CN103632214B - Maintenance under inventory degradation data deletion condition and spare part supply combined optimization method - Google Patents

Abstract

The invention discloses the maintenance under inventory degradation data deletion condition and spare part supply combined optimization method.The method includes: set up randomicity stock degradation model according to spare part out-of-service time distribution function；Based on randomicity stock degradation model, analyze a series of system maintenances such as relevant expectation expense and the spare part supply correlative charges such as expectation maintenance cost, expectation spare parts purchasing expense, expectation spare part storage cost, expectation spare part shortage failure costs；Based on randomicity stock degradation model, to minimize system maintenance and spare part supply correlative charges as target, set up maintenance and spare part supply combined optimization model；Give system prediction maintenance cycle and the bound of spare part maximum inventory amount to be optimized finally according to practical situation, and solve maintenance and the spare part the relation between supply and demand of optimization accordingly.For when there is Parts Inventory degeneration environment but lacking enough stock's degeneration Monitoring Data, the present invention more conforms to practical situation than existing combined optimization method, meanwhile, it is suitable for do not consider stock's degenerate case.

Description

Maintenance under inventory degradation data deletion condition and spare part supply combined optimization method

Technical field

The present invention relates to the prediction of the complex engineering system failure and predictive maintenance technology, especially with regard to one stock Maintenance under degraded data deletion condition and spare part supply combined optimization method.

Background technology

Interwoveness between the maintenance of system and spare part supply, maintenance is the main source of spare parts consumption, standby Part is the premise that maintenance is carried out.Lack spare part makes system cannot recover normal work by affecting maintenance Make.But this is it is not intended that spare part is The more the better, the most on the one hand spare part can increase the spare part fund input of enterprise, separately On the one hand corresponding stock management expense can be increased.And there is inevitably degeneration in storage process in spare part With scrap phenomenon.The electronic equipment stored the most on deep-sea vessels and components and parts and some mechanical element, Its for lost efficacy operation parts change in time, to ensure the properly functioning of boats and ships, but due to high humidity, The severe marine environment such as high salinity, surge, spare part is scrapped phenomenon very seriously, is being brought the same of economic loss Time, also can affect the reliability of vessel motion.

Maintenance and spare part supply combined optimization method are a kind of methods that can effectively solve above-mentioned contradiction.This Close optimization method compared to according only to maintenance or according only to the spare part supply method that is optimized of situation for, be A kind of global optimum rather than the method for local optimum, utilize combined optimization method to be obtained in that and more conform to actual need The result asked.But at some in particular cases, such as lack enough for there is Parts Inventory degeneration environment Stock's degeneration Monitoring Data situation for, owing to there is no enough data for the degeneration of Parts Inventory is advised Rule is modeled, and causes combined optimization method can be subject to certain restrictions in the specific implementation.

Summary of the invention

For the problems referred to above, it is an object of the invention to for existing maintenance and spare part supply combined optimization lack standby The situation of part inventory degradation data, it is provided that a kind of maintenance based on randomicity stock degradation model and spare part supply connection Close optimization method.The method can be set up describe part warehouse from the out-of-service time distribution function of single spare part Deposit the randomness Parts Inventory degradation model of deterioration law；Based on randomicity stock degradation model, to minimize it is System maintenance and spare part supply cost are target, set up maintenance and spare part supply combined optimization model further；Then By the restriction to some amounts to be optimized, obtain more conforming to the maintenance of practical situation and spare part supply combined optimization Result.

The present invention provides the maintenance under a kind of inventory degradation data deletion condition and spare part supply combined optimization method, Comprise the following steps:

S100, set up randomicity stock degradation model based on spare part out-of-service time distribution function；

S200, based on randomicity stock degradation model, analyze system maintenance and spare part supply correlative charges, described Expense includes system expectation maintenance cost within a preventative maintenance cycle, expectation spare parts purchasing expense, phase Hope spare part storage cost and expectation spare part shortage failure costs；

S300, based on randomicity stock degradation model, to minimize system maintenance and spare part supply cost as target, Set up maintenance and spare part supply combined optimization model；

S400, based on maintenance and spare part supply combined optimization model and the bound of given amount to be optimized, obtain Maintenance and spare part the relation between supply and demand optimal solution, described amount to be optimized includes that prospective maintenance cycle of system and spare part are Big quantity in stock.

According to embodiments of the invention, above-mentioned steps S100 farther includes:

If there is m spare part in system part warehouse, the out-of-service time distribution function of each spare part is p (t), system It is m p (t) that the distribution of the inefficacy spare part number Y in time t is approximately average, and variance is The normal distribution of m p (t) (1-p (t)), then the probability density function of Y is g_Y(y):

$g_Y\left(y\right)=\left\{\begin{array}{cc}\frac{1}{\sqrt{2\mathrm{\π}}\·\sqrt{m\·p\left(t\right)\·(1-p\left(t\right))}}{e}^{-\frac{{(y-m\·p\left(t\right))}^2}{2m\·p\left(t\right)\·(1-p\left(t\right))}}& ,m>0\\ \mathrm{\δ}\left(y\right)& ,m=0\end{array}\right.$

Wherein, δ (y) is unit impulse function, and

According to embodiments of the invention, above-mentioned steps S200 farther includes:

S210, analysis thrashing maintenance frequency:

System in a preventative maintenance cycle T, the expectation E [X] of the Failure count X of each operation parts= M (T), function M (T) are iterated when k=T calculating by following approximate formula and obtain:

$M\left(k\right)=\underset{i=0}{\overset{k-1}{\mathrm{\Σ}}}[1+M(k-i-1)]\·{\∫}_i^{i+1}{f}_a\left(t\right)\mathrm{dt},k=\mathrm{1,2},...$

M(0)=0

Corresponding variance Var [X] utilizes following approximate formula to calculate and obtains::

