1:k) = t_B(1:k), N(1:k) = t_N(1:k), D(1:k) = t_D(1:k)}; otherwise:

[0046] Set B 0 (1:k) = t_B 0 (1:k), B(1:k) = t_B(1:k), N(1:k) = t_N(1:k), D(1:k) = t_D(1:k), output LRB d , the state space matrices V, V0, Z and Z 0 , the current set of cells of unspecified states U{B0 , B, N, D}; Go to step 2.

[0047] The initial state space set includes:

[0048] The first column of matrix N is used to store the number of qualified d-MPs. At the initial moment, set N(1,1) = L, where L is the total number of d-MPs; the second column of matrix N is used to store the key positions for judging whether the d-MPs are qualified. At the initial moment, set N(1,2) = 1;

[0049] Matrix t_N temporarily stores the number and key positions of the currently qualified d-MPs;

[0050] Matrices D and t_D respectively save the indices of the currently qualified d-MP sets. Set D(1) = (1, 2, …, L);

[0051] Matrix B 0 stores the upper limit vectors of each group of unspecified states. B 0 (1) = M, where M is the maximum state vector;

[0052] Matrix B stores the lower limit vectors of each group of unspecified states. B(1) = 0;

[0053] Matrix t_B 0 stores the temporary upper limit vectors of each group of unspecified states;

[0054] Matrix t_B stores the temporary lower limit vectors of each group of unspecified states; The initial limit vectors include the upper limit vector b 0 , b 0 = B 0 (1) = M and the lower limit vector b of the unspecified state set. b = B(1) = 0.

[0055] The reliability assessment method based on state space reconstruction is specifically implemented through the following steps:

[0056] Step 5.1: Initialization; Set st = 0; For i = 1:ε, where ε is the number of rows of the state space matrix, perform the following operations:

[0057] Step 5.2: If V0(i, :) ≤ m, update the partial bound LT using the following formula, and then update LRB d :

[0058]

[0059] LRB d = LRB d + LT

[0060] If V(i, :) > Z(i, :) && Z(i, :) ≤ m, update the partition boundary UT using the following formula, and then update URB d :

[0061]

[0062] URB d = URB d - UT

[0063] Step 5.3: If DR < γ, set st = 1 and return URB d and LRB d ; Otherwise, go back to Step 6.

[0064] The specific steps of the depth SSD method are as follows:

[0065] Step 6.1: Set the number of elements in the current set of unspecified states to the number of rows of B 0 , k = size(B 0 ); Stop flag st = 0; Create four empty matrices t_N, t_D, t_B 0 and t_B;

[0066] Step 6.2: Set the number of sets of current unspecified states uns = k, and set the initial state index set of unspecified states to in = 1;

[0067] Step 6.3: Repeat Step 3.3; Further update the LRB d and URB d obtained in Step 5 using Step 3.3;

[0068] Step 6.4: If k = 0 or st = 1, output LRB d and URB d , and store the unit U{B 0 (1:k) = t_B 0 (1:k), B(1:k) = t_B(1:k), N(1:k) = t_N(1:k), D(1:k) = t_D(1:k)}, Otherwise, let B 0 (1:k) = t_B 0 (1:k), B(1:k) = t_B(1:k), N(1:k) = t_N(1:k), D(1:k) = t_D(1:k), and go to Step 2.

[0069] In the technical solution of the present invention, the reliability evaluation method for a multi-state flow network under the change of network evaluation configuration is modified. Since the method proposed in this paper enables the reliability evaluation of a multi-state flow network configuration change to have better accuracy and higher efficiency, the need to reduce repeated calculations, reduce the calculation cost, and improve the evaluation efficiency is achieved. Description of the Drawings

[0070] Figure 1 : Shows a two-terminal bridge network;

[0071] Figure 2 : MFN reliability evaluation framework;

[0072] Figure 3 : Medium MFN;

[0073] Figure 4 : Comparison of medium MFN network reliability errors;

[0074] Figure 5 : Hebei Transportation MFN;

[0075] Figure 6 : Comparison of the reliability errors of the MFN network in Hebei Province;

[0076] Figure 7 : A large MFN;

[0077] Figure 8 : Adjusted large MFN;

[0078] Figure 9 : Comparison of network reliability errors in a large MFN; Detailed Description of the Invention

[0079] In order to better understand the purpose, structure and function of the present invention, the following further describes in detail a multi-state flow network reliability evaluation method considering network configuration adjustment of the present invention with reference to the accompanying drawings.

[0080] Example 1

[0081] The SSD principle is used for MFN reliability evaluation:

[0082] Consider a two-terminal MFN that satisfies the following assumptions described in:

[0083] 1) The graph is connected and has no self-loops;

[0084] 2) The capacity of each edge is a non-negative integer value following a given distribution;

[0085] 3) The capacities s of different edges are independent;

[0086] 4) All flows in the network obey the conservation law.

[0087] MFN is defined as a direct network G(V, E), where V is a finite set of nodes and E is a set of arcs (components). The numbers of nodes and arcs are m and n respectively. Denote an arc by i, then E = {1, 2, …, n}. For each arc i ∈ E, use c i ∈ {0, 1, …, m i} to represent the capacity of this component, where m i is the maximum capacity (maximum state) of arc i. m = (m1, m2, …, m n ) is the vector of component maximum states. Consider that the operating states of components follow a distribution where represents the probability that component i works in state c. P = (p 1 , p 2 , …, p n ) T is the probability distribution matrix of a network. x = (x1, x2, …, x n ) is the component state vector. M is the maximum flow from the source node to the sink node.

