CN115766494A - Multi-state network minimum path vector searching method based on feasible circulating flow - Google Patents

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

A Vector Search Method for Minimalist Paths in Multi-state Networks Based on Feasible Cyclic Flow

technical field

The invention belongs to the field of multi-state network reliability evaluation, and relates to a multi-state network minimal path vector search method based on feasible circular flow.

Background technique

As society relies more and more on various technological networks, there is an increasing need for reliable services in real life. Therefore, as a useful tool for performance evaluation of modern technology networks, reliability analysis is widely used in real-world networks such as computer networks, power transmission networks, manufacturing networks, logistics and supply chain networks. In traditional reliability analysis, the two-state network model has been successfully applied to analyze network performance. However, two-state network models have inherent limitations due to the assumption that network components have only two states: fully operational or fully failed. In the past decade, researchers have extended the two-state network model to a multi-state network model to meet the practical needs of network components with more than two levels of performance. Based on this, several different types of reliability methods are proposed, such as Monte Carlo Simulation (MCS), Universal Generating Function (UGF) method, Multivalued Decision Diagram (MDD) method, State Space Decomposition method (SSD) and minimal path vector (aka d-MP) method.

An important index to measure the performance of a multi-state network is the two-terminal reliability with a capacity requirement of d, denoted by _Rd , which is defined as the probability that the network can transport d units of traffic from the source point s to the sink point t. Scholars have proposed direct and indirect algorithms to approximate or accurately calculate the _Rd value. The direct algorithm first finds a capacity vector whose network capacity is not less than d, and then uses the capacity vector to divide the entire capacity vector space into acceptable sets and unacceptable sets. Finally, R _d is 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 first solves all d-MPs. In the second stage, the joint probability of all d-MPs is calculated by the inclusion-exclusion theorem or the disjoint sum method to obtain the value of _Rd .

The existing methods mainly use the implicit enumeration method to solve an NP-hard Diophantine equation to find d-MP, which has the outstanding problem of low computational efficiency. Therefore, there is an urgent need for a method that can improve the search for d-MP.

Contents of the invention

In view of this, the object of the present invention is to provide a multi-state network minimal path vector search method based on feasible cyclic flow, so as to improve the calculation efficiency of finding d-MP.

To achieve the above object, the present invention provides the following technical solutions:

A multi-state network minimal path vector search method based on feasible cyclic flow, comprising the following steps:

S1: Determine the search space X of the minimal path vector d-MP;

S2: Determine the relationship between M(U), M(L) and d, where M(U) represents the maximum flow of network G under the upper bound capacity vector U of X, and M(L) represents the lower bound of network G under X The maximum flow rate under the capacity vector L, d represents the capacity demand;

S3: Find d-MP by solving the circular flow problem;

S4: Decompose the search space into several disjoint subsets;

S5: Continue searching 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 ), where W _i represents the maximum capacity of edge e _i , 1≤i≤m, and m represents the edge The total number of ; let L _i =max{dM(W(0 _i )),0}(1≤i≤m) be the lower bound capacity of edge e _i , where W(0 _i ) is the ith component of W The capacity vector obtained when it is set to 0, M(W(0 _i )) represents the maximum flow of network G under W(0 _i ); Let U _i =min{W _i ,d}(1≤i≤m) as The upper bound capacity of edge e _i , then the search space of d-MP is X={x=(x ₁ ,x ₂ ,…,x _m )|L _{i ≤xi} ≤U _i , 1≤i≤m}= {x|L≤x≤U}=[L,U], where L=(L ₁ ,L ₂ ,…,L _m ) is the lower bound capacity vector of X, U=(U ₁ ,U ₂ ,…, U _m ) is the upper bound capacity vector of X, x _i is the capacity of edge e _i , and x is the capacity vector.

Further, step S2 specifically includes: if M(U)<d, there is no minimal path vector d-MP in the search space X, and the algorithm stops; if M(L)=d, according to the definition of d-MP, if for L All i whose i-th component is greater than 0 satisfy M(L-0(e _i ))<d, then L is d-MP, where 0(e _i ) is a unit vector, that is, the i-th of 0(e _i ) Each component is 1, and the other components are all 0. M(L-0(e _i )) represents the maximum flow of network G under L-0(e _i ), and the algorithm stops; otherwise, go to step S3.

Further, step S3 specifically includes: adding a new edge em ₊₁ from the sink point t to the source point s in the initial network G to obtain a new network G*, let the capacity of the edge _em+1 be a fixed value d, That is, the lower bound capacity L _{m +1} and upper bound capacity U _m+1 of e m+1 are both equal to d, and the lower bound capacity and upper bound capacity of other sides remain unchanged; use the maximum flow algorithm to solve the circular flow in the new network G* Problem, if there is no feasible cycle flow, then there is no d-MP in the search space X, and the algorithm stops; otherwise, suppose F ^d =(f ₁ ^d ,f ₂ ^d ,…,f _m ^d ,f _m+1 ^d ) is the obtained feasible circulation flow, if (f ₁ ^d ,f ₂ ^d ,…,f _m ^d ) does not contain a directed cycle, then (f ₁ ^d ,f ₂ ^d ,…,f _m ^d ) is a d- MP.