$\mathrm{Var}\left[X\right]=D\left(k\right)=M\left(k\right)-M{\left(k\right)}^2+\underset{i=1}{\overset{k}{\mathrm{\Σ}}}\left\{\begin{array}{c}M(k-i)[M(i+1)-M\left(i\right)]\\ +M(k-i-1)[M\left(i\right)-M(i-1)]\end{array}\right\}$

Wherein, f_aT () is the probability density function running the component failure time, k is discrete magnitude, for positive integer；

Operation part count in system is n, in a preventative maintenance cycle T, and thrashing maintenance time Number X_nObedience is desired for E [X_n]=nM (T), variance is Var [X_nThe normal distribution of]=nVar [X], and X_n's Probability density function is；

S220, analysis Parts Inventory amount of degradation:

If system spare part primary quantity when preventative maintenance cycle T starts is S_a, spare part out-of-service time distribution function For p (t)=F_b(t)=p, in preventative maintenance cycle T, the probability density function of inefficacy spare part number Y is:

$g_Y\left(y\right)=\left\{\begin{array}{cc}\frac{1}{\sqrt{2\mathrm{\π}}\·\sqrt{S_{a}p(1-p)}}{e}^{-\frac{{(y-S_{a}p)}^2}{2S_{a}p(1-p)}}& ,S_a>0\\ \mathrm{\δ}\left(y\right)& ,S_a=0\end{array}\right.$

Now expect that E [Y] is approximately S_aP, i.e. E [Y]=S_a p。

S230, analysis spare part maximum inventory S and spare part primary quantity S_aBetween relation:

S_a=S-n-E[X_n(T)-X_n(T-τ)]-E[Y(T)-Y(T-τ)]

=S-n-n[M(T)-M(T-τ)]-S_a[p(T)-p(T-τ)]

Wherein, τ is the delivery time ordering spare part；

S240, expectation maintenance cost C analyzed in a preventative maintenance cycle T_MFor:

C_M(S_a,T)=c_pr·n+c_cr·E[X_n]

Wherein, c_prRepresent the expense of each preventative maintenance, c_crRepresent the expense of each correction maintenance；

S250, expectation spare parts purchasing expense C analyzed in a preventative maintenance cycle T_oFor:

C_o(S_a,T)=K+c_sp·(n+E[X_n]+E[Y])

Wherein, K represents disposable buying expenses, c_spRepresent the unit price of spare part；

S260, the expectation spare part storage cost C analyzed in a preventative maintenance cycle T_hFor:

$C_h(S_a,T)=c_h\·\left[\begin{array}{cc}\underset{0\≤X_n+Y+S_a}{\∫\∫}(S_a-\frac{x_n+Y}{2})\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right)\mathrm{dyd}{x}_n& \\ +\underset{X_n+Y>S_a}{\∫\∫}\frac{S_a}{2}\·\frac{S_a}{x_n+Y}\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right)\mathrm{dy}{\mathrm{dx}}_n& \end{array}\right]$

$=c_h\left[\begin{array}{c}{\∫}_0^{S_a}{\∫}_0^{S_a-x_n}(S_a-\frac{x_n+y}{2})\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right)\mathrm{dy}{\mathrm{dx}}_n\\ +{\∫}_0^{S_a}{\∫}_{S_a-x_n}^{\∞}\frac{S_a}{2}\·\frac{S_a}{x_n+y}\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right)\mathrm{dy}{\mathrm{dx}}_n\\ +{\∫}_{S_a}^{\∞}{\∫}_0^{\∞}\frac{S_a}{2}\·\frac{S_a}{x_n+y}\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right){\mathrm{dydx}}_n\end{array}\right]$

Wherein, c_hStorage cost for single spare parts list bit time；

S270, the expectation spare part shortage failure costs C analyzed in a preventative maintenance cycle T_sFor:

$C_s(S_a,T)=c_s\·\underset{X_n+Y>S_a}{\∫\∫}\frac{x_n+Y-S_a}{2}\·\frac{x_n+Y-S_a}{x_n+Y}\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right){\mathrm{dydx}}_n$

$=c_s\left[\begin{array}{c}{\∫}_0^{S_a}{\∫}_{S_a-x_n}^{\∞}\frac{x_n+y-S_a}{2}\·\frac{x_n+y-S_a}{x_n+y}\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right)\mathrm{dy}{\mathrm{dx}}_n\\ +{\∫}_{S_a}^{\∞}{\∫}_0^{\∞}\frac{x_n+y-S_a}{2}\·\frac{x_n+y-S_a}{x_n+y}\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right){\mathrm{dydx}}_n\end{array}\right]$

Wherein, c_sShort failure costs for single spare parts list bit time.

According to embodiments of the invention, above-mentioned steps S300 farther includes:

To minimize system maintenance and spare part supply correlative charges as target, with maximum part warehouse storage S and prevention Property maintenance cycle T is amount to be optimized, sets up following Optimized model:

$\begin{array}{cc}\min & C_{2,\∞}(S,T)=C_{2,\∞}(S_a,T)=\frac{C_M(S_a,T)+C_o(S_a,T)+C_h(S_a,T)+C_s(S_a,T)}{T}\\ s.t.& S_a\≥0,T>0\end{array}$

Wherein, S_aFor the system spare part primary quantity when a preventative maintenance cycle T starts, C_MBe one pre- Expectation maintenance cost in anti-property maintenance cycle T, C_oIt is that the expectation spare part in a preventative maintenance cycle T is adopted Purchase expense, C_hIt is the expectation spare part storage cost in a preventative maintenance cycle T, C_sIt it is a preventative dimension Repair the expectation spare part shortage failure costs in cycle T.

