CN112711828A - Maintenance and spare part supply joint optimization method under partially observable information - Google Patents

Maintenance and spare part supply joint optimization method under partially observable information Download PDF

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CN112711828A
CN112711828A CN202010973772.9A CN202010973772A CN112711828A CN 112711828 A CN112711828 A CN 112711828A CN 202010973772 A CN202010973772 A CN 202010973772A CN 112711828 A CN112711828 A CN 112711828A
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spare part
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左洪福
李鑫
柏宇星
许娟
郭家琛
周迪
刘珍珍
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Nanjing University of Aeronautics and Astronautics
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Abstract

The invention discloses a maintenance and spare part supply combined optimization method under partially observable information, which comprises the steps of firstly, establishing a vector autoregressive model based on real-time multidimensional state monitoring data of a system, and calculating residual errors of an integral data set; secondly, establishing a hidden Markov model capable of reflecting the system degradation process based on residual data, and estimating unknown state parameters and observation parameters in the hidden Markov model by using an expectation-maximization algorithm; then, updating the probability density function of the remaining service life of the system at each sampling moment in real time based on Bayesian theorem; finally, the delivery time of the spare parts is considered as a random variable rather than a traditional constant, and the optimal part replacement time and the spare part ordering time are dynamically updated with the aim of minimizing the average cost rate. The invention can fully utilize the multi-dimensional state monitoring information and the prediction information of the system, guide a maintenance engineer to flexibly adjust the distribution of the corresponding spare part delivery time according to the actual requirement of the spare part and determine the optimal maintenance decision.

Description

Maintenance and spare part supply joint optimization method under partially observable information
Technical Field
The invention belongs to the field of maintenance engineering of mechanical systems, and particularly relates to a maintenance and spare part supply joint optimization method under partially observable information.
Background
The problem of maintenance and spare part optimization for non-serviceable mechanical systems is a critical link in the implementation of maintenance activities. In the existing maintenance model, reliability distribution based on historical failure statistics is widely used for a replacement model and a spare part inventory model, solving such problems as optimal replacement time of parts and spare part order time in the case of satisfying economic indicators. In addition, there is a need to optimize the quantity of orders and inventory of spare parts to minimize storage costs while maximizing spare part availability.
The traditional approach is to use a reliability function that determines lifetime based on historical failure time of the components, which reflects the statistical properties of the component population, without taking into account the potential physical failure process between components. And performing sequential optimization decision of part replacement and spare part ordering on the basis of statistical life distribution. However, this method does not take into account the variability of individual components, nor does it describe the lifetime distribution function of a single individual. For newly developed equipment and large or expensive critical equipment, historical failure data is mostly lacked, so that the distribution function of the service life is difficult to obtain. In recent years, advanced state monitoring technology enables prediction of residual life of an individual, and lays a foundation for maintenance decision based on prediction information.
Conventional repair and spare part decision models often have some assumptions that are not reasonable enough, such as assuming that the life distribution function of the component is known, that spare parts are always available or that spare part lead times are not considered, etc. Although the literature researches the part replacement and spare part ordering sequential optimization decision based on the service life prediction information, the decision is not a global optimal solution and is difficult to implement in engineering practice.
Disclosure of Invention
The purpose of the invention is as follows: the invention provides a maintenance and spare part supply combined optimization method under partially observable information, which can make full use of the state monitoring information of a system to formulate a maintenance plan and guide a maintenance engineer to flexibly adjust the distribution of spare part delivery time according to actual needs.
The technical scheme is as follows: the invention relates to a maintenance and spare part supply combined optimization method under partially observable information, which comprises the following steps:
(1) selecting a healthy data part to establish a vector autoregressive model based on real-time multidimensional state monitoring data of a monitored mechanical system, and calculating a residual error of an integral data set to enable preprocessed data to meet normality and independence;
(2) establishing a hidden Markov model reflecting the system degradation process based on residual data, and estimating unknown state parameters in the hidden Markov model by using an expectation-maximization algorithm
Figure RE-GDA0002980211480000021
And observed parameters
Figure RE-GDA0002980211480000022
(3) Real-time updating system at each sampling moment t based on Bayesian theoremkProbability density function f (t | Π) of remaining useful life ofk)；
(4) The optimal part replacement time and the spare part ordering time are dynamically updated by considering the delivery time of the spare part as a random variable rather than a traditional constant with the aim of minimizing the average cost rate.