X被分解为q+1个不相交的子集：Further, step S4 specifically includes: let [L _i , U _i ] be the capacity set of edge e _i , that is, [L _i , U _i ]={ _xi |L _{i ≤xi} ≤U _i }, therefore, the search space X=[L,U] can be recorded as [L ₁ ,U ₁ ]×[L ₂ ,U ₂ ]×…×[L _m ,U _m ]; let E ^d ={e _i |L _i <f _i ^d ,1≤i≤m}, for each e _i ∈ E ^d , the set [L _i ,U _i ] can be divided into two disjoint subsets by f _i ^d : [L _i ,f _i ^d-1 ] And [f _i ^d , U _i ], namely [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; let

X is decomposed into q+1 disjoint subsets:

Further, step S5 specifically includes: taking each subset X ^(k) (1≤k≤q) obtained in step S4 as a new search space, turning to step S2 to continue solving until all subsets are searched.

The beneficial effect of the present invention is that: compared with the existing method, the present invention does not need to know all the minimal paths of the network, does not need to resort to the implicit enumeration method, does not need to solve the Diophantine equation, and does not generate repeated d- MP, and can improve the computational efficiency of finding d-MP.

Other advantages, objects and features of the present invention will be set forth in the following description to some extent, and to some extent, will be obvious to those skilled in the art based on the investigation and research below, or can be obtained from It is taught in the practice of the present invention. The objects and other advantages of the invention may be realized and attained by the following specification.

Description of drawings

In order to make the purpose of the present invention, technical solutions and advantages clearer, the present invention will be described in detail below in conjunction with the accompanying drawings, wherein:

Fig. 1 is the flow chart of the multi-state network minimal path vector search method based on the feasible cyclic flow of the present invention;

Fig. 2 is the network diagram of network G in the embodiment of the present invention;

Fig. 3 is a network diagram of the new network G ^* in the embodiment of the present invention.

Detailed ways

Embodiments of the present invention are described below through specific examples, and those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific implementation modes, and various modifications or changes can be made to the details in this specification based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the diagrams provided in the following embodiments are only schematically illustrating the basic concept of the present invention, and the following embodiments and the features in the embodiments can be combined with each other in the case of no conflict.

Please refer to Figures 1 to 2, the present invention provides a multi-state network minimal path vector search method based on feasible cyclic flow, as shown in Figure 1, which specifically includes the following steps:

1) Determine the search space of d-MP:

Given the maximum capacity vector W=(W ₁ ,W ₂ ,…,W _m ) of the network G, where W _i represents the maximum capacity of edge e _i , let L _i =max{dM(W(0 _i )),0 }(1≤i≤m) is the lower bound capacity of edge e _i , where W(0 _i ) is the capacity vector obtained when the i-th component of W is set to 0, and M(W(0 _i )) indicates that the network is The maximum flow rate under W(0 _i ), let U _i =min{W _i ,d}(1≤i≤m) be the upper bound capacity of edge e _i , for d-MP x=(x ₁ ,x ₂ , …, x _m ), L _i ≤ x _i ≤ U _i (1≤i≤m) is established, therefore, 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], where L=(L ₁ ,L ₂ ,…, L _m ) is called the lower bound capacity vector of X, and U=(U ₁ , U ₂ ,..., U _m ) is called the upper bound capacity vector of X.

2) Determine 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 ) is the minimum capacity vector of X, U=(U ₁ ,U ₂ ,…,U _m ) is the maximum capacity vector of X.

If M(U)<d, then for any x∈X, M(x)≤M(U)<d holds true, that is, by definition, X does not contain d-MP, then X is no longer discussed, and the algorithm stops.

If M(L)=d, then for any x>L, M(x)≥M(L)=d; meanwhile, for any x∈X, M(x)≤d, then M(x)= M(L)=d for any x>L. In this case, for any x>L, x is not d-MP because there exists at least one edge e _i such that M(x-0(e _i ))=M(L)=d, which is the same as d-MP definitions are contradictory. So you only need to verify the vector L. According to the definition of d-MP, if M(L-0(e _i ))<d, then L is d-MP, where 0(e _i ) is the unit vector, that is, 0(e _i )'s i-th component is 1, other components are 0, and the algorithm stops. Otherwise, proceed to the next step.

3) Find d-MP by solving the circular flow problem:

Add a new edge _em+1 from the sink point t to the source point s in the initial network G to obtain a new network G*, let the capacity of the edge _em+1 be a fixed value d, which is the lower bound of _em+1 The capacity L _m+1 and the upper bound capacity U _m+1 are both equal to d, and the lower bound capacity and upper bound capacity of other sides remain unchanged. The problem of feasible cyclic flow in network G* is to find a feasible flow vector x=(x ₁ ,x ₂ ,…,x _m ,x _m+1 ) that satisfies the following conditions:

L _i ≤ x _i ≤ U _i , e _i ∈ E∪{e _m+1 }

Among them, V is the set of nodes in network G, E is the set of edges in network G, Out(v) represents the set of all edges originating from point v, and In(v) represents the set of all edges pointing to point v.