[0088] R d = Pr(φ(x) ≥ d) is the reliability of demand d, meaning the probability that the flow from the source node to the sink node is not less than d, and φ(·) is the system structure function.

[0089] If for any y < x, φ(x) ≥ d and φ(y) < d, where y < x means that for all i, y i < x i , then the component state vector x can be called the minimum path vector (d-MP) of system demand d. Then, the d-MP can be used to evaluate the reliability as follows.

[0090] R d = Pr(φ(x) ≥ d) = Pr({x ≥ z 1} ∪ {x ≥ z 2} ∪ … ∪ {x ≥ z L}) (1)

[0091] where z 1 , z 2 , …, z L are all d-MPs of the current multi-state network.

[0092] An effective method for calculating this joint probability is based on SSD.

[0093] The component state vector of the multi-state network is x = (x1, x2, …, x n ), where b 0 and b are the limiting vectors of the current state space. There exists a critical vector v = (v1, v2, …, v n ). If any b i ≤ x i < v i , then such a state vector x is an unacceptable state vector. There is another critical vector If the state vector x satisfies , then the state vector x is an acceptable state vector. When all the states in the state vector x satisfy and at least one x i satisfies , then the state space vector is called an unspecified state vector.

[0094] For any component x i in the multi-functional network (MFN), there are three cases: unacceptable x i ∈ V I , b i ≤ x i < v i , unspecified x i ∈ V C , acceptable x i ∈ V A , Then, there are three state spaces: 1) Unspecified state C: If at least one component state in the state vector x satisfies x i ∈ V C , and no component state belongs to V U , then x ∈ C; 2) Unacceptable state U: If at least one component state in the state vector x satisfies x i ∈ V U , then x ∈ U; 3) Acceptable state A: If all the component states in the state vector x satisfy x i ∈ V A , then x ∈ A. These three state spaces satisfy A ∪ U ∪ C = S, and where S is the entire state space.

[0095] Initially, there is only one set of unspecified states, where C = S. Then, the critical vector v is generated by heuristic rules 0Given \(d\) and \(v\), these rules take \(d - MP\) as input. Next, the state space is decomposed into a set of acceptable states, a set of unacceptable states, and a set of disjoint unspecified state sets. The disjoint unspecified state sets will be recursively decomposed until there are no unspecified state sets remaining or the accuracy requirement is met. When decomposing each unspecified state set, a set of acceptable states \(A\) is obtained, which means the part of the state space that satisfies the network traffic flow requirements within the boundary of the current unspecified state. Finally, \(R\) d is the sum of the probabilities of all disjoint acceptable state sets. Note that the SSD process can be terminated by setting the accuracy parameter \(\gamma\) instead of decomposing all unspecified states. Then, the reliability bound for demand \(d\) is denoted as \(LRB\) d .

[0096] The present invention considers two types of network configuration adjustments:

[0097] Case 1: The SPD (state probability distribution) of a component changes while the network topology remains unchanged. Removing a component from the initial network can be regarded as a special case of Case 1 adjustment, where the probability of such a component being in a failed state is 1, indicating its complete removal.

[0098] Case 2: The network topology is changed by introducing new components. If the maximum state of a component is adjusted to be greater than its original value, this case is classified as Case 2.

[0099] Based on the SSD method, a reliability evaluation framework for MFN configuration adjustment is proposed, as Figure 2 shown, which specifically includes the following steps:

[0100] Step 1: Initialize parameters; take all \(d - MP\) (minimal path sets) in the existing network configuration and the initial probability distribution matrix \(P(0)\) as input;

[0101] Step 2: If there is an available state space matrix (there is no available matrix during the first calculation or when the network topology changes), go to Step 4; otherwise, go to Step 3.

[0102] Step 3: Use the improved state space decomposition (SSD) method to obtain the initial state space set and get \(LRB\) with a preset accuracy requirement \(\gamma\) d . The initial state space set specifically includes a state space matrix for storing key vectors and limit points of each set of acceptable and unacceptable states and cells for storing information about the remaining set of unspecified states; finally, go to Step 7. The specific steps of the improved state space decomposition method are as follows:

[0103] Using the SSD method, when the accuracy requirement is met, there is a series of acceptable and unacceptable state sets. In fact, the sum of the probabilities of the acceptable state sets is the lower reliability bound \(LRB\) of the system demand \(d\)d , upper reliability boundary URB d is 1 minus the sum of the probabilities of the unacceptable state set, and d is the unit of traffic that needs to be transmitted from the source node to the sink node. The critical vector in the state space can provide valuable information for network reliability assessment. In addition, when the network configuration changes, the sets of acceptable and unacceptable states will also be updated.

[0104] Step 3.1: Set initialization parameters, LRB d =0, URB d =1; at this time, the number of cycles k = 1, and the stop flag st = 0;

[0105] Create the initial state space set as follows:

[0106] The first column of the matrix N is used to store the number of d-MPs that meet the conditions. At the initial moment, N(1,1)=L, where L is the total number of d-MPs. The second column of the matrix N is used to store the key position for determining whether the d-MP is qualified. At the initial moment, N(1,2)=1.

