CN103150481A - Computer-based realization method of cut set non-intersection in fault tree analysis - Google Patents

Abstract

The invention provides a computer-based realization method for acquiring the minimal cut sets of a top event through the Boole function of a fault tree and carrying out non-intersection treatment on the minimal cut sets. The basic method of the non-intersection treatment comprises the steps of: dividing one event into the sum of subevents and complementary events thereof, and carrying out a plurality of such operation on intersecting events until non-intersection events are obtained through division. The computer-based realization method is realized by mapping the minimal cut sets into one set of n-dimensional vectors and defining an operation, corresponding to the intersection operation among the minimal cut sets, in the set. The computer-based realization method has the following characteristics that (1) as long as the minimal cut sets are coded, the coded minimal cut sets can be input into a computer and then subjected to the non-intersection treatment automatically by the computer, so that the automation degree is high, and no manual intervention is needed; and (2) the minimal cut set with the fewest bottom events is preferentially selected as the non-intersection minimal cut set, so that the number of the minimal cut sets is reduced after non-intersection treatment, and the top event failure probability is simpler to calculate.

Description

Not friendship and computer implemented method of cut set in fault tree analysis

Technical field

The present invention relates in a kind of fault tree analysis the not computer implemented method of friendship of cut set

Background technology

Fault tree analysis is assessment and a kind of standard method that improves reliability and security, is used widely in the fields such as aerospace, nuclear energy, electronics, electric power, chemical industry, railway, transportation and insurance.Early stage Fussell-Vesely algorithm and Semanderes algorithm can adopt boolean to absorb the minimal cut set that strategy obtains top event, and this is being feasible and effective aspect fault tree qualitative analysis.Yet, carry out quantitative test to fault tree, must be at first to the not friendship processing of minimal cut set of top event.Yet, to minimal cut set not the appearance of friendship processing to scold theorem be " NP " difficult problem, its main cause is that the identical bottom event in different cut sets may cause operand acutely to increase in the friendship union of top event, i.e. so-called " shot array " problem.Although early stage not friendship fault tree synthesis function can reduce operand to a certain extent, adopt the long-pending sum theorem of not friendship type to realize that it is loaded down with trivial details and time-consuming that cut set is not shipped the process of calculation.

In international reliability in 1996 and maintainability annual meeting, Joanne B.Dugan and Stacy A.Doyle and Roslyn M.Sinnamon and John D.Andrews have provided respectively the thinking of utilizing BDD to carry out fault tree analysis, for " shot array " problem that solves in fault tree analysis provides new outlet.Due to expression on its room and time with process the high efficiency of Boolean function, BDD is widely used in the aspect such as the logic function checking in large-scale Design of Digital System, comprehensive and model detection and day by day comes into one's own.Many scholars have carried out the research based on the fault tree analysis of BDD, and some achievements have been obtained, but the fault tree analysis based on BDD still exists many deficiencies and open question, at first be that bottom event index order is larger on the impact of BDD scale, also do not have at present a kind of method can solve sequencing problem, the fault tree module also needs manual analysis; Next be achievement method versatility poor,, operand is large, can not be fully by computer realization.

Method

Summary of the invention

The present invention aims to provide a kind of Boole function by fault tree and obtains the top event minimal cut set, and to the minimal cut set computer implemented method of friendship processing not.

The present invention has following features:

(1) obtain the minimal cut set of top event by the Boole function after; As long as these minimal cut sets are encoded, the input computing machine can be realized the not friendship processing of minimal cut set automatically by computing machine, and automaticity is high, does not need manual intervention;

(2) preferentially select the minimum minimal cut set of bottom event as not handing over minimal cut set, reduced the not friendship processing number of minimal cut set afterwards, make the calculating of top event failure probability simpler.

Description of drawings

Fig. 1 does not hand over the block diagram of changing handling procedure.

Embodiment

1. not friendship disposal route and convergence thereof

Suppose A ₁, A ₂..., A _nBe n bottom event, the Boole function of fault tree can be expressed as

F＝B ₁UB ₂U…UB _N

Wherein

Be minimal cut set, N is the minimal cut set number,

Can be

Or

(

Opposite event), n ₁, n ₂... n _N∈ 1,2 ..., n}, F are top events. and replace the union of " ∪ " presentation of events with "+" when event is non-intersect.

