CN107358046A - Consider more lifetime piece renewal reward theorem searching algorithms of structural dependence - Google Patents
Consider more lifetime piece renewal reward theorem searching algorithms of structural dependence Download PDFInfo
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Abstract
The present invention relates to aircraft engine maintenance technical field, specifically a kind of more lifetime piece renewal reward theorem searching algorithms for considering structural dependence, on the basis of economical dependence between considering aero-engine lifetime piece and structural dependence, the lowest cost is changed as optimization aim using lifetime piece in Life cycle, establishes more lifetime piece chance replacement problem Optimized models；The characteristics of for Optimized model, it is proposed that four kinds of model solution space reduction rules, the rule based on proposition propose a kind of searching algorithm based on reduction rules, and the algorithm can obtain the optimal solution of model.Finally using numerical experiment and application case to proposing that algorithm is assessed and verified.As a result show, propose that algorithm can realize the accurate solution of small-scale more lifetime piece chance replacement problems.
Description
Technical field：
The present invention relates to aircraft engine maintenance technical field, specifically a kind of more life-spans for considering structural dependence
Part renewal reward theorem searching algorithm.
Background technology：
Aero-engine is the complex equipment very high to security requirement, and maintenance cost is very high.Wherein, lifetime piece
Replacement cost is the important component of engine maintenance cost.By taking CFM56-5B engines as an example, a full set of lifetime piece is changed
Cost is about 2,600,000 dollars.Lifetime piece is that have the part for forcing life-span limitation in engine.In engine operation process, the longevity
Life part usage time does not allow to limit beyond its life-span, must be changed before life-span limitation is reached.Due to each lifetime piece
Material, structure, condition of work etc. have differences, and the life-span limitation of each lifetime piece differs greatly.Such as the increasing of CFM56-5B engines
Press rotor life is limited to 30000 flight cycles, and the High Pressure Turbine Rotor disk life-span is limited to 20000 flight cycles.It is if every
The lifetime piece for reaching the limitation of its life-span is only changed in secondary maintenance, then maintenance frequency can be caused too many, greatly increase the maintenance of engine
Cost；If whole of life part is changed in maintenance every time, a large amount of wastes of lifetime piece can be caused, can equally be made to airline
Into economic loss.As can be seen here, it is important to determine that the Optimal Replacement strategies of more lifetime pieces has to the maintenance cost for reducing engine
Meaning.
There is presently no a kind of effective derivation algorithm can adapt to asking in the more lifetime piece chance replacement problems of engine
Solution, and current research mainly surrounds the economical dependence expansion between part, it is related to existing structure between part
Journal of Sex Research is fewer.Engine generally employs Unit agent structure design.When changing some lifetime piece, it is necessary first to will start
Machine is decomposed into cell cube state, and certain assembly and disassembly cost can accordingly occur；Then need cell cube where the lifetime piece entering one
Step is decomposed, and the assembly and disassembly cost of the cell cube can accordingly occur, the assembly and disassembly cost of each unit body is often different.Current research
No matter thinking to change which lifetime piece, the fixation maintenance cost of generation is all identical, unit where not considering lifetime piece
Influence of the body to maintenance cost.
The content of the invention：
The present invention is for shortcoming and defect present in prior art, it is proposed that a kind of more life-spans for considering structural dependence
Part renewal reward theorem searching algorithm.
The present invention is reached by following measures：
A kind of more lifetime piece renewal reward theorem searching algorithms for considering structural dependence, it is characterised in that comprise the following steps：
Step 1：More lifetime piece chance replacement problem Optimized models are established, are specifically included：
Step 1-1：Define each parameter：Consideration engine entire life is Tlim, include the individual cell cubes of p (p >=1) and n (n >=p)
The chance replacement problem of individual lifetime piece, it is c that note engine, which fixes maintenance cost,b,0, its lifetime piece quantity independently of replacing；Note
K-th of cell cube of engine is Mk(k=1,2, p), it includes nkIndividual lifetime piece, thenIt assembles and disassembles cost
For cb,k；NoteFor MkI-thk(ik=1,2nk) individual lifetime piece, its life-span is limited toCost isHair
The motivation currently used time is designated as T,Usage time is designated asResidual life is designated asObviouslyThe minimum residual life for remembering all lifetime pieces is tres,min(T), i.e.,With T increase,Corresponding increase,It is corresponding to reduce, whenWhen,It must be changed, be changedWhen need to be by where the lifetime piece
Cell cube MkDecompose, generation unit body assembly and disassembly cost cb,k, therefore change lifetime pieceCaused cost is After replacing,Zero, restarts to accumulate, in order to reduce the engine maintenance number in Life cycle,
It can also in advance be changed when changing other lifetime pieces, can so save engine and fix maintenance cost and cell cube dress
