CN115766494A - Multi-state network minimum path vector searching method based on feasible circulating flow - Google Patents
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Abstract
The invention relates to a multi-state network minimum path vector searching method based on a feasible circulating flow, and belongs to the field of multi-state network reliability evaluation. The method comprises the following steps: s1: determining a search space X of a minimum path vector d-MP; s2: judging the relation among M (U), M (L) and d; s3: searching for d-MP by solving a circulating flow problem; s4: decomposing the search space into a number of disjoint subsets; s5: the search continues in the decomposed subset. The invention improves the calculation efficiency of searching the d-MP.
Description
Technical Field
The invention belongs to the field of multi-state network reliability evaluation, and relates to a multi-state network minimum path vector searching method based on a feasible cycle flow.
Background
With the increasing dependence of society on various technical networks, people have an increasing demand for reliable services in real life. Therefore, as a useful tool for the performance evaluation of modern technology networks, reliability analysis is widely applied to real-world networks such as computer networks, power transmission networks, production and manufacturing networks, logistics and supply chain networks, and the like. In conventional reliability analysis, a two-state network model has been successfully applied to analyze network performance. However, since network components are assumed to have only two states: fully operational or fully failed, two-state network models have their inherent limitations. In the last decade, researchers have extended the two-state network model to a multi-state network model to meet the actual need for network components with more than two levels of performance. On the basis, several different types of reliability methods are proposed, such as a monte carlo simulation Method (MCS), a Universal Generation Function (UGF) method, a multi-valued decision diagram (MDD) method, a state space decomposition method (SSD) and a minimum path quantity (also called d-MP) method.
An important metric for multi-state network performance is the reliability of the two terminals with capacity requirement d, using R d Is expressed as the probability that the network can deliver d units of traffic from the source s to the sink t. Scholars propose direct and indirect algorithms to approximate or accurately calculate R d The value is obtained. The direct algorithm firstly finds a capacity vector with the network capacity not less than d, then utilizes the capacity vector to divide the whole capacity vector space into an acceptable set and an unacceptable set, and finally, R d Equal to the sum of the probabilities of all acceptable sets. The indirect algorithm is a two-stage algorithm. In the first stage, the algorithm firstly solves all d-MP, and in the second stage, the joint probability of all d-MP is calculated by using a repulsion theorem or an disjoint sum method to obtain R d The value of (c).
The existing method mainly solves a lost-to-noise graph equation which is difficult to solve NP by using an implicit enumeration method to search for the d-MP, and has the outstanding problem of low calculation efficiency. Therefore, a method for improving d-MP search is needed.
Disclosure of Invention
In view of this, the present invention provides a method for searching a minimum path vector of a multi-state network based on a feasible cycle flow, so as to improve the calculation efficiency of searching for a d-MP.
In order to achieve the purpose, the invention provides the following technical scheme:
a multi-state network minimum path vector searching method based on feasible cycle flow comprises the following steps:
s1: determining a search space X of a minimum path vector d-MP;
s2: determining the relation between M (U), M (L) and d, wherein M (U) represents the maximum flow of the network G under an upper bound capacity vector U of X, M (L) represents the maximum flow of the network G under a lower bound capacity vector L of X, and d represents the capacity demand;
s3: searching for d-MP by solving a circulating flow problem;
s4: decomposing the search space into a number of disjoint subsets;
s5: the search continues in the decomposed subset.
Further, step S1 specifically includes: let the maximum capacity vector of the network G be W = (W) 1 ,W 2 ,…,W m ) Wherein W is i Represents an edge e i I is more than or equal to 1 and less than or equal to m, and m represents the total number of edges; let L i ＝max{d-M(W(0 i ) 0) (i is not less than 1 and not more than m) is an edge e i A lower bound capacity of (1), wherein W (0) i ) Is the capacity vector obtained when the ith component of W is set to 0, M (W (0) i ) Denotes that the network G is at W (0) i ) The maximum flow rate of; order U i ＝min{W i D (i is more than or equal to 1 and less than or equal to m) is an edge e i Then the search space of d-MP is X = { X = (X =) (X capacity) 1 ,x 2 ,…,x m )|L i ≤x i ≤U i ，1≤i≤m}＝{x|L≤x≤U}＝[L,U]Wherein, L = (L) 1 ,L 2 ,…,L m ) Lower bound capacity vector of X, U = (U) 1 ,U 2 ,…,U m ) An upper bound capacity vector of X, X i Is an edge e i X is a capacity vector.