For reducing operand and being convenient to fault analysis, minimal cut set should not make the quantity of cut set the least possible after the friendship processing, and in other words, the bottom event that consists of each cut set should be the least possible.Make n _m=min{n ₁, n ₂... n _N, wherein, m ∈ 1,2 ..., N} is got by the formula in theory of probability

$F=B_m+{B_m}^{\&prime;}\&cap;\underset{k<N,k\&NotEqual;m}{\&cup;}{B}_k=B_m+\underset{k<N,k\&NotEqual;m}{\&cup;}(B_k\&cap;{B_m}^{\&prime;})---\left(1\right)$

B after the equation right-hand member in one _k∩ B _m' be the common factor of some bottom events or its opposite event, if some bottom events event opposite to it occurs simultaneously, B _k∩ B _m'=φ (empty set), otherwise B _k∩ B _m' remain minimal cut set.After removing empty set

Remain the union of minimal cut set, be designated as v ₂, and note U ₁=B _m,

F＝U ₁+V ₂，

U wherein ₁, V ₂Mutually disjoint, V ₂The union of minimal cut set. said process has been completed once not friendship processing, U ₁Not hand over to change the event that processing obtains, V ₂It is the event that not yet realizes not handing over change.

After the processing of above-mentioned not friendship, if V ₁=φ, or V ₁Only be comprised of a minimal cut set, the friendship process does not finish; If V ₁The middle minimal cut set that comprises is more than 1, and repeating above-mentioned not friendship processing procedure can obtain

F＝U ₁+U ₂+V ₃

Usually, if r V after the friendship processing not _rThe minimal cut set that comprises is still more than 1,

F＝U ₁+U ₂+…U _r+V _r+1

V wherein _rThe union of minimal cut set, and U ₁, U ₂..., U _r, V _r+1Mutually disjoint.

Due to cut set be all shape as

The union of some events, so U ₁, U ₂..., U _rIn all comprise at least a shape as

Event, and shape as

The number maximum 2 of event ⁿIndividual.Each time not after the friendship processing, the cut set that mutually disjoints and concentrate increase at least a shape as

Event, therefore when enough F must be expressed as the mutually disjointing union of cut set greatly time of friendship number of processes not.

2. the computer realization of friendship processing not

Formula in equation (1) in applied probability

$F=B_m+\underset{k<N,k\&NotEqual;m}{\&cup;}(B_k\&cap;{B_m}^{\&prime;})$

$=B_m+\underset{k<N,k\&NotEqual;m}{\mathrm{\&cup;}}\left(\underset{j=1}{\overset{n_k}{\mathrm{\&cap;}}}{\stackrel{-}{A}}_{k_j}\mathrm{\&cap;}{\left(\underset{t=1}{\overset{n_m}{\mathrm{\&cap;}}}{\stackrel{-}{A}}_{m_t}\right)}^{\&prime;}\right)$

$=B_m+\underset{k<N,k\&NotEqual;m}{\mathrm{\&cup;}}\left(\underset{j=1}{\overset{n_k}{\mathrm{\&cap;}}}{\stackrel{-}{A}}_{k_j}\mathrm{\&cap;}{\left(\underset{t=1}{\overset{n_m}{\mathrm{\&cap;}}}{\stackrel{-}{A}}_{m_t}\right)}^{\&prime;}\right)$

$=B_m+\underset{k<N,k\&NotEqual;m}{\mathrm{\&cup;}}\underset{t=1}{\overset{n_m}{\mathrm{\&cup;}}}\left({{\stackrel{-}{A}}_{m_t}}^{\&prime;}\underset{j=1}{\overset{n_k}{\mathrm{\&cap;}}}{\stackrel{-}{A}}_{k_j}\right)---\left(2\right)$

Therefore the fundamental operation of friendship processing is not exactly the calculation of shipping of bottom event or its opposite event and minimal cut set.For the common factor in after equation (2) right-hand member one If have

Make

Can divide two kinds of situations this moment, namely ${{\stackrel{\&OverBar;}{A}}_{m_t}}^{\&prime;}={\stackrel{\&OverBar;}{A}}_{k_{\hat{j}}}$ Perhaps ${{\stackrel{\&OverBar;}{A}}_{m_t}}^{\&prime;}={{\stackrel{\&OverBar;}{A}}_{k_{\hat{j}}}}^{\&prime;}.$ The previous case ${{\stackrel{\&OverBar;}{A}}_{i_t}}^{\&prime;}\underset{j=1}{\overset{n_k}{\&cap;}}{\stackrel{\&OverBar;}{A}}_{i_j}=\underset{j=1}{\overset{n_k}{\&cap;}}{\stackrel{\&OverBar;}{A}}_{i_j},$ Latter event ${{\stackrel{\&OverBar;}{A}}_{i_t}}^{\&prime;}\underset{j=1}{\overset{n_k}{\&cap;}}{\stackrel{\&OverBar;}{A}}_{i_j}=\mathrm{\&phi;}.$ If

Consist of

Bottom event number ratio Many 1, namely bottom event or its opposite event are not all the factor event in minimal cut set, and the result of shipping calculation is the common factor of bottom event and minimal cut set.