This is splitted into, but certain lifetime piece can be caused to waste, engine is in TlimInterior maintenance frequency is designated as m, and all previous maintenance opportunity is designated as
Tj(j=1,2, m) and, m is different, TjDifference, the lifetime piece changed every time during maintenance are different, then the full Life Cycle of engine
Lifetime piece totle drilling cost C in phase is just different；Solve more lifetime piece chance replacement problems and be just to determine more lifetime piece renewal reward theorems, i.e.,
Determine m, Tj(j=1,2, m), every time maintenance when the lifetime piece changed so that C is minimum, its can using formalization representation as
Formula (1)：
In formula,The solution vector formed for decision variable；N
Represent natural number；
The solution space that all solution vectors are formed is represented,When representing jth time maintenanceWhether change, change value be 1, otherwise for
0；Calculating such as formula (2) shown in：
In formula, T0=0,RepresentInitial usage time；
Step 1-2：More lifetime piece chance replacement problems belong to combinatorial optimization problem, the solution space scale such as formula of the problem
(3) shown in：
Solution space expression is tree, and the root node of tree represents original state (T=T0=0), it is designated as N0；N0Any son
Node represents the 1st maintenance (T=T1), it is designated as N1；By that analogy；Solution space tree removes N0Outside each node represent
Single Maintenance, any node N corresponding to jth time maintenancejComprising decision variable be maintenance opportunity TjAnd all lifetime pieces are changed
SituationM corresponds to from N0To the length in the path of leaf node, from N0Correspond to the problem to the path of any leaf node
One solution；
Step 2：The searching algorithm based on reduction rules is performed, specifically includes herein below：
Step 2-1：Optimized model about rule of simplification, including feasibility reduction rules, lifetime piece change reduction rules,
Maintenance opportunity determines reduction rules, cost reduction rules；
Step 2-2：The searching algorithm based on reduction rules is performed, wherein feasibility reduction rules, lifetime piece changes yojan
Rule and maintenance opportunity determine create-rule of the reduction rules as solution space tree child nodes, and cost reduction rules are used as and solved
Space tree solution procedure child nodes search stopping rules；
Assuming that in certain maintenance, lifetime pieceWith(k≠k*) be required to change, then cell cube MkWithAll
Need to be decomposed, then during solution, problem can be effectively reduced by the two cell cubes are merged into a cell cube carrying out processing
Solution space, improve solution efficiency, above-mentioned four classes reduction rules also set up to the combining unit body, and solution space tree is given below
Child node generation method, for more clearly pine torch node generation method, it is specified below：In maintenance opportunity TjAt the moment, arrive
The collection of longevity lifetime piece is combined intoThe collection of cell cube is combined into where to longevity lifetime pieceIn not to the collection of longevity lifetime piece
It is combined intoWhole of life part is not combined into the cell cube collection in longevityInterim Replacing Scheme lifetime piece collection is combined into RmustWith
Rchioce, child node collection is combined into R.Determine arbitrary node NjChild node Nj+1The step of set, is as follows：
Step1 makesBy SRESElement according to suitable from small to large
Sequence forms a line, and determines that reduction rules can be defined below gathering respectively by maintenance opportunity：
Then
Step2 will gatherIn the residual life of lifetime piece be ranked up according to order from small to large, remember respectively
For r0,r2,···,rq, thenInitialize l=0 (0≤l≤q), noteForMiddle residual life is small
In equal to rlLifetime piece set, i.e.,
Step3 determining unit body setThe middle collection for changing lifetime piece is combined into：
Step4 determines the set not included to the cell cube of longevity lifetime piece,
Being numbered by cell cube willIn cell cube form a line, then cell cube number isInitialize b=1,
Remember b be fromIn the number of cell cube selected,
Step5 fromMiddle b cell cube of selection makes as hair object is torn openThen shareKind side
Case, these schemes are formed a line, are designated as respectivelyInitialize c=1, PcFor c kind schemes, note scheme PcIt is corresponding
Cell cube collection be combined into
Step6 remembersForThe set of middle whole of life part,ForResidual life is most short in middle cell cube
Lifetime piece set,WillIn the residual life of lifetime piece be ranked up according to order from small to large, point
R is not designated as it1,r2,···,rM, thenInitialize m=0, noteForMiddle residual life is less than or equal to rm
Lifetime piece set, i.e.,
Step7 determines the lifetime piece Replacing Scheme of the cell cube not comprising lifetime piece：It can determine that the longevity
Order part Replacing Scheme child node NS：Change RchioceAnd RmustIn lifetime piece,
Step8 judges whetherMeet, then continue；
Otherwise, by Replacing Scheme node NSAdd in Replacing Scheme child node set R, order
Step9 makes m=m+1, judges whether to meet m ＞ M, if so, then continue, otherwise jump to Step7,
Step10 makes c=c+1, judges whetherIf set up, continue, otherwise jump to Step6,
Step11 makes b=b+1, judges whether b ＞ B；If set up, continue；Otherwise Step5 is jumped to,
Step12 makes l=l+1, judges whether l ＞ q；If set up, algorithm terminates, and returns to Replacing Scheme child node set
R；Otherwise, Step3 is jumped to；
According to child node generation method, the derivation algorithm of more lifetime piece chance replacement problems is proposed, is comprised the following steps that.