Further, step S2 specifically includes: if M (U)<d, the minimum path vector d-MP does not exist in the search space X, and the algorithm is stopped; if M (L) = d, according to the definition of d-MP, if M (L-0 (e) is satisfied for all i's of which i's component of L is greater than 0 i ))<d, then L is d-MP, where 0 (e) i ) Is a unit vector, i.e. 0 (e) i ) Is 1, and the other components are all 0 i ) Denotes that the network G is at L-0 (e) i ) Stopping the algorithm at the lower maximum flow rate; otherwise, go to step S3.
Further, step S3 specifically includes: adding a new edge e from the sink t to the source s in the initial network G m+1 Obtaining a new network G, let edge e m+1 Has a fixed value d, i.e. e m+1 Lower bound capacity L of m+1 And upper bound capacity U m+1 D, the lower bound capacity and the upper bound capacity of other edges are kept unchanged; solving a circulating flow problem in a new network G by using a maximum flow algorithm, if no feasible circulating flow exists, no d-MP exists in a search space X, and stopping the algorithm; otherwise, assume F d ＝(f 1 d ,f 2 d ,…,f m d ,f m+1 d ) Is the determined possible recycling flow if (f) 1 d ,f 2 d ,…,f m d ) Not including directed circles, then (f) 1 d ,f 2 d ,…,f m d ) Is a d-MP.
Further, step S4 specifically includes: let [ L ] be i ,U i ]Is an edge e i A set of capacities of, i.e. [ L ] i ,U i ]＝{x i |L i ≤x i ≤U i Therefore, search space X = [ L, U =]Can be recorded as [ L 1 ,U 1 ]×[L 2 ,U 2 ]×…×[L m ,U m ](ii) a Let E d ＝{e i |L i <f i d I is 1. Ltoreq. M, for each e i ∈E d Set of [ L i ,U i ]Can be f i d Partitioning into two disjoint subsets: [ L ] i ,f i d-1 ]And [ f i d ,U i ]I.e., [ L ] i ,U i ]＝[L i ,f i d-1 ]∪[f i d ,U i ]，[L i ,f i d-1 ]∩[f i d ,U i ]= phi; order to
X is decomposed into q +1 disjoint subsets:
further, step S5 specifically includes: each subset X obtained in step S4 (k) And (k is more than or equal to 1 and less than or equal to q) as a new search space, and sequentially turning to the step S2 to continue solving until all the subsets are searched.
The invention has the beneficial effects that: compared with the prior art, the method does not need to know all the minimum paths of the network, does not need to use an implicit enumeration method, does not need to solve a loss-of-use graph equation, does not generate repeated d-MP, and can improve the calculation efficiency of searching the d-MP.
Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention. The objectives and other advantages of the invention may be realized and attained by the means of the instrumentalities and combinations particularly pointed out hereinafter.
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For the purposes of promoting a better understanding of the objects, aspects and advantages of the invention, reference will now be made to the following detailed description taken in conjunction with the accompanying drawings in which:
FIG. 1 is a flow chart of a multi-state network minimum path vector searching method based on feasible cycle flow according to the present invention;
FIG. 2 is a network diagram of a network G according to an embodiment of the present invention;
FIG. 3 is a diagram of a new network G according to an embodiment of the present invention * A network diagram of (a).
Detailed Description
The following embodiments of the present invention are provided by way of specific examples, and other advantages and effects of the present invention will be readily apparent to those skilled in the art from the disclosure herein. The invention is capable of other and different embodiments and of being practiced or of being carried out in various ways, and its several details are capable of modification in various respects, all without departing from the spirit and scope of the present invention. It should be noted that the drawings provided in the following embodiments are only for illustrating the basic idea of the present invention in a schematic way, and the features in the following embodiments and examples may be combined with each other without conflict.