According to embodiments of the invention, above-mentioned steps S400 farther includes:

1) given system relevant parameter, described parameter includes calculating system maintenance and spare part supplies correlative charges Parameters, given system runs the probability density function f of component failure time_aT () and spare part out-of-service time are distributed Function p (t)=F_b(t), and calculate the operation parts expectation out-of-service time

2) bound the discretization amount to be optimized of amount to be optimized are set:

Bound [the T of preventative maintenance cycle T is set_min,T_max], and according to the interval △ preventative dimension of T discretization Repair cycle T, it is thus achieved that discrete time node T_i=T_min+ i △ T, i=0,1,2 ...；

Set the bound [S of spare part maximum inventory S_min,S_max], and maximum according to interval △ S discretization spare part Quantity in stock S, it is thus achieved that discrete stock node S_j=S_min+ j △ S, j=0,1,2 ...；

3) i=0 is taken；

4) discrete time node T is compared_iWith upper bound T_maxIf, T_i>T_max, then step 12) is entered, otherwise Enter step 5)；

5) discrete time node T is calculated according to the following formula_iCorresponding M (T_i) and D (T_i):

$M\left(T_i\right)=\underset{0}{\overset{T_i-1}{\mathrm{\Σ}}}[1+M(T_i-i-1)]\·{\∫}_i^{i+1}{f}_a\left(t\right)\mathrm{dt}$

M(0)=0

$D\left(T_i\right)=M\left(T_i\right)-M{\left(T_i\right)}^2+\underset{i=1}{\overset{T_i}{\mathrm{\Σ}}}\left\{\begin{array}{c}M(T_i-i)[M(i+1)-M\left(i\right)]\\ +M(T_i-i-1)[M\left(i\right)-M(i-1)]\end{array}\right\}$

6) j=0 is taken；

7) relatively discrete stock node S_jIf, S_j>S_max, then enter step 10), otherwise enter step 8)；

8) discrete stock node S is calculated according to the following formula_jCorresponding spare part beginning inventory S_a,

S_a=S-n-E[X_n(T_i)-X_n(T_i-τ)]-E[Y(T_i)-Y(T_i-τ)]

=S-n-n[M(T_i)-M(T_i-τ)]-S_a[p(T_i)-p(T_i-τ)]

By S_aBring following maintenance into and spare part supply combined optimization model is calculated corresponding C_2,∞(S_j,T_i):

$C_{2,\∞}(S_j,T_i)=\frac{C_M(S_a,T_i)+C_o(S_a,T_i)+C_h(S_a,T_i)+C_s(S_a,T_i)}{T}$

Wherein, C_MIt is the expectation maintenance cost in a preventative maintenance cycle T, C_oIt it is a preventative dimension Repair the expectation spare parts purchasing expense in cycle T, C_hIt it is the expectation spare part storage in a preventative maintenance cycle T Expense, C_sIt is the expectation spare part shortage failure costs in a preventative maintenance cycle T,

9) take j=j+1, return step 7)；

10) optimal solution S of spare part maximum inventory S is calculated^i,*:

$S^{i,*}=\left\{S_j\right|\underset{S_j}{\min }{C}_{2,\∞}(S_j,T_i)\}$

Now T_iIt it is set-point；

11) take i=i+1, return step 4)；

12) at the { [S obtained^i,*,T_i,C_2,∞(S^i,*,T_i)] in set, find out so that system maintenance and spare part supply One group of solution of correlative charges minimum is as maintenance and the optimum results of spare part supply associating relation:

$(S^{*},T^{*})=\left\{(S^{i,*},T_i)\right|\underset{i}{\min }{C}_{2,\∞}(S^{i,*},T_i)\}$

Wherein, S^*For spare part maximum inventory optimum results, T^*For corresponding preventative maintenance cycle optimum results, Terminate.

According to embodiments of the invention, in above-mentioned steps S400, the lower bound T of described preventative maintenance cycle T_min Value relevant with repair part ordering delivery time τ, upper bound T_maxValue with run parts expectation out-of-service time μ Relevant.

Further, T_min=τ, T_max=1.5μ。

According to embodiments of the invention, in above-mentioned steps S400, the lower bound S of described spare part maximum inventory S_min Value relevant with operation part count n of system, upper bound S_maxOperation part count n of value and system Relevant with running parts expectation out-of-service time μ.

Further, S_min=n, S_max=n+2·n·M(μ)。

Compared with prior art, the present invention brings following beneficial effect:

The combined optimization method that the present invention provides can in actual use, based on existing data, and Some the relevant prioris losing efficacy spare part, estimate the out-of-service time distribution of single spare part, from single spare part Out-of-service time distribution function set out, set up describe Parts Inventory deterioration law randomness Parts Inventory degeneration mould Type, and this model is incorporated in the calculating of maintenance and spare parts management correlative charges, thus more conformed to The joint optimization result of practical situation, and then provide the strongest scientific basis for later stage correct decisions.And this Invention is particularly suited for lacking enough stock's degeneration Monitoring Data for there is Parts Inventory degeneration environment Situation, simultaneously it is suitable for do not consider the situation that stock degenerates.

Other features and advantages of the present invention will illustrate in the following description, and, partly from description In become apparent, or by implement the present invention and understand.The purpose of the present invention and other advantages can be passed through Structure specifically noted in description, claims and accompanying drawing realizes and obtains.

Accompanying drawing explanation

Accompanying drawing is for providing a further understanding of the present invention, and constitutes a part for description, with the present invention Embodiment be provided commonly for explain the present invention, be not intended that limitation of the present invention.In the accompanying drawings:

Fig. 1 is the workflow diagram of the combined optimization method that the present invention provides；

Fig. 2 is the schematic diagram of " spare part is superfluous " situation；

Fig. 3 is the schematic diagram of " spare part shortage " situation；

Fig. 4 is the numerical search algorithm stream in combined optimization method step S400 of the embodiment that the present invention provides Cheng Tu；

Fig. 5 is the equal pitch contour schematic diagram of the optimum results of the embodiment that the present invention provides.