Step1 initialization optimal values Cmin, optimal leaf segment point set Smin, movable joint point list SL, root node N0, CminIt is set to one
Individual larger number, SminFor sky, SLFor sky；
Step2 judges whether tres,min(T0)≥Tlim.If it is, Cmin=0, Smin={ N0, algorithm terminates；If not,
By N0Add SL；
Step3 judges SLWhether it is empty.If it is, algorithm terminates；If not, specify SLIn last add node
For present node Nc；
Step4 is using child node generation method generation present node NcChild node, child node forms set R；
Step5 judges each child node in R successivelyWhether meetIf meet the node from R
Remove, otherwise continue；
Step6 judges whether the whole of life part in R in each node can use and arrives T successivelylim, if it is, according to formula
(2-1) calculates target function value corresponding to the element, by target function value and CminIt is compared, updates CminAnd Smin, and from R
In remove the element；Otherwise by remaining node to slip-knot point set SLIn, return to step 3；
The algorithm has carried out a large amount of yojan to solution space tree it can be seen from the solution procedure of algorithm, and one surely
Enough get the optimal solution of the more lifetime piece chance replacement problems of engine.
Feasibility reduction rules are described in step 2-1 of the present invention：To more lifetime piece chances shown in formula (1)
Replacement problem, if Nj-1For arbitrary node in solution space tree, if the node has child node Nj, then haveMeanwhile if
So
Following two lemma can easily be drawn by the constraints of formula (1)：
Lemma 1：With(i ≠ j) is cell cube MkIn two same type lifetime pieces, if in T=TaWhen, meet
ConditionThen as T ＞ TaWhen, it is all to be adapted toRenewal reward theorem be applied to
Lemma 2：To more lifetime piece chance replacement problems shown in formula (1), it is assumed that there are two feasible solution pathsWithIf existence time point Ta
So that(k=1,2, p；ik=1,2, nk), then pathAnd the feasible solution of the problem.
Lifetime piece described in step 2-1 of the present invention, which changes reduction rules, includes lifetime piece replacing reduction rules 1 and life-span
Part changes reduction rules 2, and wherein lifetime piece replacing reduction rules 1 are：More lifetime piece chances shown in formula (1) are changed
Problem, if NjFor arbitrary node in solution space tree, in same cell cube MkIn, if lifetime piece meets
So certainly exist not comprising node NjFeasible solution, and the feasible solution however be inferior to include node NjFeasible solution；The rule is said
Bright, when carrying out lifetime piece replacing to engine, in a certain cell cube, the replacing of lifetime piece should be according to residual life from small
To big order.
Lifetime piece described in step 2-1 of the present invention, which changes reduction rules, includes the lifetime piece replacing He of reduction rules 1
Lifetime piece changes reduction rules 2, and wherein lifetime piece replacing reduction rules 2 are：To more lifetime piece chances shown in formula (1)
Replacement problem, if NjFor arbitrary node in solution space tree, in same cell cube MkIn, if there is
So certainly exist not comprising node NjFeasible solution, and the feasible solution is not inferior to include node NjFeasible solution；By lifetime piece more
Change reduction rules 2 to understand, when repairing engine, in same cell cube Mk(k=1,2, p) in, arrive the longevity
Lifetime piece must be changed；Meanwhile the lifetime piece being replaced in cell cube needs to meet condition：
Maintenance opportunity determines reduction rules
Maintenance opportunity determines that reduction rules are described in step 2-1 of the present invention：To more lifetime piece machines shown in formula (1)
Meeting replacement problem, if NjFor arbitrary node in solution space tree, if there is So certainly exist not comprising node NjFeasible solution, and the feasible solution however be inferior to wrap
N containing nodejFeasible solution, reduction rules are determined from maintenance opportunity, work as NjIt is more lifetime piece chance replacement problem solution space trees
Any node, corresponding maintenance opportunity is Tj.If NjChild node N be presentj+1, then Nj+1Maintenance opportunity can be defined as Tj+1
=Tj+tRes, min(Tj)。
Cost reduction rules are described in step 2-1 of the present invention：NaIt is a node (nonleaf node) in solution space tree,
It is all to include node NaObject function solution assessment of cost lower limit Cl(Na) can be calculated by formula (4)：
In formula--- rounding is integer；Cost reduction rules：To more lifetime piece chance replacement problems shown in formula (1),
If CminFor object function global optimum, NaFor a node (nonleaf node) in solution space, if Cl(Na) ＞ Cmin, then
Include node NaSolution be not optimal solution；By the rule, if the assessment of cost floor value C of the solution comprising present nodel
(Na) exceed object function globally optimal solution Cmin, then the searching route comprising present node should be terminated.
The present invention on the basis of economical dependence between considering aero-engine lifetime piece and structural dependence, with
It is optimization aim that lifetime piece, which changes the lowest cost, in Life cycle, establishes more lifetime piece chance replacement problem optimization moulds
Type；The characteristics of for Optimized model, it is proposed that four kinds of model solution space reduction rules, the rule based on proposition propose that one kind is based on
The searching algorithm of reduction rules, the algorithm can obtain the optimal solution of model, can realize that small-scale more lifetime piece chances are changed
The accurate solution of problem.
Brief description of the drawings：
Accompanying drawing 1 is solution space tree schematic diagram in the present invention.
Accompanying drawing 2 is that lifetime piece changes the exemplary plot of reduction rules 1 in the present invention.
Accompanying drawing 3 is that lifetime piece changes the exemplary plot of reduction rules 2 in the present invention.