[0107] The matrix t_N temporarily stores the number and key positions of the currently qualified d-MPs;

[0108] The matrices D and t_D store the indices of the current qualified d-MP set respectively, and set D(1) = (1, 2, ..., L);

[0109] Matrix B 0 Stores the upper bound vector for each set of unspecified states, B 0 (1) = M, where M is the maximum state vector;

[0110] Matrix B stores the lower limit vector of each group of unspecified states, B(1)=0;

[0111] Matrix t_B 0 Store a temporary upper bound vector for each set of unspecified states;

[0112] The matrix t_B stores the temporary lower limit vector for each set of unspecified states; the initial limit vector includes the upper limit vector b for the set of unspecified states 0 , b 0 =B 0 (1) = M and the lower limit vector b of the unspecified state set, b = B(1) = 0.

[0113] Step 3.2: Set the number of unspecified state sets in the current generation to uns, uns = k, and the initial unspecified state set index to in = 1. Set the index τ of the storage state space to 1;

[0114] Step 3.3: Start looping k times. For nu = 1:uns, where "for" is the loop instruction and nu is the loop variable, with its value looping from 1 to uns times; perform the following operations:

[0115] Step 3.3.1: Update the limit vector, b 0 = B 0 (nu), b = B(nu), w = 0, w stores the number of current qualified d-MP sets;

[0116] Step 3.3.2: Decompose the d-MP with index D(nu) into qualified d-MPs and unqualified d-MPs. Let the l-th d-MP in the current multi-state network be denoted as z l , there is:

[0117] z l (N(nu,2)) ≤ b 0 (N(nu,2)), qualified;

[0118] z l (N(nu,2)) > b 0 (N(nu,2)), unqualified;

[0119] Save the index e of the current qualified d-MP set, save the number of the current qualified d-MP set, and update w;

[0120] Step 3.3.3: Obtain all the qualified d-MPs in the current set and calculate two vectors respectively using equations (3) and (4) and v i :

[0121]

[0122] where b i is the i-th element of the lower limit vector, and v i is the i-th element of the critical vector v of the unacceptable state set.

[0123] Step 3.3.3.a: Select z l , such that z l minimizes equation (5); during this process, if there are multiple z l that simultaneously minimize equation (5), then select the one that maximizes equation (6) among them. If there are still multiple options, choose any one from them:

[0124]

[0125] x i is the i-th element of the state vector, and b 0 i is the i-th element of the upper limit vector

[0126] Step 3.3.3.b: Obtain the qualified bound on the $i$-th edge from Equation (7).

[0127] is the critical vector $v$ of the acceptable state set 0 and the $i$-th element of;

[0128] Step 3.3.3.c: Update the state space matrix, let $V(\tau,:)=v$, $V_0(\tau,:)=v$ 0 , $Z(\tau,:)=b$ and $Z$ 0 $(\tau,:)=b$ 0 .

[0129] Step 3.3.4: If $st = 0$, perform the following steps, otherwise jump to Step 3.3.5:

[0130] Step 3.3.4.a: Set the partial bound $LT = 0$; Calculate the probability from to of

[0131]

[0132] LRB d $=LRB$ d $+LT (9)$

[0133] $p$ i is the probability that the $i$-th edge meets the reliability requirement, and its product is the probability $LT$ of the current qualified state space; $LT$ is a part of the reliability that the newly calculated reliability lower bound should increase;

[0134] Step 3.3.4.b: Set the partial boundary $UT = 0$, and update $UT$ and $URB$ d :

[0135]

[0136] $URB$ d $=URB$ d $-UT (13)$

[0137] Step 3.3.4.c: Set the precision $DR$ of the reliability as $DR = URB$ d $-LRB$ d ; If $DR \lt \gamma$, then $st = 1$.

[0138] Step 3.3.5: Let the set $i$ that satisfies be $a$ q, q = 1, 2, …, s, i ∈ {1, 2, …, n};

[0139] If no such i exists, then set s = 0; if for q = 1, 2, …, s, i = 1, 2, …, n, s ≥ 1 exists, then perform the following operations:

[0140]

[0141] t_N(q + in - 1, 1) = w (16)

[0142] t_N(q + in - 1, 2) = a q (17)

[0143] t_D(q + in - 1, :) = e (18)

[0144] Update the settings k ← k - 1 + s, in ← in + s, τ ← τ + 1; if nu = nus, end the for loop.

[0145] Step 3.4: If k = 0, stop and output LRB d and URB d ; otherwise:

[0146] If st = 1, output LRB d and URB d , and then store the cell U{B 0 (1:k) = t_B 0 (1:k), B(1:k) = t_B(1:k), N(1:k) = t_N(1:k), D(1:k) = t_D(1:k)}; otherwise:

[0147] Set B 0 (1:k) = t_B 0 (1:k), B(1:k) = t_B(1:k), N(1:k) = t_N(1:k), D(1:k) = t_D(1:k), output LRB d , the state - space matrices V, V0, Z and Z 0 , the current set of cells of unspecified states U{B 0 , B, N, D}; go to Step 2.

[0148] However, the improved SSD method here is serial, and it can also operate as a parallel mechanism, similar to the serial - parallel boundary algorithm in the prior art.

[0149] Step 4: If the current network configuration adjustment is the second case, i.e., changing the network topology by introducing new components or adjusting the maximum state of a component to be greater than its original value, search for the d-MP of all new network configurations, then go to Step 3 to update the output LRB d , the state space matrices V, V0, Z, and Z 0 , the current set of cells with unspecified states U{B 0 ,B,N,D}; otherwise, go to Step 5.

[0150] Step 5: Based on the current state space matrices V, V0, Z, and Z 0 , update the LRB through the state space reconstruction-based reliability assessment (SSR) method d . If the LRB d meets the accuracy requirement γ, go to Step 7; otherwise, go to Step 6.