Realize the calculation of shipping with above-mentioned characteristic of bottom event or its opposite event and minimal cut set for computer, define an event

Mapping to the n-dimensional vector space

As follows:

Wherein

j＝1，2，…，n

It is set

To set P={ (x ₁, x ₂..., x _n), x _i=1 ,-1,0, i=1,2 ..., the mapping one by one of n}, and

To arbitrarily

(x ₁, x ₂, x _n) ∈ P, defining a kind of computing, might as well be designated as

, make:

If x _i=0,

If x _i≠ 0, x _i=± 1 o'clock

If x _i≠ 0,

The time

Like this, the calculation of shipping of bottom event or its opposite event and minimal cut set just can be by mapping

With they be converted in P shape as

Vector and P in another vector

Computing.

Computing can also be generalized to (y arbitrarily ₁, y ₂..., y _n), (x ₁, x ₂..., x _n) ∈ P, i.e. definition

As seen two vectors in P

Operation result is a subset of vector in P.

4. friendship handling procedure not

The friendship handling procedure is not as follows:

1) obtain the Boole function of top event according to fault tree, obtain thus all minimal cut sets, and change into corresponding n-dimensional vector collection, be designated as V ₀, put i=0;

2)

At V _iSeek the n-dimensional vector that nonzero component is minimum, be designated as U _i

3) storage U _i, calculate

4) if V _i+1Middle vectorial number returns to 2 greater than 1);

5) if V _i+1In a vector is arranged, U _i+1=V _i+1Storage;

6) stop.

The block diagram of friendship handling procedure is not seen Fig. 1.

Claims (3)

1. not friendship and computer implemented method of cut set in a fault tree analysis, is characterized in that: supposition A ₁, A ₂..., A _nBe n the bottom event of top event F, the Boole function of fault tree can be expressed as F=B ₁UB ₂U ... UB _N, wherein N is the minimal cut set number,

Minimal cut set, n ₁, n ₂... n _N∈ 1,2 ..., n},

Can be Or

(

Opposite event), method comprises two steps, i.e. not friendship and computer realization; In the friendship process for reducing operand and being convenient to fault analysis, the bottom event of making every effort to consist of each cut set is the least possible, therefore makes n _m=min{n ₁, n ₂... n _N, wherein, m ∈ 1,2 ..., N}, and F is divided into

Realize the not event of friendship; Repeat above-mentioned not friendship processing procedure r time, until V _r+1The minimal cut set number that comprises is not more than 1,

F＝U ₁+U ₂+…U _r+V _r+1

If V _r+1≠ φ makes U _r+1=V _r+1, U ₁, U ₂..., U _r, U _r+1, otherwise U ₁, U ₂..., U _rDo not hand over minimal cut set for one group that is F.

2. not friendship and computer implemented method of cut set in fault tree analysis according to claim 1, its computer implemented method is: definition set

To set P={ (x ₁, x ₂..., x _n), x _i=1 ,-1,0, i=1,2 ..., the mapping of n}

As follows:

Wherein

J=1,2 ..., n, and to any

(x ₁, x ₂..., x _n) ∈ P, the definition computing

As follows:

If x _i=0,

If x _i≠ 0, x _i=± 1 o'clock

If x _i≠ 0,

The time

To any Definition

。

3. not friendship and computer implemented method of cut set in fault tree analysis according to claim 1, its computer implemented program is:

1) obtain the Boole function of top event according to fault tree, obtain thus all minimal cut sets, and change into corresponding n-dimensional vector collection, be designated as V ₀, put i=0;

2)

At V _iSeek the n-dimensional vector that nonzero component is minimum, be designated as U _i

3) storage U _i, calculate

4) if V _i+1Middle vectorial number returns to 2 greater than 1);

5) if V _i+1In a vector is arranged, U _i+1=V _i+1Storage;

6) stop.

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