Accompanying drawing 4 be in the present invention repair opportunity determine reduction rules exemplary plot.
Embodiment：
The present invention is further illustrated below in conjunction with the accompanying drawings.
The present invention considers the structural dependence and economical dependence between more lifetime pieces, it is proposed that one kind considers structure
More lifetime piece renewal reward theorem searching algorithms of correlation, specifically include herein below：
First, more lifetime piece chance replacement problem Optimized models are established, consideration engine entire life is Tlim, comprising p (p >=
1) the chance replacement problem of individual cell cube and n (n >=p) individual lifetime piece.It is c to remember that engine fixes maintenance costb,0, its independently of
The lifetime piece quantity of replacing.K-th of cell cube for remembering engine is Mk(k=1,2, p), it includes nkIndividual lifetime piece, thenIt is c that it, which assembles and disassembles cost,b,k.NoteFor MkI-thk(ik=1,2nk) individual lifetime piece, its life-span is limited toCost isThe engine currently used time is designated as T,Usage time is designated asResidual life is designated asObviouslyThe minimum residual life for remembering all lifetime pieces is tres,min(T), i.e.,With T increase,Corresponding increase,It is corresponding to reduce.WhenWhen,It must be changed.ChangeWhen need to be by where the lifetime piece
Cell cube MkDecompose, generation unit body assembly and disassembly cost cb,k, therefore change lifetime pieceCaused cost is After replacing,Zero, restarts to accumulate.In order to reduce the engine maintenance number in Life cycle,
It can also in advance be changed when changing other lifetime pieces, can so save engine and fix maintenance cost and cell cube dress
This is splitted into, but certain lifetime piece can be caused to waste.Engine is in TlimInterior maintenance frequency is designated as m, and all previous maintenance opportunity is designated as
Tj(j=1,2, m).M is different, TjDifference, the lifetime piece changed every time during maintenance are different, then the full Life Cycle of engine
Lifetime piece totle drilling cost C in phase is just different.
Solve more lifetime piece chance replacement problems and be just to determine more lifetime piece renewal reward theorems, that is, determine m, Tj(j=1,
2, m), every time maintenance when the lifetime piece changed so that C is minimum, and it can be using formalization representation as formula (1).
In formula,The solution vector formed for decision variable；
N represents natural number；
The solution space that all solution vectors are formed is represented,When representing jth time maintenanceWhether change, change value be 1, otherwise for
0；Calculating such as formula (2) shown in.
In formula, T0=0,RepresentInitial usage time.
Solution space analysis is carried out, more lifetime piece chance replacement problems belong to combinatorial optimization problem.According to each decision variable
Definition, is not difficult to obtain the solution space scale of the problem, as shown in formula (3).
As can be seen that solution Space Scale is extremely huge, major influence factors are cell cube number p, lifetime piece sum
N, and engine entire life Tlim.It is even if empty by complete ergodic solutions when engine entire life and less big lifetime piece quantity
Between obtain optimal solution be also nearly impossible.
It is seen that all previous maintenance of engine has natural time sequencing.Therefore, problem solution space being capable of natural table
Up to for tree, as shown in Figure 1.The root node of tree represents original state (T=T0=0), it is designated as N0；N0Any child node represent the 1st
Secondary maintenance (T=T1), it is designated as N1；By that analogy.Solution space tree removes N0Outside each node represent Single Maintenance.Jth
Any node N corresponding to secondary maintenancejComprising decision variable be maintenance opportunity TjAnd all lifetime pieces change situationM pairs
Ying Yucong N0To the length in the path of leaf node.From N0Correspond to a solution of the problem to the path of any leaf node.
The searching algorithm based on reduction rules is performed, in order to improve the validity of search, the present invention proposes a kind of based on about
The searching algorithm of simple rule.The algorithm mainly reduces the scale of solution space by following three aspects：(1) feasible solution is only traveled through；
(2) it is determined that some node child node when, the child node of optimal solution may be obtained by only retaining；(3) terminate in time to non-optimal
The search of node.Optimized model reduction rules first, wherein feasibility reduction rules：To more lifetime pieces shown in formula (1)
Chance replacement problem, if Nj-1For arbitrary node in solution space tree, if the node has child node Nj, then haveMeanwhile if
So
Following two lemma can easily be drawn by the constraints of formula (1)：
Lemma 1：With(i ≠ j) is cell cube MkIn two same type lifetime pieces, if in T=TaWhen, meet
ConditionThen as T ＞ TaWhen, it is all to be adapted toRenewal reward theorem be applied to
Lemma 2：To more lifetime piece chance replacement problems shown in formula (1), it is assumed that there are two feasible solution pathsWithIf existence time point Ta
So that(k=1,2, p；ik=1,2, nk), then pathAnd the feasible solution of the problem.(1) lifetime piece changes reduction rules：Including
Lifetime piece changes reduction rules 1：To more lifetime piece chance replacement problems shown in formula (1), if NjArbitrarily to be saved in solution space tree
Point, in same cell cube MkIn, if lifetime piece meets So certainly exist not comprising node NjFeasible solution, and should
Feasible solution however be inferior to include node NjFeasible solution.