[0151] The state space reconstruction-based reliability assessment method is specifically implemented through the following steps:

[0152] Due to the transition of the SPDs (state probability density) of components in the second case, the maximum state vector m = (m1, m2, …, m n ) may change, and such a change may cause parts of the state space matrix to have no effect on the LRB d .

[0153] According to equations (8) and (9) in Step 3.3.4.a, the LRB d is the sum of the probabilities of each set of acceptable states, which is determined by V0 and Z 0 . For example, the i-th row vectors in the two matrices (V0(i, :) and Z 0 (i, :)) are the key vectors v 0 and b 0 of the i-th set of acceptable states, and we have:

[0154] Property 1: The i-th set of acceptable states contributes to the LRB j (LT ≠ 0) only when for any j ∈ {1, 2, …, n}, V0(i, j) ≤ m d (if the j-th element of the i-th row of V0 is less than or equal to the j-th element of m, then proceed to the next step).

[0155] Proof: Assume that in the contribution set of acceptable states, there exists a state that satisfies V0(i, j) > m j . The maximum state vector m determines x i ≤ m j . Then, we have x i < V0(i, j), which contradicts the assumed condition.

[0156] Similar to the acceptable state, according to equations (10)-(13), URB d is determined by V, Z, and Z 0 and there is:

[0157] Property 2: The i-th group of unacceptable states contributes to URB j (UT≠0) only when for any j∈{1,2,…,n}, Z(i,j)≤m d .

[0158] Proof: Assume that in a set of contributing unacceptable states, there is a Z(i,j) such that Z(i,j)>m j . Then, we have V(i,j)>m j . According to the probability distribution matrix of the network, we have Pr(x i ≥Z(i,j)) = 0 and Pr(x i ≥V(i,j)) = 0. Therefore, UT = 0 contradicts the assumed condition.

[0159] Based on the above Property 1 and Property 2, an SSR method is proposed to obtain LRB d and URB d as follows:

[0160] Step 5.1: Initialization; set st = 0; for i = 1:ε, where ε is the number of rows of the state space matrix, perform the following operations:

[0161] Step 5.2: If V0(i,:)≤m, update the partial bound LT using equation (19), and then update LRB d using equation (9):

[0162]

[0163] If V(i,:)>Z(i,:) && Z(i,:)≤m, update the partial boundary UT using equation (20), and then update URB d using equation (13):

[0164]

[0165] Step 5.3: DR = URB d - LRB d . If DR < γ, set st = 1 and return URB d and LRB d . Otherwise, go back to step 6.

[0166] Step 5.1 is based on Property 1, which can remove the set of valueless states in the acceptable states from the evaluation. We observe that the vectors V(i, :) and Z(i, :) derived from the improved SSD method may be equal in some cases. However, the probability of such a set of unacceptable states is 0. Therefore, in Step 5.2, these acceptable states are first eliminated by setting the condition V(i, :) > Z(i, :), and another condition Z(i, :) ≤ m is based on Property 2 to further remove the valueless set of unacceptable states.

[0167] Step 6: If the obtained DR > γ, then combine the output of the SSR method and the cell U{B 0 , B, N, D}, and use the proposed deep state space decomposition method to obtain the LRB that meets the accuracy requirements d , while updating the state space matrices V, V0, Z, and Z 0 , and the cell U{B 0 , B, N, D}; finally, go to Step 7.

[0168] Use the state space matrix obtained in Step 5 to calculate the reliability bound. The bound derived from the current state space matrix may not meet the accuracy requirements, where st = 0. Therefore, the deep state space decomposition (DSSD) method is proposed to further obtain the bound that meets the accuracy requirements while updating the state space matrix. The specific steps of the DSSD method are as follows:

[0169] Input: All d-MPs; the bounds of the SSR method: LRB d and URB d ; the current probability distribution matrix P; the cell U{B 0 , B, N, D}; the preset accuracy requirement γ.

[0170] Output: LRB d and URB d and the additional state space matrices V′, V0′, Z′, and Z 0 ′.

[0171] Step 6.1: Set the number of the current set of unspecified states to the number of rows of B 0 , k = size(B 0 ); the stop flag st = 0; create four empty matrices t_N, t_D, t_B 0 and t_B.

[0172] Step 6.2: Set the number of the current set of unspecified states, uns = k. The initial state index set of the unspecified states is set to in = 1.

[0173] Step 6.3: Repeat Step 3.3; the LRB d and URB updated in Step 5d Update further using Step 3.3;

[0174] Step 6.4: If k = 0 or st = 1, output LRB d and URB d , and store the memory cell U{B 0 (1:k) = t_B 0 (1:k), B(1:k) = t_B(1:k), N(1:k) = t_N(1:k), D(1:k) = t_D(1:k)}, otherwise let B 0 (1:k) = t_B 0 (1:k), B(1:k) = t_B(1:k), N(1:k) = t_N(1:k), D(1:k) = t_D(1:k), and go to Step 2.

[0175] Based on the reliability bound of the SSR method, the DSSD method takes the cells of the current unspecified state set U as input and decomposes these unspecified states, thus making the reliability bound converge further. At the same time, the new critical vectors are updated as the state space matrix. In this way, Algorithm 2 will have more abundant information in subsequent calculations.

[0176] Step 7: If the network needs to adjust the configuration, go to Step 2, otherwise terminate the evaluation.