Further, the step (1) is realized as follows:
for multidimensional state monitoring data collected by the sensor, it is expressed as
Figure RE-GDA0002980211480000023
d is the dimension of the data; the health data part is assumed to follow a stationary vector autoregressive process:
Figure RE-GDA0002980211480000024
wherein epsilonnSubject to N for independent and same distributiond(0，∑)；
Figure RE-GDA0002980211480000025
Is the order of the model;
Figure RE-GDA0002980211480000026
is an autocorrelation matrix;
Figure RE-GDA0002980211480000027
is an average value;
Figure RE-GDA0002980211480000028
is a covariance;
using estimated VAR model parameters
Figure RE-GDA0002980211480000029
Calculating residual Y of overall monitoring datan；
Figure RE-GDA00029802114800000210
Wherein the content of the first and second substances,
Figure RE-GDA00029802114800000211
further, the step (2) is realized as follows:
establishing 3 states capable of reflecting the degradation of the monitored mechanical system, including a hidden Markov model of a state 0, a state 1 and a state 2, wherein the state 0 represents a healthy state, the state 1 represents an unhealthy state, and the state 2 represents a failed state; estimating state parameters in hidden Markov models using expectation-maximization algorithms
Figure RE-GDA00029802114800000212
And observed parameters
Figure RE-GDA00029802114800000213
By ΠkRepresents the sampling time t at the k-th timekGiven observation data yΔ，y2Δ，...，
Figure RE-GDA0002980211480000031
Posterior probability of system in state 1 under the conditions:
Πk＝Pr(Xk＝1|ξ＞kΔ，yΔ，y2Δ，...，ykΔ) (3)
further, the step (3) is realized as follows:
the posterior probability pi of each sampling point is determined by Bayes' theoremkThe update can be iteratively updated by:
Figure RE-GDA0002980211480000032
at sampling time tkThe conditional reliability function for the remaining life of the system is:
Figure RE-GDA0002980211480000033
the probability density function is:
Figure RE-GDA0002980211480000034
at sampling time tk(i.e., k-th sample, total time k Δ), the joint optimization function for repair and spare parts ordering under part of observable information is:
Figure RE-GDA0002980211480000035
wherein E (CC) is the expected total cost of a life cycle, and E (CL) is the expected time length of a life cycle, so thatThe minimum value of the average cost rate of one period is the corresponding optimal replacement time
Figure RE-GDA0002980211480000036
And spare part order time
Figure RE-GDA0002980211480000041
Constraint conditions
Figure RE-GDA0002980211480000042
Indicating that the spare part ordering time should be before the current sampling time and the replacement time; if the optimized spare part ordering time is after the next sampling time, the next sampling is continued without adopting a spare part ordering strategy, and the optimal replacement time and the optimal spare part ordering time are updated; and if the spare part ordering time is before the next sampling, taking maintenance measures according to the optimized replacement time and the spare part ordering time.