Detailed description of the invention

Embodiments of the present invention are described in detail, whereby to the present invention such as below with reference to drawings and Examples What application technology means solves technical problem, and the process that realizes reaching technique effect can fully understand and evidence To implement.As long as it should be noted that do not constitute conflict, each embodiment in the present invention and respectively implementing Each feature in example can be combined with each other, the technical scheme formed all protection scope of the present invention it In.

It addition, can be at the meter of such as one group of computer executable instructions in the step shown in the flow chart of accompanying drawing Calculation machine system performs, and, although show logical order in flow charts, but in some situation Under, can be to be different from the step shown or described by order execution herein.

As it is shown in figure 1, be the workflow diagram of the combined optimization method that the present invention provides.It can be seen that should Method, from the out-of-service time distribution function of single spare part, is set up and is described the random of Parts Inventory degenerate case Property inventory degradation model；It is then based on randomicity stock degradation model, analyzes system maintenance and spare part supply phase Pass expense；To minimize system maintenance and spare part supply cost as target, set up maintenance and spare part supply associating Optimized model, finally combines the practical situation initialization system preventative maintenance cycle and spare part maximum inventory is to be optimized The bound of amount, thus draw the maintenance more tallied with the actual situation and spare part supply associating relation.The method is concrete Comprise the following steps:

S100, set up randomicity stock degradation model based on spare part out-of-service time distribution function:

In system, the out-of-service time distribution function of each spare part is p (t), and the inefficacy of each spare part can be considered as one Secondary Bernoulli Jacob tests, then the inefficacy spare part number for the part warehouse that there is m spare part, in moment t Y obeys binomial distribution B～(m, p (t)).If m is sufficiently large, it is m p (t) that the distribution of Y may be approximately equal to average, Variance is the normal distribution of m p (t) (1-p (t))；If m is zero, Y identically vanishing.Therefore, the probability of Y Density function is g_Y(y):

$g_Y\left(y\right)=\left\{\begin{array}{cc}\frac{1}{\sqrt{2\mathrm{\π}}\·\sqrt{m\·p\left(t\right)\·(1-p\left(t\right))}}{e}^{-\frac{{(y-m\·p\left(t\right))}^2}{2m\·p\left(t\right)\·(1-p\left(t\right))}}& ,m>0\\ \mathrm{\δ}\left(y\right)& ,m=0\end{array}\right.$

Wherein, δ (y) is unit impulse function, and understands

S200, based on randomicity stock degradation model, analyze system maintenance and spare part supply correlative charges, described Expense includes system expectation maintenance cost within a preventative maintenance cycle, expectation spare parts purchasing expense, phase Hope spare part storage cost and expectation spare part shortage failure costs.This step S200 can be subdivided into following steps:

S210, analysis thrashing maintenance frequency:

Generally system uses preventative maintenance strategy in batch, i.e. after certain runs component failure, loses efficacy it Replace (in the presence of corresponding spare part), and often through after a while, the operation parts in system are all prevented Property ground replace.Can be considered that recovery, as new, therefore sets T as system shortsightedness in system after this preventative replacement Property maintenance cycle, also referred to as update cycle.The inefficacy replacement process of each operation parts is also renewal process, so The expectation E [X] of its Failure count X in System Preventive Maintenance Cycle T namely runs the updated of parts The renewal function M (T) of journey, i.e. E [X]=M (T), and function M (T) can utilize following approximate formula at k=T Time iterative computation obtain:

$M\left(k\right)=\underset{i=0}{\overset{k-1}{\mathrm{\Σ}}}[1+M(k-i-1)]\·{\∫}_i^{i+1}{f}_a\left(t\right)\mathrm{dt},k=\mathrm{1,2},...$

M(0)=0

Wherein, f_aT () is the probability density function running the component failure time, k is discrete magnitude, for positive integer.

Corresponding variance Var [X] can utilize following approximate formula to calculate and obtain:

$\mathrm{Var}\left[X\right]=D\left(k\right)=M\left(k\right)-M{\left(k\right)}^2+\underset{i=1}{\overset{k}{\mathrm{\Σ}}}\left\{\begin{array}{c}M(k-i)[M(i+1)-M\left(i\right)]\\ +M(k-i-1)[M\left(i\right)-M(i-1)]\end{array}\right\}$

In the power electronic system of reality application, quantity n running parts is the biggest, according to big number Theorem, system thrashing maintenance frequency X in a preventative maintenance cycle T_nGenerally obey and be desired for E[X_n]=nM (T), variance is Var [X_nThe normal distribution of]=nVar [X], and X_nProbability density function be

S220, analysis Parts Inventory amount of degradation:

If system includes S when a preventative maintenance cycle T starts_aIndividual separate spare part, i.e. spare part Primary quantity is S_a, the out-of-service time distribution function of these spare parts is F_b(t).It is random that integrating step S100 is set up Property inventory degradation model has: m=S_a, p (t)=F_b(t)=p, then spare part is in System Preventive Maintenance Cycle T The probability density function of inefficacy spare part number Y is:

$g_Y\left(y\right)=\left\{\begin{array}{cc}\frac{1}{\sqrt{2\mathrm{\π}}\·\sqrt{S_{a}p(1-p)}}{e}^{-\frac{{(y-S_{a}p)}^2}{2S_{a}p(1-p)}}& ,S_a>0\\ \mathrm{\δ}\left(y\right)& ,S_a=0\end{array}\right.$

Now expect that E [Y] is approximately S_aP, i.e. E [Y]=S_a p。

S230, the relation analyzed between spare part maximum inventory and spare part primary quantity:

Generally system stock uses cyclic check spare part supply strategy, i.e. arrives it in each preventative maintenance cycle Position the iT-τ, i=1,2 in front τ moment ... carrying out repair part ordering, τ is the delivery time point ordering spare part, and orders Principle is that spare part quantity is added to maximum inventory S.Therefore within a preventative maintenance cycle, spare part May there are two kinds of situations in Expenditure Levels: spare part superfluous (as shown in Figure 2) or spare part shortage are (such as Fig. 3 Shown in), wherein spare part surplus can cause unnecessary spare part fund input and increase stock management expense, and standby Part shortage can make system-down, can affect system production run, bring economic loss time serious.Therefore spare part Maximum inventory S and spare part primary quantity S_aBetween transformational relation have:

S_a=S-n-E[X_n(T)-X_n(T-τ)]-E[Y(T)-Y(T-τ)]

=S-n-n[M(T)-M(T-τ)]-S_a[p(T)-p(T-τ)]

Lower surface analysis is owing to running component failure and the system maintenance that produces and spare part supply cost.If in system The quantity running parts is n, and system includes S when preventative maintenance cycle T starts_aIndividual separate Spare part, in a preventative maintenance cycle T, expectation maintenance cost, expectation spare parts purchasing needed for system maintenance take Obtained by step S240～S270 respectively with, expectation spare part storage cost and expectation spare part shortage failure costs:

Expectation maintenance cost C in S240, a preventative maintenance cycle T_MFor:

C_M(S_a,T)=c_pr·n+c_cr·E[X_n]=c_pr·n+c_cr·n·M(T)

Wherein, c_prRepresent the expense of each preventative maintenance, c_crRepresent the expense of each correction maintenance.

Expectation spare parts purchasing expense C in S250, a preventative maintenance cycle T_oFor:

C_o(S_a,T)=K+c_sp·(n+E[X_n]+E[Y])=K+c_sp[n+nM(T)+S_ap]

Wherein, K represents disposable buying expenses (such as freight charges etc.), c_spRepresent the unit price of spare part.It can be seen that Spare parts purchasing quantity in one preventative maintenance cycle T is by demand n of preventative replacement, and lost efficacy the need replaced Seek E [X_n], and the degenerate loss E [Y] that causes of spare part self determines together.

Expectation spare part storage cost C in S260, a preventative maintenance cycle T_hFor:

$C_h(S_a,T)=c_h\·\left[\begin{array}{cc}\underset{0\≤X_n+Y+S_a}{\∫\∫}(S_a-\frac{x_n+Y}{2})\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right)\mathrm{dyd}{x}_n& \\ +\underset{X_n+Y>S_a}{\∫\∫}\frac{S_a}{2}\·\frac{S_a}{x_n+Y}\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right)\mathrm{dy}{\mathrm{dx}}_n& \end{array}\right]$

$=c_h\left[\begin{array}{c}{\∫}_0^{S_a}{\∫}_0^{S_a-x_n}(S_a-\frac{x_n+y}{2})\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right)\mathrm{dy}{\mathrm{dx}}_n\\ +{\∫}_0^{S_a}{\∫}_{S_a-x_n}^{\∞}\frac{S_a}{2}\·\frac{S_a}{x_n+y}\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right)\mathrm{dy}{\mathrm{dx}}_n\\ +{\∫}_{S_a}^{\∞}{\∫}_0^{\∞}\frac{S_a}{2}\·\frac{S_a}{x_n+y}\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right){\mathrm{dydx}}_n\end{array}\right]$

Wherein, c_hStorage cost for single spare parts list bit time.

Expectation spare part shortage failure costs C in S270, a preventative maintenance cycle T_sFor:

$C_s(S_a,T)=c_s\·\underset{X_n+Y>S_a}{\∫\∫}\frac{x_n+Y-S_a}{2}\·\frac{x_n+Y-S_a}{x_n+Y}\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right){\mathrm{dydx}}_n$

$=c_s\left[\begin{array}{c}{\∫}_0^{S_a}{\∫}_{S_a-x_n}^{\∞}\frac{x_n+y-S_a}{2}\·\frac{x_n+y-S_a}{x_n+y}\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right)\mathrm{dy}{\mathrm{dx}}_n\\ +{\∫}_{S_a}^{\∞}{\∫}_0^{\∞}\frac{x_n+y-S_a}{2}\·\frac{x_n+y-S_a}{x_n+y}\·T\·g_{X_n}\left(x_n\right)g_Y\left(y\right){\mathrm{dydx}}_n\end{array}\right]$

Wherein, c_sShort failure costs for single spare parts list bit time.

S300, based on randomicity stock degradation model, set up maintenance and spare part supply combined optimization model:

To minimize system overall expenses rate as target, with maximum part warehouse storage S and preventative maintenance cycle T For amount to be optimized, obtain following combined optimization model:

$\begin{array}{cc}\min & C_{2,\∞}(S,T)=C_{2,\∞}(S_a,T)=\frac{C_M(S_a,T)+C_o(S_a,T)+C_h(S_a,T)+C_s(S_a,T)}{T}\\ s.t.& S_a\≥0,T>0\end{array}$

According to step S250, in the case of other parameter determinations, S and S_aOne_to_one corresponding, i.e. S_aOr After person S determines, correspondingly S or S_aCan determine therewith.Therefore, the C of combined optimization model_2,∞(S, T) etc. It is same as C_2,∞(S_a,T)。

Analyze the extreme case of above-mentioned Optimized model:

Work as S_aWhen → 0, can obtain:

$\underset{S_a\→0}{\lim }{C}_{2,\∞}(S_a,T)=\frac{1}{T}\left\{\begin{array}{c}{c}_{\mathrm{pr}}n+c_{\mathrm{cr}}\mathrm{nM}\left(T\right)+K+c_{\mathrm{sp}}[n+\mathrm{nM}\left(T\right)]\\ +c_h[{\∫}_0^{S_a}(S_a-\frac{x_n}{2})\·T\·g_{X_n}\left(x_n\right){\mathrm{dx}}_n+{\∫}_{S_a}^{\∞}\frac{S_a}{2}\·\frac{S_a}{x_n}\·T\·g_{X_n}\left(x_n\right){\mathrm{dx}}_n]\\ +c_s[{\∫}_{S_a}^{\∞}\frac{x_n-S_a}{2}\·\frac{x_n-S_a}{x_n}\·T\·g_{X_n}\left(x_n\right){\mathrm{dx}}_n]\end{array}\right\}$

This formula with 2003With Hudoklin in the premise not considering Parts Inventory degenerate case Lower set up model is the same, illustrates that the Optimized model of the present invention can be contained and did not originally account for Parts Inventory and move back Situation about changing.