[0177] Step 3 of this method is to use the improved SSD method to obtain LRB d and the stored state space matrices V, V0, Z and Z 0 during the first calculation of reliability, and then use them as the input of the SSR method (i.e., Step 5). In addition, the network topology may change by introducing new components in the second adjustment case. The new components will generate different d-MPs, making the current state space matrix unavailable. In this case, all updated d-MPs need to be searched, and the improved SSD method is used again. When the preset accuracy requirement γ is met, the improved SSD method will stop. Then, there are several groups of unspecified states left. The information of these unspecified states is stored as the input of the DSSD method (i.e., Step 6).

[0178] Considering Case 1, in Step 5, the SSR method uses the current state space matrix as input. Then, the state space matrix is reconstructed, and LRB d and URB d are calculated by adding state space sets one by one until the accuracy requirement is met or all state space sets are used.

[0179] If when using all the reconstructed state spaces, LRB dThe accuracy cannot meet the requirements. Then, the information of the current boundary and the unspecified state U{B 0 ,B,N,D} is input from Step 3 to Step 6 to obtain qualified reliability. At the same time, the state space matrix and the information of the unspecified state are updated.

[0180] Embodiment 2

[0181] In this embodiment, Figure 1 taking the bridged network shown as an example to further illustrate the method proposed in Embodiment 1.

[0182] As Figure 1 shown, s is the source point and t is the sink point. There are 6 edges in the figure, namely a1 to a6. Table 1 shows the probabilities of each edge in states 0 - 3:

[0183] Table 1

[0184] Status 0 1 2 3 a1 0.05 0.10 0.25 0.60 a2 0.05 0.35 0.60 —— a3 0.05 0.95 —— —— a4 0.05 0.95 —— —— a5 0.05 0.95 —— —— a6 0.05 0.25 0.70 ——

[0185] Table 1 is the initial network configuration of the bridged network, and the adjusted configuration is shown in Table 2;

[0186] Table 2

[0187] Index of the set of acceptable states <![CDATA[v 0 > <![CDATA[b 0 > 1 (1,1,0,0 1,1) (3,2,1,1 1,2) 2 (1,0,1,0,1,2) (3,0,1,1,1,2) 3 (2,1,1,0,0,1) (3,2,1,1,0,2) 4 (1,2,0,1,1,0) (3,2,1,1,1,0)

[0188] There are five 2-MPs: {1,0,1,0,1,2}, {2,2,0,0,0,0}, {2,1,1,0,0,1}, {1,1,0,0,1,1}, {1,2,0,1,1,0}. Consider the preset accuracy γ = 0.01.

[0189] Step 1: Initialize the parameters according to the network initial configuration. Take all 2-MPs and the initial probability distribution matrix P as the input;

[0190] Step 2: There is no available state space matrix, go to Step 3.

[0191] Step 3: Execute the improved SSD method to obtain LRB d = 0.9067; The state space matrix is as follows:

[0192]

[0193] There are still 3 groups of unspecified states, so there is U{B 0 ,B,N,D}, where B 0 = [3,2,0,1,0,2; 3,2,1,1,0,0; 3,2,1,0,1,0], B = [2,1,0,0,0,0; 2,1,1,0,0,0; 1,2,0,0,1,0], N =

[0194] [2, 3; 2, 6; 2, 4] and D = [2, 3; 2, 3; 2, 5].

[0195] Step 4: Adjust the network configuration and go back to Step 2.

[0196] Step 2: There is an available state space matrix, go to Step 4.

[0197] Step 4: The current adjustment belongs to the first case, go to Step 5.

[0198] Step 5: Execute the SSR method:

[0199] Step 5.1: m = (3 2 1 0 1 1); Set st = 0.

[0200] Step 5.2: If V0(1, :) = (1 1 0 0 1 1) ≤ m is satisfied, then, LT = 0.1176, LRB d = LRB d + LT = 0.1176. If V(1, :) > Z(1, :) && Z(1, :) ≤ m is satisfied, then UT = 0.3, URB d = URB d - UT = 0.7 then DR = 0.5824.

[0201] Step 5.3: If V0(2, :) = (1 0 1 0 1 2) ≤ m is not satisfied, then, LT = 0 and LRB d = LRB d + LT = 0.1176. If V(2, :) > Z(2, :) && Z(2, :) ≤ m, then UT = 0.21 and URB d = URB d - UT = 0.49 then DR = 0.3724.

[0202] Step 5.4: If V0(3, :) = (2 1 1 0 0 1) ≤ m is satisfied, then, LT = 0.0151 and LRB d = LRB d + LT = 0.1327. If V(3, :) > Z(3, :) && Z(3, :) ≤ m is satisfied, then UT = 0.168 and URB d = URB d - UT = 0.322 then DR = 0.1893.

[0203] Step 5.5: If V0(4, :) = (1 2 0 1 1 0) ≤ m is not satisfied, then, LT = 0 and LRB d = LRB d+LT = 0.1327. If V(4,:) > Z(4,:) && Z(4,:) ≤ m, then UT = 0.0448 and URB d = URB d -UT = 0.2772 then DR = 0.1445.

[0204] Step 6: When DR > γ, execute the DSSD method (LRB d , URB d , P, U{B 0 , B, N, D}).

[0205] Step 6.1: For the first set of unspecified states, B(1,:) and B 0 (1,:), there are LT = 0.0432 and LRB d = LRB d +LT = 0.1759. URB d = URB d -UT = 0.2772 - 0.0576 = 0.2196. Record the key vector. No set of unspecified states is generated.

[0206] Step 6.2: For the second set of unspecified states, B(2,:) and B 0 (2,:), there are LT = 0.0043 and LRB d = LRB d +LT = 0.1802. URB d = URB d -UT = 0.2196 - 0.05058 = 0.2138. Record the key vector. No set of unspecified states is generated.