The rule declaration, when carrying out lifetime piece replacing to engine, in a certain cell cube, the replacing of lifetime piece
Should be according to the order of residual life from small to large.Lifetime piece changes reduction rules 2：To more lifetime pieces shown in formula (1)
Chance replacement problem, if NjFor arbitrary node in solution space tree, in same cell cube MkIn, if there is
So certainly exist not comprising node NjFeasible solution, and the feasible solution is not inferior to include node NjFeasible solution.
Reduction rules 2 are changed from lifetime piece, when being repaired to engine, in same cell cube Mk(k=1,
2, p) in, the lifetime piece to the longevity must be changed；Meanwhile the lifetime piece being replaced in cell cube needs to meet bar
Part：The rule can
Illustrated with Fig. 3.
(3) maintenance opportunity determines reduction rules：To more lifetime piece chance replacement problems shown in formula (1), if NjIt is empty for solution
Between set in arbitrary node, if there isThat
Certainly exist not comprising node NjFeasible solution, and the feasible solution however be inferior to include node NjFeasible solution.
Reduction rules are determined from maintenance opportunity, work as NjIt is any section of more lifetime piece chance replacement problem solution space trees
Point, corresponding maintenance opportunity are Tj.If NjChild node N be presentj+1, then Nj+1Maintenance opportunity can be defined as Tj+1=Tj+
tres,min(Tj)。
(4) cost reduction rules, NaIt is a node (nonleaf node) in solution space tree, it is all to include node NaMesh
The assessment of cost lower limit C of scalar functions solutionl(Na) can be calculated by formula (4)：
In formula--- rounding is integer.
Cost reduction rules：To more lifetime piece chance replacement problems shown in formula (1), if CminFor the object function overall situation most
The figure of merit, NaFor a node (nonleaf node) in solution space, if Cl(Na) ＞ Cmin, then include node NaSolution be not optimal
Solution.
The rule is clearly what is set up.By the rule, if the assessment of cost lower bound of the solution comprising present node
Value Cl(Na) exceed object function globally optimal solution Cmin, then the searching route comprising present node should be terminated.
The searching algorithm based on reduction rules is performed, it is the derivation algorithm based on reduction rules：Feasibility reduction rules,
Lifetime piece changes reduction rules and maintenance opportunity determines create-rule of the reduction rules as solution space tree child nodes, and cost is about
Simple rule searches stopping rules as in solution space tree solution procedure child nodes.
At present when being repaired to engine, it is necessary first to be broken down into cell cube, then carry out follow-up dimension again
Repair activity.Assuming that in certain maintenance, lifetime pieceWithIt is required to change, then cell cube MkWithAll
Need to be decomposed, then during solution, problem can be effectively reduced by the two cell cubes are merged into a cell cube carrying out processing
Solution space, improve solution efficiency.Above-mentioned four classes reduction rules are also set up to the combining unit body.
The child node generation method of solution space tree is given below.For more clearly pine torch node generation method, make such as
Lower regulation：In maintenance opportunity TjMoment, the collection to longevity lifetime piece are combined intoThe collection of cell cube is combined into where to longevity lifetime piece In be not combined into the collection of longevity lifetime pieceWhole of life part is not combined into the cell cube collection in longevityFace
When Replacing Scheme lifetime piece collection be combined into RmustAnd Rchioce, child node collection is combined into R.Determine arbitrary node NjChild node Nj+1Set
The step of it is as follows：
Step1 makesBy SRESElement according to suitable from small to large
Sequence forms a line, and determines that reduction rules can be defined below gathering respectively by maintenance opportunity：
Then
Step2 will gatherIn the residual life of lifetime piece be ranked up according to order from small to large, remember respectively
For r0,r2,···,rq, thenInitialize l=0 (0≤l≤q), noteForMiddle residual life is less than
Equal to rlLifetime piece set, i.e.,
Step3 determining unit body setThe middle collection for changing lifetime piece is combined into：
Step4 determines the set not included to the cell cube of longevity lifetime piece,
Being numbered by cell cube willIn cell cube form a line, then cell cube number isInitialize b=1,
Remember b be fromIn the number of cell cube selected.
Step5 fromMiddle b cell cube of selection makes as hair object is torn openThen shareKind side
Case, these schemes are formed a line, are designated as respectivelyInitialize c=1, PcFor c kind schemes, note scheme PcIt is corresponding
Cell cube collection be combined into
Step6 remembersForThe set of middle whole of life part,ForResidual life is most short in middle cell cube
Lifetime piece set,WillIn the residual life of lifetime piece be ranked up according to order from small to large, point
R is not designated as it1,r2,···,rM, thenInitialize m=0, noteForMiddle residual life is less than or equal to
rmLifetime piece set, i.e.,
Step7 determines the lifetime piece Replacing Scheme of the cell cube not comprising lifetime piece：It can determine that the longevity
Order part Replacing Scheme child node NS：Change RchioceAnd RmustIn lifetime piece.
Step8 judges whetherMeet, then continue；
Otherwise, by Replacing Scheme node NSAdd in Replacing Scheme child node set R, order
Step9 makes m=m+1, judges whether to meet m ＞ M, if so, then continue, otherwise jump to Step7.