S400, bound based on maintenance and spare part supply combined optimization model and given amount to be optimized solve The maintenance optimized and spare part the relation between supply and demand.This step S400 can be subdivided into following steps:

S410, from step S300, preventative maintenance cycle T and spare part maximum inventory S for optimizing The amount to be optimized of journey, arranges the bound of amount to be optimized according to the actual requirements:

S410.1, preventative maintenance cycle T:

Preventative maintenance cycle T can carry out discretization in actual applications.In the present embodiment, first, if The lower bound putting preventative maintenance cycle T is the delivery time point τ ordering spare part, and the upper bound is the operation parts of 1.5 times Expect out-of-service time μ, i.e. τ≤T≤1.5 μ；Secondly, according to being spaced △ T to the preventative maintenance cycle in this interval T carries out discretization, it is thus achieved that corresponding discrete time node T_i:

S410.2, spare part maximum inventory S:

In the case of preventative maintenance cycle T is certain, if

$P\left(S_a\right)=\frac{\∂}{{\∂S}_a}{C}_{2,\∞}(S_a,T),$

Then have

$P\left(0\right)=-\frac{1}{4}{c}_s\mathrm{nM}\left(T\right)\underset{S_a\&RightArrow;0}{\lim }\frac{1}{S_a}+\frac{c_{\mathrm{sp}}{p}_T}{T}-c_s\frac{p}{4(1-p)}\&CenterDot;\mathrm{nM}\left(T\right)-\frac{1}{2}{c}_s=-\&infin;<0$

$P(\&infin;)>c_h\frac{(1+p)}{4p}\underset{S_a\&RightArrow;\&infin;}{\lim }{S}_a+\frac{c_{\mathrm{sp}}{p}_T}{T}+\frac{3}{4}{c}_h-c_h\frac{(1+p)}{4p}\mathrm{nM}\left(T\right)=+\&infin;>0$

In the case of preventative maintenance cycle T is certain, always there is optimum spare part in the explanation of above-mentioned two formula Primary quantityBy aforementioned spare part maximum inventory S and spare part primary quantity S_aBetween transformational relation understand, S and S_aOne_to_one corresponding, therefore for optimum spare part primary quantityAlways there is optimum spare part maximum inventory S^*。 Thus, the present invention can consider by enumerative technique optimal solution S to spare part maximum inventory S^*Carry out numerical search.

In the present embodiment, first, the lower bound arranging spare part maximum inventory S is n, and the upper bound is n+2 n M (μ), I.e. n≤S≤n+2 n M (μ)；Secondly, according to interval △ S, spare part maximum inventory S is carried out discrete in this interval Change, it is thus achieved that corresponding discrete time node S_i:

S_j=n+j △ S, j=0,1,2 ....

S420, by numerical search obtain maintenance and spare part supply joint optimization result:

As shown in Figure 4, it is that the present invention obtains maintenance and the numerical search algorithm stream of spare part supply joint optimization result Cheng Tu.Detailed process is as follows:

1) parameter initialization: given system relevant parameter n, τ, c_pr, c_cr, K, c_sp, c_hAnd c_sValue, give Determine f_a(t) and F_b(t), and calculate the operation parts expectation out-of-service time

2) bound the discretization amount to be optimized of amount to be optimized are set:

The bound arranging preventative maintenance cycle T is [τ, 1.5 μ], and according to the interval △ preventative dimension of T discretization Repair cycle T, it is thus achieved that discrete time node T_i=τ+i △ T, i=0,1,2 ...；

The bound [n, n+2 n M (μ)] of spare part maximum inventory S is set, and standby according to interval △ S discretization Part maximum inventory S, it is thus achieved that discrete stock node S_j=n+j △ S, j=0,1,2 ...；

3) i=0 is taken；

4) discrete time node T is compared_iWith its upper bound 1.5 μ, if T_i> 1.5 μ, then enter step 12), no Then enter step 5)；

5) discrete time node T is calculated_iCorresponding M (T_i) and D (T_i):

$M\left(T_i\right)=\underset{0}{\overset{T_i-1}{\mathrm{\&Sigma;}}}[1+M(T_i-i-1)]\&CenterDot;{\&Integral;}_i^{i+1}{f}_a\left(t\right)\mathrm{dt}$

M(0)=0

$D\left(T_i\right)=M\left(T_i\right)-M{\left(T_i\right)}^2+\underset{i=1}{\overset{T_i}{\mathrm{\&Sigma;}}}\left\{\begin{array}{c}M(T_i-i)[M(i+1)-M\left(i\right)]\\ +M(T_i-i-1)[M\left(i\right)-M(i-1)]\end{array}\right\}$

6) j=0 is taken;

7) relatively discrete stock node S_jWith its upper bound n+2 n M (μ), if S_j> n+2 n M (μ), then enter Enter step 10), otherwise enter step 8)；

8) discrete stock node S is calculated according to the following formula_jCorresponding spare part beginning inventory S_a:

S_a=S-n-E[X_n(T_i)-X_n(T_i-τ)]-E[Y(T_i)-Y(T_i-τ)]

=S-n-n[M(T_i)-M(T_i-τ)]-S_a[p(T_i)-p(T_i-τ)]

By S_aBring following maintenance into and spare part supply combined optimization model is calculated corresponding C_2,∞(S_j,T_i):

$C_{2,\&infin;}(S_j,T_i)=\frac{C_M(S_a,T_i)+C_o(S_a,T_i)+C_h(S_a,T_i)+C_s(S_a,T_i)}{T}$

9) take j=j+1, return step 7)；

10) optimal solution of spare part maximum inventory S is calculatedNow T_iBe to Definite value；

11) take i=i+1, return step 4)；

12) at the { [S obtained^i,*,T_i,C_2,∞(S^i,*,T_i)] in set, find out so that system maintenance and spare part supply phase One group of solution of pass expense minimum is as maintenance and the optimum results of spare part supply associating relation, it may be assumed that

$(S^{*},T^{*})=\left\{(S^{i,*},T_i)\right|\underset{i}{\min }{C}_{2,\&infin;}(S^{i,*},T_i)\}$

Terminate.