[0207] Step 6.3: For the third set of unspecified states, B(2,:) and B 0 (2,:), we have LT = 0.0144. And LRB d = LRB d +LT = 0.1946. URB d = URB d -UT = 0.2138 - 0.0192 = 0.1964. Record the key vector. No set of unspecified states is generated. DR < γ, set st = 1.

[0208] Step 6.4: Using the recorded key vector, the state space matrix is updated as follows:

[0209]

[0210] Time complexity analysis of the method of the present invention: Considering the second case where the new network configuration requires repeated execution of the SSD method, the present invention improves the SSD method. Instead of comparing each state of z l and b 0 , it focuses on only one state affected by the previous decomposition, as shown in Equation (17). The time complexity of this process is reduced from O(n) to O(1). Therefore, the time complexity of the proposed improved SSD method is O(ε * max(ρ, n 2 ))), which is faster than the SSD method in the prior art.

[0211] For the first case, currently there is a method that uses O(ε * n) to obtain the LRB d , and the time complexity of the SSR method of the present invention is O(ε′ * n), where ε′ is the number of acceptable state sets contributing to the LRB d . According to Property 1, ε′ ≤ ε. Therefore, for the same LRB d accuracy, the SSR method is faster than the prior art method. When considering the required accuracy of the LRB d , the worst case is the combined use of the SSR method and the DSSD method. Then, the time complexity is where is the total number of unspecified sets generated during the SSD process of the DSSD method. Compared with the time complexity of the prior art method, we have Therefore, the time complexity of the proposed method to obtain the LRB d that meets the accuracy requirements is faster than using the traditional SSD method.

[0212] Embodiment 3

[0213] Due to the adoption of the above technical solutions, the technical effects of the present invention can be verified by the following experimental data:

[0214] In this embodiment, the performance comparison between the method proposed in this aspect and the reported methods is studied. All algorithms are coded in MATLAB 2017a, and the experiments are carried out on a personal computer with Windows 10, Core i7 13700KF@5.8GHz, and 128GB RAM.

[0215] First, consider a case such as Figure 3The medium-sized network shown with 12 nodes and 17 components. Assume that each component in the medium-sized network with 6 states (0 to 5) is independent and identically distributed (i.i.d.), and the original probability distribution is as follows: p(0) = (0.05, 0.15, 0.2, 0.25, 0.15, 0.2). Based on the cost constraint, the performance of one of the network components is deliberately reduced. The network adjustment scheme involves selecting one component in each adjustment cycle to satisfy p = (0.34, 0.55, 0, 0, 0, 0). For example, in the first simulation, the SPD of component 1 is converted to p = (0.34, 0.55, 0, 0, 0, 0). Then, in the second simulation, the SPD of component 2 is changed while component 1 is restored to p(0).

[0216] As shown in Table 4, the CPU time (in seconds) is recorded when using the method of the present invention (denoted as T P ), Liu's repeated SSD method (denoted as T R ), and the traditional serial SSD bound algorithm (denoted as T S ) to calculate the reliability of all system requirements d in the full network adjustment scheme, under different precision requirements γ.

[0217] Table 4

[0218]

[0219] The proposed method is superior to the repeated SSD method, especially as the precision requirement γ increases. Compared with the traditional SSD bound algorithm, the proposed method shows a significant advantage in computational efficiency.

[0220] In addition to recording the time required to calculate the reliability using these different methods, the network reliability errors obtained by the three methods are further recorded. The reliability error is defined as follows:

[0221]

[0222] S > 0 indicates that the error is greater than the preset precision requirement. The larger the value of S, the higher the error level; on the contrary, when S < 0, it indicates that the error is less than the predefined threshold γ; the smaller the value of S, the smaller the error.

[0223] As Figure 4As shown, the green cells represent the case where the error S is less than 0, which means that the error is within the preset threshold γ. The red cells represent the error greater than 0, indicating that the error significantly exceeds the accuracy requirement. The gray cells represent the case where there is no available network traffic. The comparison of network reliability errors under different accuracy requirements shows that the proposed method can achieve higher accuracy and stability in most cases, and it outperforms other methods in various cases. Liu's repeated SSD method shows larger errors in multiple cases, indicating lower accuracy. The traditional SSD method is relatively stable under lower accuracy requirements. Overall, compared with the other two methods, the proposed method is superior in maintaining accuracy and meeting strict accuracy requirements.

[0224] Figure 5 The Hebei traffic network with 33 components is shown. It is assumed that each component is independent and identically distributed, and it follows p(0) = (0.1, 0.3, 0.6). We focus on the maintenance of components 2, 12, and 27, where each component can be adjusted to three different SPDs configurations: p1 = (1, 0, 0), p2 = (0.2, 0.8, 0), and p3 = (0.1, 0.4, 0.5). Different SPDs represent different traffic restriction schemes during maintenance. Managers need to evaluate the impact of various combinations of maintenance and traffic restriction schemes on network reliability. Therefore, there are 27 possible adjustment configurations in total. The d-level reliability bounds for all different accuracy requirements are calculated, and the CPU time is recorded.

[0225] Table 5

[0226]

[0227]

[0228] As shown in Table 5, the data show that as the accuracy requirement γ becomes more stringent, the proposed method (T P ), repeated SSD method (T R ) and the traditional SSD boundary algorithm (T S ) significantly increase in computational time. However, the growth rates of different methods are different, highlighting the computational advantage of this method.