Step10 makes c=c+1, judges whetherIf set up, continue, otherwise jump to Step6.
Step11 makes b=b+1, judges whether b ＞ B；If set up, continue；Otherwise Step5 is jumped to.
Step12 makes l=l+1, judges whether l ＞ q；If set up, algorithm terminates, and returns to Replacing Scheme child node set
R；Otherwise, Step3 is jumped to.
According to child node generation method, the derivation algorithm of more lifetime piece chance replacement problems is proposed, is comprised the following steps that.
Step1 initialization optimal values Cmin, optimal leaf segment point set Smin, movable joint point list SL, root node N0, CminIt is set to one
Individual larger number, SminFor sky, SLFor sky.
Step2 judges whether tres,min(T0)≥Tlim.If it is, Cmin=0, Smin={ N0, algorithm terminates；If not,
By N0Add SL。
Step3 judges SLWhether it is empty.If it is, algorithm terminates；If not, specify SLIn last add node
For present node Nc。
Step4 is using child node generation method generation present node NcChild node, child node forms set R.
Step5 judges each child node in R successivelyWhether meetIf meet the node from R
Remove, otherwise continue.
Step6 judges whether the whole of life part in R in each node can use and arrives T successivelylim, if it is, according to formula
(2-1) calculates target function value corresponding to the element, by target function value and CminIt is compared, updates CminAnd Smin, and from R
In remove the element；Otherwise by remaining node to slip-knot point set SLIn, return to step 3.
The algorithm has carried out a large amount of yojan to solution space tree it can be seen from the solution procedure of algorithm, and one surely
Enough get the optimal solution of the more lifetime piece chance replacement problems of engine.
Proof of algorithm is carried out below, by numerical experiment, using the method for generating more lifetime piece chance replacement problems at random
To proposing that algorithm is assessed.Make cb,0~U (100000,300000), cb,k~U (4500,20000),P=1, (Tlim,n)∈
60000+10000k | k=0,1,2,3 } × 5+k | k=0,1 ..., 7 }.To any (Tlim, n) generation 10 is asked at random
Topic.Algorithm is realized using Java.It is respectively adopted on a common computer and proposes that algorithm solves to every group of problem.Record calculation
The mean consumption time t that method solves.Table is experimental result.
As seen from table, with Life cycle TlimWith lifetime piece number n increase, the time of problem solving also increases therewith
It is long, when problem scale is larger, internal memory occurs and overflows.For example, work as TlimWhen=60000, the solution time increases with n increase
Long, as n=11, internal memory overflows.
The experimental result of table 1
Below by taking certain engine of airline as an example, propose that algorithm is changed to more lifetime piece chances using the present invention and ask
Topic is solved.The model engine includes 20 lifetime pieces, and the cost of more transducer set lifetime piece is about 2,600,000 dollars.Table is
The model engine contains the cell cube inventory of lifetime piece, and table is the lifetime piece inventory of the model engine.If cb,0=160000
Dollar, Tlim=60000 flight cycles.Table is solving result.
The common elapsed time 14.64min of inventive algorithm, achieves 1 optimal solution, corresponding target function value is
8142147 dollars.Application case shows that the present invention proposes that algorithm is changed suitable for more lifetime piece chances of aero-engine and asked
Topic.
The cell cube inventory of table 2
The lifetime piece inventory of table 3
The solving result of table 4
The present invention on the basis of economical dependence between considering aero-engine lifetime piece and structural dependence, with
It is optimization aim that lifetime piece, which changes the lowest cost, in Life cycle, establishes more lifetime piece chance replacement problem optimization moulds
Type；The characteristics of for Optimized model, it is proposed that four kinds of model solution space reduction rules, the rule based on proposition propose that one kind is based on
The searching algorithm of reduction rules, the algorithm can obtain the optimal solution of model.Finally using numerical experiment and application case to carrying
Go out algorithm to be assessed and verified.As a result show, propose that algorithm can realize small-scale more lifetime piece chance replacement problems
It is accurate to solve.