Wherein, S^*For spare part maximum inventory optimum results, T^*For corresponding preventative maintenance cycle optimum results.

Above formula represents that when spare part maximum inventory and preventative maintenance cycle be corresponding S^i,*And T_iTime, system maintenance And spare part supply correlative charges rate C_2,∞(S^i,*,T_i) minimize.

Below with Slovenian Railway Bureau announce power vehicle arc-extinguish chamber data as object, the present invention is described Be embodied as step, and the optimum results obtained.This experiment Mathematica9.0 is the soft of Optimization Solution Part platform carries out emulation and solves.

(1) experimental subject

FoundationThe Slovenian Railway Bureau data case that and Hudoklin provides, obtains following three The value of class data correlated variables: operation number of components n=120 in system；Run the general of component failure time Rate density functionWherein mean μ₁=44 weeks, standard deviation sigma₁=12 weeks；Spare part is paid Period tau=12 week, corresponding unit costs is c_pr=58.2units, c_cr=800.5units, K=20units, c_sp=43units, c_h=0.6units, c_s=5196units；Spare part out-of-service time distribution F_bT () is for obeying average μ₂=40, standard deviation sigma₂The normal distribution of=15.

(2) based on randomicity stock degradation model, set up maintenance and spare part supply combined optimization model, go forward side by side Row Optimization Solution.

Concrete condition is as follows:

1) parameter initialization: given system relevant parameter n, τ, c_pr, c_cr, K, c_sp, c_hAnd c_sValue, give Determine f_a(t) and F_b(t), and calculate operation parts expectation out-of-service time μ=44；

2) discretization preventative maintenance cycle T: selected △ T=1, with τ as lower bound, with 1.5 μ=66 as the upper bound；

Discretization maximum part warehouse storage S: selected △ S=1, with n as lower bound, with n+2 n M (μ)=240.988 For the upper bound；

3) i=0 is taken；

4) if T_i=τ+i △ T > 1.5 μ, then enter step 12)；Otherwise enter step 5)；

5) discrete time node T is calculated_iCorresponding M (T_i) and D (T_i)；

6) j=0 is taken;

7) if S_j> 240.988, then enter step 10), otherwise enter step 8)；

8) discrete stock node S is calculated according to the following formula_jCorresponding spare part beginning inventory S_a,

By S_aBring maintenance into and spare part supply combined optimization model is calculated corresponding C_2,∞(S_j,T_i):

9) take j=j+1, return step 7)；

10) optimal solution S of spare part maximum inventory S is calculated^i,*:

11) take i=i+1, return step 4)；

12) maintenance and spare part supply iptimum relationship are found: find one group of minimum solution as maintenance and spare part supply Iptimum relationship $(S^{*},T^{*})=\left\{(S^{i,*},T_i)\right|\underset{i}{\min }{C}_{2,\&infin;}(S^{i,*},T_i)\}$ ；Terminate.

The final result of calculation of the present embodiment is (S^*,T^*)=(136,22), C_2,∞(S^*,T^*)=713.668, such as Fig. 5 Shown in, namely when spare part maximum inventory S is 136, and corresponding prospective maintenance cycle T is 22 weeks, This maintenance is standby can make system overall expenses rate minimize to mode, can reach 713.668units.

Although the embodiment that disclosed herein is as above, but described content is only to facilitate understand the present invention And the embodiment used, it is not limited to the present invention.Technology in any the technical field of the invention Personnel, on the premise of without departing from the spirit and scope that disclosed herein, can implement in form and Any amendment and change being made in details, but the scope of patent protection of the present invention, still must want with appended right Book is asked to be defined in the range of standard.

Claims (8)

1. the maintenance under inventory degradation data deletion condition and a spare part supply combined optimization method, comprises the following steps:

S100, set up randomicity stock degradation model based on spare part out-of-service time distribution function；

S200, based on randomicity stock degradation model, analyzing system maintenance and spare part supply correlative charges, described expense includes system expectation maintenance cost within a preventative maintenance cycle, expectation spare parts purchasing expense, expectation spare part storage cost and expectation spare part shortage failure costs；Described step S200 includes following little step:

S210, analysis thrashing maintenance frequency:

System in a preventative maintenance cycle T, the expectation E [X] of the Failure count X of each operation parts=M (T), function M (k) by following approximate formula when k=T iterative computation obtain:

Variance Var [X] utilizes following approximate formula to calculate and obtains:

Wherein, f_aT () is the probability density function running the component failure time, k is discrete magnitude, for positive integer；

Operation part count in system is n, in a preventative maintenance cycle T, and thrashing maintenance frequency X_nObedience is desired for E [X_n]=nM (T), variance is Var [X_nThe normal distribution of]=nVar [X], and X_nProbability density function be

S220, analysis Parts Inventory amount of degradation:

If system spare part primary quantity when preventative maintenance cycle T starts is S_a, spare part out-of-service time distribution function is p (t)=F_b(t)=p, in preventative maintenance cycle T, the probability density function of inefficacy spare part number Y is:

Now E [Y]=S_ap；

S230, analysis spare part maximum inventory S and spare part primary quantity S_aBetween relation:

Wherein, τ is the delivery time ordering spare part；

S240, expectation maintenance cost C analyzed in a preventative maintenance cycle T_MFor:

C_M(S_a, T) and=c_pr·n+c_cr·E[X_n]

Wherein, c_prRepresent the expense of each preventative maintenance, c_crRepresent the expense of each correction maintenance；

S250, expectation spare parts purchasing expense C analyzed in a preventative maintenance cycle T_oFor:

C_o(S_a, T) and=K+c_sp·(n+E[X_n]+E[Y])

Wherein, K represents disposable buying expenses, c_spRepresent the unit price of spare part；

S260, the expectation spare part storage cost C analyzed in a preventative maintenance cycle T_hFor:

Wherein, c_hStorage cost for single spare parts list bit time；

S270, the expectation spare part shortage failure costs C analyzed in a preventative maintenance cycle T_sFor:

Wherein, c_sShort failure costs for single spare parts list bit time；

S300, based on randomicity stock degradation model, to minimize system maintenance and spare part supply cost as target, set up maintenance and spare part supply combined optimization model；

S400, based on maintenance and spare part supply combined optimization model and the bound of given amount to be optimized, obtain maintenance and spare part the relation between supply and demand optimal solution, described amount to be optimized includes prospective maintenance cycle and the spare part maximum inventory of system.

2. combined optimization method as claimed in claim 1, it is characterised in that described step S100 farther includes:

If system part warehouse exists m spare part, the out-of-service time distribution function of each spare part is p (t), it is m p (t) that the distribution of system inefficacy spare part number Y in time t is approximately average, variance is the normal distribution of m p (t) (1-p (t)), then the probability density function of Y is g_Y(y):

Wherein, δ (y) is unit impulse function, and

3. combined optimization method as claimed in claim 1, it is characterised in that described step S300 farther includes:

To minimize system maintenance and spare part supply correlative charges as target, it is amount to be optimized with maximum part warehouse storage S and preventative maintenance cycle T, sets up following Optimized model:

Wherein, S_aFor the system spare part primary quantity when a preventative maintenance cycle T starts, C_MIt is the expectation maintenance cost in a preventative maintenance cycle T, C_oIt is the expectation spare parts purchasing expense in a preventative maintenance cycle T, C_hIt is the expectation spare part storage cost in a preventative maintenance cycle T, C_sIt it is the expectation spare part shortage failure costs in a preventative maintenance cycle T.

4. combined optimization method as claimed in claim 1, it is characterised in that described step S400 farther includes:

1) given system relevant parameter, described parameter includes calculating system maintenance and the parameters of spare part supply correlative charges, and given system runs the probability density function f of component failure time_a(t) and spare part out-of-service time distribution function p (t)=F_b(t), and calculate the operation parts expectation out-of-service time

2) bound the discretization amount to be optimized of amount to be optimized are set:

Bound [the T of preventative maintenance cycle T is set_min, T_max], and according to interval delta T discretization preventative maintenance cycle T, it is thus achieved that discrete time node T_i=T_min+ i Δ T, i=0,1,2 ...；

Set the bound [S of spare part maximum inventory S_min, S_max], and according to interval delta S discretization spare part maximum inventory S, it is thus achieved that discrete stock node S_j=S_min+ j Δ S, j=0,1,2 ...；

3) i=0 is taken；

4) discrete time node T is compared_iWith upper bound T_maxIf, T_i＞ T_max, then step 12 is entered), otherwise enter step 5)；

5) discrete time node T is calculated according to the following formula_iCorresponding M (T_i) and D (T_i):

6) j=0 is taken；

7) relatively discrete stock node S_jIf, S_j＞ S_max, then step 10 is entered), otherwise enter step 8)；

8) discrete stock node S is calculated according to the following formula_jCorresponding spare part beginning inventory S_a,

S_a=S-n-E [X_n(T_i)-X_n(T_i-τ)]-E[Y(T_i)-Y(T_i-τ)]

=S-n-n [M (T_i)-M(T_i-τ)]-S_a[p(T_i)-p(T_i-τ)]

By S_aBring following maintenance into and spare part supply combined optimization model is calculated corresponding C_{2, ∞}(S_j,T_i):

Wherein, C_MIt is the expectation maintenance cost in a preventative maintenance cycle T, C_oIt is the expectation spare parts purchasing expense in a preventative maintenance cycle T, C_hIt is the expectation spare part storage cost in a preventative maintenance cycle T, C_sIt is the expectation spare part shortage failure costs in a preventative maintenance cycle T,

9) take j=j+1, return step 7)；

10) optimal solution S of spare part maximum inventory S is calculated^{i , *}:

Now T_iIt it is set-point；

11) take i=i+1, return step 4)；

12) at the { [S obtained^i,*,T_i,C_{2, ∞}(S^i,*,T_i)] in set, find out so that system maintenance and spare part supply one group of minimum solution of correlative charges as maintenance and the optimum results of spare part supply associating relation:

Wherein, S^*For spare part maximum inventory optimum results, T^*For corresponding preventative maintenance cycle optimum results, terminate.

5. combined optimization method as claimed in claim 4, it is characterised in that:

In described step S400, the lower bound T of described preventative maintenance cycle T_minValue relevant with repair part ordering delivery time τ, upper bound T_maxValue with run parts expectation out-of-service time μ relevant.

6. combined optimization method as claimed in claim 5, it is characterised in that:

T_min=τ, T_max=1.5 μ.

7. combined optimization method as claimed in claim 4, it is characterised in that:

In described step S400, the lower bound S of described spare part maximum inventory S_minValue relevant with operation part count n of system, upper bound S_maxValue and operation part count n of system and to run parts expectation out-of-service time μ relevant.

8. to remove the combined optimization method as described in 7 such as right, it is characterised in that:

S_min=n, S_max=n+2 n M (μ).