[0229] For lower accuracy requirements (γ=10 -2 ), ratio T R / T P =4.33 and T S / T P = 11.10 shows that the proposed method has obvious computational efficiency compared with the other two methods. As γ becomes more accurate, these ratios increase steadily and reach -5 When reaching T R / TP = 8.88 and T S / T P = 12.75. This trend indicates that the proposed method scales more effectively under higher precision requirements and thus maintains a computational advantage compared to the repeated SSD and traditional SSD boundary algorithms in scenarios requiring high precision.

[0230] Figure 6 A detailed comparison was made of the network reliability errors of the proposed method, Liu's repeated SSD, and traditional SSD under different precision requirements and network traffic d. The proposed method showed better consistency in keeping the error below the threshold, especially under more stringent precision requirements. Liu's repeated SSD method often exceeded the error threshold under high precision requirements, indicating that it may not be suitable for high-precision applications. The traditional SSD method performed moderately well, showing stability but not achieving the same level of consistent accuracy as the proposed method under the most stringent precision thresholds.

[0231] The third network is a large network with 30 components. All components are i.i.d and they follow p(0) = (0.3, 0.3, 0.4). In this example, state probability distributions (SPDs) and topological adjustments were considered. In the first stage, as Figure 7 shown, three SPDs, namely p1 = (1, 0, 0), p2 = (0.2, 0.8, 0), and p3 = (0.1, 0.4, 0.5), were assigned to components 6 and 17. Thus, there were 9 adjustments in the first stage. Then, in the second stage, the original network topology changed. As Figure 8 shown, the new component 6 connects nodes 6 and 11. The new component 17 connects nodes 11 and the sink node. The three state probability distributions from the first stage were also used in this stage.

[0232] Table 6

[0233]

[0234] As shown in Table 6, as γ becomes more precise, the computational time of all methods increases, although at different rates, which emphasizes the scalability of the proposed method. When γ = 10 -2 , the computational intensity of the repeated SSD method is only slightly higher than that of the proposed method, while the traditional SSD algorithm requires significantly more time. As γ tightens to 10 -5 , both ratios increase, where T R / T P = 3.51 and T S / T P= 11.25. This trend indicates that as the requirement for higher precision increases, the computational cost of all methods will increase, but the proposed method always maintains a computational advantage, especially in high-precision scenarios. T R / T P and T S / T P The steady increase of

[0235] such as Figure 9 shown indicates the continuous improvement of the efficiency of the proposed method compared to the other two algorithms, indicating that it is particularly beneficial for applications that require high precision.

[0236] It can be understood that the present invention is described through some embodiments. Those skilled in the art know that without departing from the spirit and scope of the present invention, various changes or equivalent replacements can be made to these features and embodiments. Additionally, under the teaching of the present invention, these features and embodiments can be modified to adapt to specific situations and materials without departing from the spirit and scope of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application belong to the scope protected by the present invention.

Claims

1. A multi-state flow network reliability assessment method considering network configuration adjustment, characterized in that: The following steps are involved: Step 1: Initialize parameters; take all minimal path sets d-MP in the existing network configuration and the initial probability distribution matrix P as input; Step 2: If there is an available state space matrix, go to step 4, otherwise go to step 3; Step 3: Based on all d-MPs in the current network configuration, use the improved state space decomposition method to obtain the initial state space set and obtain the lower reliability bound LRB with the preset accuracy requirement γ d ; The initial state space set specifically includes a state space matrix for storing key vectors and limit points of each set of acceptable and unacceptable states and cells for storing information of the remaining set of unspecified states; finally, go to step 7; Step 4: Network configuration adjustment If the network topology is changed by introducing new components, or the maximum state of a component is adjusted to be greater than its original value, search for the d-MP of all new network configurations and go to step 3 to update the output LRB d , the state space matrix, the current set of cells with unassigned states; Otherwise go to step 5; Step 5: Based on the current state space matrix, calculate the LRB through the reliability assessment method based on state space reconstruction d , if LRB d If the accuracy requirement is met, go to step 7, otherwise go to step 6; Step 6: If the obtained reliability accuracy is greater than the accuracy requirement, the proposed deep SSD method is used to combine the output of the SSR method and cell U to obtain the LRB that meets the accuracy requirement. d ; Update the state space matrix and cell U at the same time, and finally go to step 7; Step 7: If the network configuration needs to be adjusted, go to step 2, otherwise terminate the evaluation.