Referring to fig. 1 to 2, the present invention provides a method for searching a minimum path vector of a multi-state network based on a feasible cycle flow, as shown in fig. 1, which specifically includes the following steps:
1) Determining the search space of the d-MP:
maximum capacity vector W = (W) for a given network G 1 ,W 2 ,…,W m ) Wherein W is i Represents an edge e i Maximum capacity of (1), order L i ＝max{d-M(W(0 i ) 0) (i is not less than 1 and not more than m) is an edge e i A lower bound capacity of (2), wherein W (0) i ) Is the capacity vector obtained when the ith component of W is set to 0, M (W (0) i ) Denotes that the network is at W (0) i ) Maximum flow of i ＝min{W i D (i is more than or equal to 1 and less than or equal to m) is an edge e i For d-MP x = (x) 1 ,x 2 ,…,x m ) In particular, L i ≤x i ≤U i (1 ≦ i ≦ m) is true, and thus the search space for d-MP is X = { X = (X) = 1 ,x 2 ,…,x m )|L i ≤x i ≤U i ，1≤i≤m}＝{x|L≤x≤U}＝[L,U]Wherein, L = (L) 1 ,L 2 ,…,L m ) Lower bound capacity vector, called X, U = (U) 1 ,U 2 ,…,U m ) An upper bound capacity vector called X.
2) Determining the relationship between M (U), M (L) and d:
search space X = { X | L ≦ X ≦ U } = [ L, U]Also called L = (L) 1 ,L 2 ,…,L m ) Minimum capacity vector of X, U = (U) 1 ,U 2 ,…,U m ) The maximum capacity vector of X.
If M (U) < d, then for any X ∈ X, M (X) ≦ M (U) < d holds, i.e., by definition X is known not to contain d-MP, then X is not discussed and the algorithm stops.
If M (L) = d, then for any x>L, M (x) ≥ M (L) = d; at the same time, for any X ∈ X, M (X) ≦ d, then M (X) = M (L) = d applies to any X ∈ X>And L. In this case, for any x>Neither L, x is d-MP, since there is at least one edge e i Satisfies M (x-0 (e) i ) M (L) = d, which contradicts the definition of d-MP. So only the vector L needs to be verified, according to the definition of d-MP, if M (L-0 (e) i ))<d, then L is d-MP, where 0 (e) i ) Is a unit vector, i.e. 0 (e) i ) The ith component of (1) and all other components are 0, and the algorithm stops. Otherwise, the next step is performed.
3) Finding d-MP by solving the problem of circulating flow:
adding a new edge e from the sink t to the source s in the initial network G m+1 Obtaining a new network G, let edge e m+1 Has a fixed value d, i.e. e m+1 Lower bound capacity L of m+1 And an upper bound capacity U m+1 Are all equal to d, the lower bound capacity and the upper bound capacity of the other edges remain unchanged. The problem of the feasible circulation flow in the network G is to find a feasible flow vector x = (x) satisfying the following condition 1 ,x 2 ,…,x m ,x m+1 )：
L i ≤x i ≤U i ，e i ∈E∪{e m+1 }
Where V is the set of points In the network G, E is the set of edges In the network G, out (V) represents the set of all edges emanating from point V, and In (V) represents the set of all edges pointing to point V.
Searching a feasible circulating flow in G by using a maximum flow algorithm, if the feasible circulating flow does not exist, stopping the algorithm if the d-MP does not exist in X; otherwise, assume F d ＝(f 1 d ,f 2 d ,…,f m d ,f m+1 d ) Is the calculated possible circulation flow, if (f) 1 d ,f 2 d ,…,f m d ) Does not include a directional loop, then (f) 1 d ,f 2 d ,…,f m d ) Is a d-MP.
4) The search space is decomposed into several disjoint subsets:
let [ L ] be i ,U i ]Is an edge e i A set of capacities of [ i.e. [ L ] i ,U i ]＝{x i |L i ≤x i ≤U i Therefore, the set X = [ L, U =]Can be recorded as [ L 1 ,U 1 ]×[L 2 ,U 2 ]×…×[L m ,U m ]. Because of d-flow (f) 1 d ,f 2 d ，…，f m d ) From the set X = [ L, U =]For 1. Ltoreq. I. Ltoreq.m, L i ≤f i d ≤U i If true; further, let E d ＝{E i |L i <f i d I is 1. Ltoreq. M, and E d Not an empty set (if E) d If it is an empty set, i is greater than or equal to 1 and less than or equal to m, L i ＝f i d It holds that L is a d-stream, which does not comply with the condition M (L)<d) .1. The For each e i ∈E d Set of [ L ] i ,U i ]Can be f i d Partitioning into two disjoint subsets: [ L ] i ,f i d-1 ]And [ f i d ,U i ]I.e., [ L ] i ,U i ]＝[L i ,f i d-1 ]∪[f i d ,U i ]，[L i ,f i d-1 ]∩[f i d ,U i ]= Φ. Order to
The set X may be divided into q +1 disjoint subsets, as follows:
wherein,
subset X (1) ，X (2) ，…，X (q) ，X (q+1) Are disjoint.