Further, the step (4) is realized as follows:
at sampling time tkThere are five possible scenarios between the spare part order time, the spare part arrival time, the replacement time, and the time to failure: a component failure before the spare part order time point; a component failure occurs between the point in time when the spare part has issued an order and the spare part has not been delivered; failure of the component between the time the spare part has been delivered and the optimal replacement time; the spare part has issued an order request but not delivered before the optimal replacement time point; spare parts have been delivered, and component failure occurs after optimal replacement time;
the expected spare part shortage times in the five cases are respectively represented by ES1, ES2, ES3, ES4 and ES5, and the expected spare part shortage times in each condition are calculated as follows:
Figure RE-GDA0002980211480000043
Figure RE-GDA0002980211480000044
ES3＝0 (14)
Figure RE-GDA0002980211480000045
ES5＝0 (16)
the expected spare part shortage times in the five cases are respectively represented by EH1, EH2, EH3, EH4 and EH5, and the expected spare part holding time under each condition is calculated as follows:
EH1＝0 (17)
EH2＝0 (18)
Figure RE-GDA0002980211480000046
EH4＝0 (20)
Figure RE-GDA0002980211480000047
the expected spare part shortage time ES for a life cycle is:
ES＝ES1+ES2+ES3+ES4+ES5 (22)
the expected spare part hold time EH for a life cycle is:
EH＝EH1+EH2+EH3+EH4+EH5 (23)
the expected total cost e (cc) for a life cycle is then:
Figure RE-GDA0002980211480000051
the expected length of time in a life cycle, e (cl), is:
Figure RE-GDA0002980211480000052
has the advantages that: compared with the prior art, the invention has the beneficial effects that: the invention can make a maintenance plan by fully utilizing the state monitoring information of the system, guides a maintenance engineer to flexibly adjust the distribution of spare part delivery time according to actual needs and further makes an optimal maintenance decision according with actual engineering application.
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FIG. 1 is a flow chart of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings.
As shown in fig. 1, the present invention provides a joint optimization method for maintenance and spare part supply under partially observable information, which specifically comprises the following steps:
step 1: based on real-time multidimensional state monitoring data of a monitored mechanical system, a healthy data part is selected to establish a vector autoregressive model, and the residual error of the whole data set is calculated, so that preprocessed data meet normality and independence.
For multidimensional state monitoring data collected by the sensor, it is expressed as
Figure RE-GDA0002980211480000053
K is the dimension of the data. The health data portion is assumed to follow a stationary Vector Autoregressive (VAR) process:
Figure RE-GDA0002980211480000054
wherein epsilonnSubject to N for independent and same distributionK(0，∑)；
Figure RE-GDA0002980211480000055
Is the order of the model;
Figure RE-GDA0002980211480000056
is an autocorrelation matrix;
Figure RE-GDA0002980211480000057
is an average value;
Figure RE-GDA0002980211480000058
is the covariance. The multidimensional state monitoring data refers to monitoring data collected by a plurality of sensors, the data collected by one sensor is one-dimensional, and generally a system can be provided with a plurality of sensors at different positions.
Using estimated VAR model parameters
Figure RE-GDA0002980211480000061
Residual error Y capable of calculating overall monitoring datan：
Figure RE-GDA0002980211480000062
Wherein the content of the first and second substances,
Figure RE-GDA0002980211480000063
step 2: establishing a hidden Markov model reflecting the system degradation process based on residual data, and estimating unknown state parameters in the hidden Markov model by using an expectation-maximization algorithm
Figure RE-GDA0002980211480000064
And observed parameters
Figure RE-GDA0002980211480000065
Real-time updating system at each sampling moment t based on Bayesian theoremkProbability density function f (t | Π) of remaining useful life ofk)。
Based on the overall residual data, the degradation process of the system is assumed to conform to a Hidden Markov Model (HMM) of 3 states (state space S ═ {0, 1, 2 }): a healthy state (state 0), a warning state or unhealthy state (state 1), and a failed state (state 2). State 0 and state 1 are non-observable states, i.e., the states are hidden. Only state 2 can be observed directly. Suppose the system always starts in a healthy state, i.e., P (X)00) 1. The transient transfer rate matrix is:
Figure RE-GDA0002980211480000066
wherein λ is01，λ02，λ12E (0, + ∞) is an unknown state parameter in the model, representing the state transition rate.
When xi ═ inf { t ≧ 0: xt2 represents the failure time of the system, and Δ is the sampling interval. Observed value
Figure RE-GDA0002980211480000067
Independent given the system state. By yΔ，y2Δ，...，
Figure RE-GDA0002980211480000068
Representing d-dimensional observations, then y in state xkΔObey Nd(μx，∑x) And x is a d-element normal distribution of 0 and 1, and the probability density function is as follows:
Figure RE-GDA0002980211480000069
wherein the content of the first and second substances,
Figure RE-GDA00029802114800000610
the unknown observation parameters in the model respectively represent the mean value and the covariance in each state.