2. A multi-state flow network reliability assessment method considering network configuration adjustment according to claim 1, characterized in that: The specific steps of the improved state space decomposition method are as follows: Step 3.1: Set initialization parameters, lower reliability bound LRB d =0, upper reliability boundary URB d =1; at this time, the number of loops k = 1, and the stop flag st = 0; create an initialization state space set to store the initial temporary parameters; Step 3.2: Set the number of unspecified state sets in the current generation to uns, uns = k, and the initial unspecified state set index to in = 1; set the index τ of the storage state space to 1; Step 3.3: Start looping k times, for nu=1:uns, for is the loop instruction, nu is the loop variable, its value is 1 and loops to uns times; perform the following operations: Step 3.3.1: Update the restriction vector, b 0 =B 0 (nu), b = B(nu), w = 0, w stores the number of the current qualified d-MP set; Step 3.3.2: d-MP is based on the upper bound vector b 0 Decompose into qualified d-MP and unqualified d-MP, save the index e of the current qualified d-MP set, save the number of the current qualified d-MP set, and update w; Step 3.3.3: Get all qualified d-MPs in the current set, and let the lth d-MP in the current multi-state network be represented by z l , compare the lower limit vector with z l , obtain the critical vector v of the unacceptable state set; Select z l , so that z l Minimize the following: b i is the i-th element of the lower limit vector. If there are multiple z l At the same time, minimize H2(z l ), then choose the one that maximizes the following: Among them, x i is the i-th element of the state vector, b 0 i is the i-th element of the upper limit vector. If there are still multiple options, select one of them at random; Compare the above selected z l and the lower limit vector, to obtain the critical vector v of the acceptable state set 0 ; Update the state space matrix, let V(τ,:)=v, V0(τ,:)=v 0 , Z(τ,:)=b and Z 0 (τ,:)=b 0 ; Step 3.3.4: If st = 0, perform the following steps, otherwise jump to step 3.3.5: Step 3.3.4.a: Set partial limit LT = 0; calculate from arrive Probability i=1,2,…,n: LRB d =LRB d +LT p i is the probability that the i-th edge meets the reliability requirement; Step 3.3.4.b: Set the partial boundary UT = 0, update UT and URB d : URB d =URB d -UT (13) Step 3.3.4.c: Set the reliability precision DR = URB d -LRB d ; If DR < γ, then st = 1; Step 3.3.5: Assume that The i set is a q , q=1,2,…,s, i∈{1,2,…,n}; If there is no such i, set s=0; if for q=1,2,…,s,i=1,2,…,n, there exists s≥1, then update the parameters of the initial state space set, and update the settings k←k-1+s, in←in+s, τ←τ+1; if nu=nus, end the for loop; Step 3.4: If k = 0, stop and output LRB d and URB d ;otherwise: If st=1, output LRB d and URB d , then store the cell U{B 0 (1:k) = t_B 0 (1:k), B(1:k)=t_B(1:k), N(1:k)=t_N(1:k), D(1:k)=t_D(1:k)}; otherwise: Setting B 0 (1:k) = t_B 0 (1:k), B(1:k)=t_B(1:k), N(1:k)=t_N(1:k), D(1:k)=t_D(1:k), output LRB d , the state space matrices V, V0, Z and Z 0 , the current set of cells with unspecified status U{B 0 ,B,N,D}; go to step 2.

3. A multi-state flow network reliability assessment method considering network configuration adjustment according to claim 2, characterized in that: The initial space state set includes: The first column of the matrix N is used to store the number of d-MPs that meet the conditions. At the initial moment, N(1,1)=L, where L is the total number of d-MPs. The second column of the matrix N is used to store the key position for determining whether the d-MP is qualified. At the initial moment, N(1,2)=1. The matrix t_N temporarily stores the number and key positions of the currently qualified d-MPs; The matrices D and t_D store the indices of the current qualified d-MP set respectively, and set D(1) = (1, 2, ..., L); Matrix B 0 Stores the upper bound vector for each set of unspecified states, B 0 (1) = M, where M is the maximum state vector; Matrix B stores the lower limit vector of each group of unspecified states, B(1)=0; Matrix t_B 0 Store a temporary upper bound vector for each set of unspecified states; The matrix t_B stores the temporary lower limit vector for each set of unspecified states; the initial limit vector includes the upper limit vector b for the set of unspecified states 0 , b 0 =B 0 (1) = M and the lower limit vector b of the unspecified state set, b = B(1) = 0.

4. A multi-state flow network reliability assessment method considering network configuration adjustment according to claim 3, characterized in that: The reliability assessment method based on state space reconstruction is specifically implemented by the following steps: Step 5.1: Initialization; set st = 0; for i = 1:ε, where ε is the number of rows in the state space matrix, do the following: Step 5.2: If V0(i,:)≤m, update the partial bound LT using the following formula, and then update LRB d : LRB d =LRB d +LT If V(i,:)>Z(i,:)&&Z(i,:)≤m, update the sub-boundary UT using the following formula, and then update the URB d : URB d =URB d -UT Step 5.3: If DR < γ, set st = 1 and return URB d and LRB d ; Otherwise, return to step 6.

5. A multi-state flow network reliability assessment method considering network configuration adjustment according to claim 4, characterized in that: The specific steps of the deep SSD method are as follows: Step 6.1: Set the number of currently unspecified state sets to B 0 The number of rows, k = size(B 0 ); stop flag st = 0; create four empty matrices t_N, t_D, t_B 0 and t_B; Step 6.2: Set the number of sets of currently unspecified states uns = k, and the unspecified initial state index set in = 1; Step 6.3: Repeat step 3.3; update the LRB obtained in step 5 d and URB d Use step 3.3 to further update; Step 6.4: If k = 0 or st = 1, output LRB d and URB d , and the storage unit U{B 0 (1:k) = t_B 0 (1:k), B(1:k)=t_B(1:k), N(1:k)=t_N(1:k), D(1:k)=t_D(1:k)}, otherwise let B 0 (1:k) = t_B 0 (1:k), B(1:k)=t_B(1:k), N(1:k)=t_N(1:k), D(1:k)=t_D(1:k), and go to step 2.

6. A multi-state flow network reliability assessment method considering network configuration adjustment according to claim 5, characterized in that: The d-MP is based on the upper limit vector b 0 The breakdown into qualified d-MP and unqualified d-MP is as follows: z l (N(nu,2))≤b 0 (N(nu,2)),pass through; z l (N(nu,2))>b 0 (N(nu,2)), failed.

7. A multi-state flow network reliability assessment method considering network configuration adjustment according to claim 6, characterized in that: The comparison of the above selected z l and the lower limit vector, obtain the critical vector v of the acceptable state set 0 The details are as follows:

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