5) Continuing the search in the decomposed subset:
each subset X obtained in the step 4) (k) And (k is more than or equal to 1 and less than or equal to q) as a new search space, and sequentially turning to the step 2) to continue solving until all the subsets are searched.
The present invention will be described in detail with reference to the following specific examples:
one specific embodiment is shown in fig. 2, where a distribution network is abstracted to obtain the network G in fig. 2. The network comprises 4 nodes and 5 transport edges, wherein the node s represents a source point, the node t represents a sink point, and the nodes 1 and 2 represent transfer places. Fig. 2 shows the maximum capacity of each edge in the network, and the maximum capacity vector W = (3,2,2,1,2) can be known from fig. 2.
All polymorphic miniroads of the transport network are calculated below with the method of the invention, assuming a demand level d =3.
According to the method steps, the solving process is as follows:
1) Determining a search space for 3-MP:
for edge e 1 ,W(0 1 ) = (0,2,2,1,2), then by max flow algorithm, M (W (0) can be obtained 1 ) ) =2. Thus, L 1 ＝max{3-M(W(0 1 ) 0} = max {3-2,0} = 1). Likewise, L is calculated 2 ＝0，L 3 ＝1，L 4 ＝0，L 5 And =1. When i is more than or equal to 1 and less than or equal to 5, U i ＝min{W i ,3}＝W i . Thus, the search space for 3-MP is X = { X = (X) 1 ,x 2 ,…,x m )|1≤x 1 ≤3,0≤x 2 ≤2,1≤x 3 ≤2,0≤x 4 ≤1,1≤x 5 ≤2}＝{x|L≤x≤U}＝[L,U]Where L = (1, 0, 1) is the lower bound capacity vector of X and U = (3, 2,1, 2) is the upper bound capacity vector of X.
2) Determining the relationship between M (U), M (L) and 3:
x = [ L, U ] = [ (1, 0, 1), (3, 2,1, 2) ], where L = (1, 0, 1), U = (3, 2,1, 2) ].
M(U)＝4>3，
M(L)＝1<3。
3) Finding 3-MP by solving the recycle stream problem:
adding a new edge e pointing from the sink t to the source s in the initial network G 6 To obtain a new network G, which is shown in FIG. 3, let edge e 6 Has a fixed value of 3, i.e. e m+1 Lower bound capacity L of 6 And upper bound capacity U 6 All equal to 3, and the lower and upper bound capacities of the other edges remain unchanged from the value in X.
Using maximum flow algorithm to search feasible circulation flow in network G to obtain a feasible circulation flow (2, 1,2, 3), since (2, 1, 2) does not contain directional loops, (2, 1, 2) is a 3-MP.
4) The search space is decomposed into several disjoint subsets:
E d ＝{e 1 ，e 2 ，e 4 ，e 5 the search space X is decomposed into 5 disjoint subsets: x (1) ＝[(1,0,1,0,1),(1,2,2,1,2)]，X (2) ＝[(2,0,1,0,1),(3,0,2,1,2)]，X (3) ＝[(2,1,1,0,1),(3,2,2,0,2)]，X (4) ＝[(2,1,1,1,1),(3,2,2,1,1)]，X (5) ＝[(2,1,1,1,2),(3,2,2,1,2)]。
5) Continuing the search in the decomposed subset:
will be a subset X (1) ，X (2) ，X (3) ，X (4) And respectively taking the search spaces as new search spaces, and sequentially turning to the step 2) to continue solving until all the subsets are searched. Finally, 4 3-MPs are obtained: (2, 1, 2), (1, 2,1,0, 2), (2, 1,2,0, 1) and (3, 0,2, 1).
Finally, although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that various changes and modifications may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.
Claims (6)
1. A multi-state network minimum path vector searching method based on a feasible cyclic flow is characterized by comprising the following steps:
s1: determining a search space X of a minimum path vector d-MP;
s2: determining the relation between M (U), M (L) and d, wherein M (U) represents the maximum flow of the network G under an upper bound capacity vector U of X, M (L) represents the maximum flow of the network G under a lower bound capacity vector L of X, and d represents the capacity demand;
s3: searching for d-MP by solving a circulating flow problem;
s4: decomposing a search space into a number of disjoint subsets;
s5: the search continues in the decomposed subset.