Claims (6)
1. a kind of more lifetime piece renewal reward theorem searching algorithms for considering structural dependence, it is characterised in that comprise the following steps：
Step 1：More lifetime piece chance replacement problem Optimized models are established, are specifically included：
Step 1-1：Define each parameter：Consideration engine entire life is Tlim, include the individual cell cubes of p (p >=1) and n (n >=p) individual longevity
The chance replacement problem of part is ordered, it is c that note engine, which fixes maintenance cost,b,0, its lifetime piece quantity independently of replacing；Note is started
K-th of cell cube of machine is Mk(k=1,2, p), it includes nkIndividual lifetime piece, thenIt assembles and disassembles cost
cb,k；NoteFor MkI-thk(ik=1,2nk) individual lifetime piece, its life-span is limited toCost isStart
The machine currently used time is designated as T,Usage time is designated asResidual life is designated asObviouslyThe minimum residual life for remembering all lifetime pieces is tres,min(T), i.e.,With T increase,Corresponding increase,It is corresponding to reduce, whenWhen,It must be changed, be changedWhen need to be by where the lifetime piece
Cell cube MkDecompose, generation unit body assembly and disassembly cost cb,k, therefore change lifetime pieceCaused cost is After replacing,Zero, restarts to accumulate, in order to reduce the engine maintenance number in Life cycle,
It can also in advance be changed when changing other lifetime pieces, can so save engine and fix maintenance cost and cell cube dress
This is splitted into, but certain lifetime piece can be caused to waste, engine is in TlimInterior maintenance frequency is designated as m, and all previous maintenance opportunity is designated as
Tj(j=1,2, m) and, m is different, TjDifference, the lifetime piece changed every time during maintenance are different, then the full Life Cycle of engine
Lifetime piece totle drilling cost C in phase is just different；
Solve more lifetime piece chance replacement problems and be just to determine more lifetime piece renewal reward theorems, that is, determine m, Tj(j=1,2,
The lifetime piece changed when m), repairing every time so that C is minimum, and it can be using formalization representation as formula (1)：
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<mi> </mi>
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In formula,The solution vector formed for decision variable；N represents nature
Number；
The solution space that all solution vectors are formed is represented,When representing jth time maintenanceWhether change, change value be 1, otherwise for
0；Calculating such as formula (2) shown in：
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<mi>e</mi>
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In formula, T0=0,RepresentInitial usage time；
Step 1-2：More lifetime piece chance replacement problems belong to combinatorial optimization problem, solution space scale such as formula (3) institute of the problem
Show：
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<mn>0</mn>
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<msup>
<mn>2</mn>
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<mi>n</mi>
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Solution space expression is tree, and the root node of tree represents original state (T=T0=0), it is designated as N0；N0Any child node
Represent the 1st maintenance (T=T1), it is designated as N1；By that analogy；Solution space tree removes N0Outside each node represent once
Maintenance, any node N corresponding to jth time maintenancejComprising decision variable be maintenance opportunity TjAnd all lifetime pieces change situationM corresponds to from N0To the length in the path of leaf node, from N0Correspond to one of the problem to the path of any leaf node
Solution；
Step 2：The searching algorithm based on reduction rules is performed, specifically includes herein below：
Step 2-1：Optimized model about rule of simplification, reduction rules, maintenance are changed including feasibility reduction rules, lifetime piece
Opportunity determines reduction rules, cost reduction rules；
Step 2-2：The searching algorithm based on reduction rules is performed, wherein feasibility reduction rules, lifetime piece changes reduction rules
Create-rule of the reduction rules as solution space tree child nodes is determined with maintenance opportunity, cost reduction rules are used as in solution space
Set solution procedure child nodes and search stopping rules；
Assuming that in certain maintenance, lifetime pieceWithIt is required to change, then cell cube MkWithIt is required for
Decomposed, then during solution, the two cell cubes are merged into a cell cube and carry out handling the solution that can effectively reduce problem
Space, solution efficiency is improved, above-mentioned four classes reduction rules are also set up to the combining unit body, and the son section of solution space tree is given below
Point generation method, for more clearly pine torch node generation method, is specified below：In maintenance opportunity TjMoment, to the longevity in longevity
The collection of life part is combined intoThe collection of cell cube is combined into where to longevity lifetime piece In be not combined into the collection of longevity lifetime pieceWhole of life part is not combined into the cell cube collection in longevityInterim Replacing Scheme lifetime piece collection is combined into RmustWith
Rchioce, child node collection is combined into R.Determine arbitrary node NjChild node Nj+1The step of set, is as follows：
Step1 makesBy SRESElement according to from small to large order arrange
Cheng Yilie, determine that reduction rules can be defined below gathering respectively by maintenance opportunity：
Then
Step2 will gatherIn the residual life of lifetime piece be ranked up according to order from small to large, be designated as r respectively0,
r2,···,rq, thenInitialize l=0 (0≤l≤q), noteForMiddle residual life is less than or equal to
rlLifetime piece set, i.e.,
Step3 determining unit body setThe middle collection for changing lifetime piece is combined into：
Step4 determines the set not included to the cell cube of longevity lifetime piece,By list
First body numbering willIn cell cube form a line, then cell cube number isB=1 is initialized, note b is
FromIn the number of cell cube selected,
Step5 fromMiddle b cell cube of selection makes as hair object is torn openThen shareKind scheme, will
These schemes form a line, and are designated as respectivelyInitialize c=1, PcFor c kind schemes, note scheme PcCorresponding list
First body collection is combined into
Step6 remembersForThe set of middle whole of life part,ForThe most short lifetime piece of residual life in middle cell cube
Set,WillIn the residual life of lifetime piece be ranked up according to order from small to large, be designated as respectively
r1,r2,···,rM, thenInitialize m=0, noteForMiddle residual life is less than or equal to rmLife-span
The set of part, i.e.,
Step7 determines the lifetime piece Replacing Scheme of the cell cube not comprising lifetime piece：It can determine that lifetime piece
Replacing Scheme child node NS：Change RchioceAnd RmustIn lifetime piece,
Step8 judges whetherMeet, then continue；It is no
Then, by Replacing Scheme node NSAdd in Replacing Scheme child node set R, order
Step9 makes m=m+1, judges whether to meet m ＞ M, if so, then continue, otherwise jump to Step7,
Step10 makes c=c+1, judges whetherIf set up, continue, otherwise jump to Step6,
Step11 makes b=b+1, judges whether b ＞ B；If set up, continue；Otherwise Step5 is jumped to,
Step12 makes l=l+1, judges whether l ＞ q；If set up, algorithm terminates, and returns to Replacing Scheme child node set R；It is no
Then, Step3 is jumped to；
According to child node generation method, the derivation algorithm of more lifetime piece chance replacement problems is proposed, is comprised the following steps that.