Use the maximum flow algorithm to find a feasible cycle flow in G*, if there is no feasible cycle flow, then there is no d-MP in X, and the algorithm stops; otherwise, suppose F ^d =(f ₁ ^d ,f ₂ ^d ,…,f _m ^d ,f _m+1 ^d ) is the obtained feasible circulation flow, if (f ₁ ^d ,f ₂ ^d ,…,f _m ^d ) does not contain a directed cycle, then (f ₁ ^d ,f ₂ ^d ,… , f _m ^d ) is a d-MP.

4) Decompose the search space into several disjoint subsets:

集合X可以划分为q+1个不相交的子集，具体过程如下：Let [L _i , U _i ] be the capacity set of edge e _i , that is, [L _i , U _i ]={ _xi |L _{i ≤xi} ≤U _i }, therefore, the set X=[L,U] can be It is recorded as [L ₁ , U ₁ ]×[L ₂ ,U ₂ ]×…×[L _m ,U _m ]. Since the d-flow (f ₁ ^d ,f ₂ ^d ,...,f _m ^d ) comes from the set X=[L,U], for 1≤i≤m, L _i ≤f _i ^d ≤U _i holds; further Say, let E ^d ={E _i |L _i <f _i ^d ,1≤i≤m}, and E ^d is not an empty set (if E ^d is an empty set, then for 1≤i≤m, L _i =f _i ^d holds true, that is, L is a d-flow, which does not satisfy the condition M(L)<d). For each e _i ∈ E ^d , the set [L _i , U _i ] can be divided by f _i ^d into two disjoint subsets: [L _i , f _i ^d-1 ] and [f _i ^d , U _i ], namely [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. make

The set X can be divided into q+1 disjoint subsets, the specific process is as follows:

in,

The subsets X ⁽¹⁾ , X ⁽²⁾ , ..., X ^(q) , X ^(q+1) are disjoint.

5) Continue searching in the decomposed subset:

Take each subset X ^(k) (1≤k≤q) obtained in step 4) as a new search space, turn to step 2) and continue to solve until all subsets are searched.

The present invention is described in detail below in conjunction with specific embodiment:

A specific embodiment is shown in FIG. 2 . A distribution network is abstracted to obtain the network G in FIG. 2 . The network consists of 4 nodes and 5 transport edges, where node s represents the source point, t represents the sink point, and nodes 1 and 2 represent the transshipment point. Figure 2 shows the maximum capacity of each edge in the network, from Figure 2 we can know the maximum capacity vector W = (3,2,2,1,2).

Next, use the method of the present invention to calculate all polymorphic minimal roads in the transportation network, assuming that the demand level d=3.

According to the method steps of the present invention, the solution process is as follows:

1) Determine the search space of 3-MP:

For edge e ₁ , W(0 ₁ )=(0,2,2,1,2), then by the maximum flow algorithm, M(W(0 ₁ ))=2 can be obtained. Therefore, L ₁ =max{3-M(W(0 ₁ )), 0}=max{3-2,0}=1. Similarly, L ₂ =0, L ₃ =1, L ₄ =0, L ₅ =1 are calculated. When 1≤i≤5, U _i =min{W _i ,3}=W _i . Therefore, the search space of 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=(1,0,1,0,1) is the lower bound capacity vector of X , U=(3,2,2,1,2) is the upper bound capacity vector of X.

2) Determine the relationship between M(U), M(L) and 3:

X=[L,U]=[(1,0,1,0,1),(3,2,2,1,2)], where L=(1,0,1,0,1),U =(3,2,2,1,2)].

M(U)=4>3,

M(L)=1<3.

3) Find 3-MP by solving the circular flow problem:

Add a new edge e ₆ from the sink point t to the source point s in the initial network G to obtain a new network G*, as shown in Figure 3, let the capacity of the edge e ₆ be a fixed value 3, that is, e Both the lower bound capacity L ₆ and the upper bound capacity U ₆ of _m+1 are equal to 3, and the lower bound capacity and upper bound capacity of other sides keep the values in X unchanged.

Use the maximum flow algorithm to find a feasible cyclic flow in the network G*, and obtain a feasible cyclic flow (2,1,1,1,2,3), because (2,1,1,1,2) does not contain a directed circle, so (2,1,1,1,2) is 3-MP.

4) Decompose the search space 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) Continue searching in the decomposed subset:

Take the subsets X ⁽¹⁾ , X ⁽²⁾ , X ⁽³⁾ and X ⁽⁴⁾ as new search spaces respectively, turn to step 2) and continue to solve until all the subsets are searched. You end up with 4 3-MPs: (2,1,1,1,2), (1,2,1,0,2), (2,1,2,0,1) and (3,0,2 ,1,1).

Finally, it is noted that the above embodiments are only used to illustrate the technical solutions of the present invention without limitation. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be carried out Modifications or equivalent replacements, without departing from the spirit and scope of the technical solution, should be included in the scope of the claims of the present invention.

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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