2. The method according to claim 1, wherein step S1 specifically comprises: let the maximum capacity vector of the network G be W = (W) 1 ,W 2 ,…,W m ) Wherein W is i Represents an edge e i I is more than or equal to 1 and less than or equal to m, and m represents the total number of edges; let L i ＝max{d-M(W(0 i ) 0) (i is more than or equal to 1 and less than or equal to m) is an edge e i A lower bound capacity of (2), wherein W (0) i ) Is the capacity vector obtained when the ith component of W is set to 0, M (W (0) i ) Denotes that the network G is at W (0) i ) The lower maximum flow rate; order U i ＝min{W i D (i is more than or equal to 1 and less than or equal to m) is an edge e i The search space of d-MP is X = { X = (X) 1 ,x 2 ,…,x m )|L i ≤x i ≤U i ，1≤i≤m}＝{x|L≤x≤U}＝[L,U]Wherein, L = (L) 1 ,L 2 ,…,L m ) Lower bound capacity vector of X, U = (U) 1 ,U 2 ,…,U m ) An upper bound capacity vector of X, X i Is an edge e i X is a capacity vector.
3. The method according to claim 2, wherein step S2 specifically comprises: if M (U)<d, if the minimum path vector d-MP does not exist in the search space X, stopping the algorithm; if M (L) = d, according to the definition of d-MP, if M (L-0 (e) is satisfied for all i's of the ith component of L greater than 0 i ))<d, then L is d-MP, where 0 (e) i ) Is a unit vector, i.e. 0 (e) i ) The ith component of (c) is 1, and the other components are all 0, M (L-0 (e) i ) Denotes that the network G is at L-0 (e) i ) Stopping the algorithm at the lower maximum flow rate; otherwise, go to step S3.
4. The method according to claim 3, wherein step S3 specifically comprises: adding a new edge e pointing from the sink t to the source s in the initial network G m+1 Obtaining a new network G, let edge e m+1 Has a fixed value d, i.e. e m+1 Lower bound capacity L of m+1 And upper bound capacity U m+1 D, the lower bound capacity and the upper bound capacity of other edges are kept unchanged; solving the circular flow problem in the new network G by using the maximum flow algorithmIf no feasible circulating flow exists, the d-MP does not exist in the search space X, and the algorithm is stopped; otherwise, assume F d ＝(f 1 d ,f 2 d ,…,f m d ,f m+1 d ) Is the determined possible recycling flow if (f) 1 d ,f 2 d ,…,f m d ) Does not include a directional loop, then (f) 1 d ,f 2 d ,…,f m d ) Is a d-MP.
5. The method according to claim 4, wherein the step S4 specifically comprises: let [ L i ,U i ]Is an edge e i A set of capacities of, i.e. [ L ] i ,U i ]＝{x i |L i ≤x i ≤U i Therefore, search space X = [ L, U =]Is marked as [ L 1 ,U 1 ]×[L 2 ,U 2 ]×…×[L m ,U m ](ii) a Let E d ＝{e i |L i <f i d I is 1. Ltoreq. M for each e i ∈E d Set of [ L i ,U i ]Quilt f i d Partitioning into two disjoint subsets: [ L ] i ,f i d-1 ]And [ f i d ,U i ]I.e., [ L i ,U i ]＝[L i ,f i d-1 ]∪[f i d ,U i ]，[L i ,f i d-1 ]∩[f i d ,U i ]= Φ; order to
X is decomposed into q +1 disjoint subsets:
6. the method according to claim 5, wherein step S5 specifically comprises: each subset X obtained in step S4 (k) And (k is more than or equal to 1 and less than or equal to q) as a new search space, and sequentially turning to the step S2 to continue solving until all the subsets are searched.
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| CN102694395A (en) * | 2012-05-29 | 2012-09-26 | 河南理工大学 | Method for evaluating reliability of two terminals of power transmission network |
| US20170063694A1 (en) * | 2015-08-25 | 2017-03-02 | National Tsing Hua University | Reliability evaluating method for multi-state flow network with learning effect and non-transitory storage device thereof |
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| CN102694395A (en) * | 2012-05-29 | 2012-09-26 | 河南理工大学 | Method for evaluating reliability of two terminals of power transmission network |
| US20170063694A1 (en) * | 2015-08-25 | 2017-03-02 | National Tsing Hua University | Reliability evaluating method for multi-state flow network with learning effect and non-transitory storage device thereof |
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