Step1 initialization optimal values Cmin, optimal leaf segment point set Smin, movable joint point list SL, root node N0, CminBe set to one compared with
Big number, SminFor sky, SLFor sky；
Step2 judges whether tres,min(T0)≥Tlim.If it is, Cmin=0, Smin={ N0, algorithm terminates；If not, by N0Add
Enter SL；
Step3 judges SLWhether it is empty.If it is, algorithm terminates；If not, specify SLIn last add node for work as
Front nodal point Nc；
Step4 is using child node generation method generation present node NcChild node, child node forms set R；
Step5 judges each child node in R successivelyWhether meetIf satisfaction removes the node from R,
Otherwise continue；
Step6 judges whether the whole of life part in R in each node can use and arrives T successivelylim, if it is, according to formula (2-1)
Target function value corresponding to the element is calculated, by target function value and CminIt is compared, updates CminAnd Smin, and remove from R
The element；Otherwise by remaining node to slip-knot point set SLIn, return to step 3；
The algorithm has carried out a large amount of yojan to solution space tree it can be seen from the solution procedure of algorithm, and can necessarily obtain
Get the optimal solution of the more lifetime piece chance replacement problems of engine.
2. a kind of more lifetime piece renewal reward theorem searching algorithms for considering structural dependence according to claim 1,
It is characterized in that feasibility reduction rules are described in step 2-1：More lifetime piece chances shown in formula (1) are changed and asked
Topic, if Nj-1For arbitrary node in solution space tree, if the node has child node Nj, then haveMeanwhile if
So
3. a kind of more lifetime piece renewal reward theorem searching algorithms for considering structural dependence according to claim 1, its
It is characterised by that lifetime piece described in step 2-1 changes reduction rules and includes the lifetime piece replacing He of reduction rules 1
Lifetime piece changes reduction rules 2, and wherein lifetime piece replacing reduction rules 1 are：To more lifetime piece machines shown in formula (1)
Meeting replacement problem, if NjFor arbitrary node in solution space tree, in same cell cube MkIn, if lifetime piece meets
So certainly exist not comprising node NjFeasible solution, and the feasible solution however be inferior to include node NjFeasible solution；The rule is said
Bright, when carrying out lifetime piece replacing to engine, in a certain cell cube, the replacing of lifetime piece should be according to residual life from small
To big order.
4. a kind of more lifetime piece renewal reward theorem searching algorithms for considering structural dependence according to claim 1,
It is characterized in that lifetime piece described in step 2-1, which changes reduction rules, includes the lifetime piece replacing He of reduction rules 1
Lifetime piece changes reduction rules 2, and wherein lifetime piece replacing reduction rules 2 are：To more lifetime piece machines shown in formula (1)
Meeting replacement problem, if NjFor arbitrary node in solution space tree, in same cell cube MkIn, if there is
So certainly exist not comprising node NjFeasible solution, and the feasible solution is not inferior to include node NjFeasible solution；By lifetime piece more
Change reduction rules 2 to understand, when repairing engine, in same cell cube Mk(k=1,2, p) in, arrive the longevity
Lifetime piece must be changed；Meanwhile the lifetime piece being replaced in cell cube needs to meet condition：
Maintenance opportunity determines reduction rules
5. a kind of more lifetime piece renewal reward theorem searching algorithms for considering structural dependence according to claim 1, its feature
It is that opportunity is repaired described in step 2-1 determines that reduction rules are：To more lifetime piece chance replacement problems shown in formula (1), if
NjFor arbitrary node in solution space tree, if there is
So certainly exist not comprising node NjFeasible solution, and the feasible solution however be inferior to include node NjFeasible solution, during by repairing
Machine determines that reduction rules are understood, works as NjIt is any node of more lifetime piece chance replacement problem solution space trees, during corresponding maintenance
Machine is Tj.If NjChild node N be presentj+1, then Nj+1Maintenance opportunity can be defined as Tj+1=Tj+tres,min(Tj)。
6. a kind of more lifetime piece renewal reward theorem searching algorithms for considering structural dependence according to claim 1, its feature
It is that cost reduction rules are described in step 2-1：NaIt is a node (nonleaf node) in solution space tree, it is all to include section
Point NaObject function solution assessment of cost lower limit Cl(Na) can be calculated by formula (4)：
In formula--- rounding is integer；Cost reduction rules：To more lifetime piece chance replacement problems shown in formula (1), if Cmin
For object function global optimum, NaFor a node (nonleaf node) in solution space, if Cl(Na) ＞ Cmin, then include section
Point NaSolution be not optimal solution；By the rule, if the assessment of cost floor value C of the solution comprising present nodel(Na) super
Object function globally optimal solution C is crossedmin, then the searching route comprising present node should be terminated.
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