Assuming N sets of collected condition monitoring failure data, use F1，...，FNAnd (4) showing. Failure data FiBy using
Figure RE-GDA0002980211480000071
Denotes, TiThe total number of samples of the ith group of data is xii＝tiWherein T isiΔ＜ti≤(Ti+1) Δ. Assuming that M sets of truncated data are collected, S is used1，...，SjAnd (4) showing. Likewise, truncated data SjBy using
Figure RE-GDA0002980211480000072
Indicating time of failure ξi＞TiAnd delta. With O ═ F1，...，FN，S1，...，SMDenotes the observed data, k ═ (a, Ψ | O) as the corresponding likelihood function, where a ═ λ (λ ═ O)01，λ02，λ12)，Ψ＝(μ0，μ1，∑0，∑1) Is the parameter to be estimated. Sample path (X) due to hidden Markov model state processt: t ≧ 0) is not observable, so the analytical expression for the maximum likelihood function is difficult to solve. The expectation-maximization (EM) algorithm may be solved by iteratively maximizing the pseudo-likelihood function. Order to
Figure RE-GDA0002980211480000073
And
Figure RE-GDA0002980211480000074
for the initial value of the parameter to be estimated, the EM algorithm comprises the following steps:
e-step: calculating a pseudo-likelihood function:
Figure RE-GDA0002980211480000075
wherein
Figure RE-GDA0002980211480000076
For a complete data set, i.e. an observation data set o each set of failure data FiAnd truncated data SjNon-observable sample path information for the state process is augmented.
M-step: selecting Λ*，Ψ*So that
Figure RE-GDA0002980211480000077
Parameter Λ updated per step*，Ψ*Then used as the first generationE-step, so that E-step and M-step iterate operation until Euclidean norm
Figure RE-GDA0002980211480000078
Where ε is an arbitrarily small positive number.
By maximizing the expected value of each update step, the estimated values of the state parameters and the observation parameters of each update step, i.e. the points at which the derivative of each parameter is 0, can be obtained. Updated at each step
Figure RE-GDA0002980211480000079
The explicit expression of (c) is as follows:
Figure RE-GDA00029802114800000710
Figure RE-GDA0002980211480000081
Figure RE-GDA0002980211480000082
updated at each step
Figure RE-GDA0002980211480000083
The explicit expression of (c) is as follows:
Figure RE-GDA0002980211480000084
Figure RE-GDA0002980211480000085
wherein:
Figure RE-GDA0002980211480000086
Figure RE-GDA0002980211480000087
Figure RE-GDA0002980211480000088
Figure RE-GDA0002980211480000089
Figure RE-GDA00029802114800000810
parameter in completion status
Figure RE-GDA00029802114800000811
And observed parameters
Figure RE-GDA00029802114800000812
After the estimation. By ΠkIndicating that the observation data y is given at the time k ΔΔ，y2Δ，...，
Figure RE-GDA00029802114800000813
Posterior probability of system in state 1 under the conditions:
Πk＝Pr(Xk＝1|ξ＞kΔ，yΔ，y2Δ，...，ykΔ) (12)
the posterior probability pi of each sampling point is determined by Bayes' theoremkThe update can be iteratively updated by:
Figure RE-GDA00029802114800000814
wherein the initial value pi0＝0，Pij(t)＝P(Xt＝j|X0I) is the transition probability matrix. Transition probability matrix Pij(t) torque available for transfer rateThe array is solved by Kolmogorov backward differential equations.
Thus, at the sampling time tkThe conditional reliability function for the remaining life of the system is:
Figure RE-GDA0002980211480000091
the probability density function is:
Figure RE-GDA0002980211480000092
and step 3: the optimal part replacement time and the spare part ordering time are dynamically updated by considering the delivery time of the spare part as a random variable rather than a traditional constant with the aim of minimizing the average cost rate.
At sampling time tk(i.e., kth sample, total time k Δ), there are five possible scenarios between spare part order time, spare part arrival time, replacement time, and failure time:
1) a component failure before the spare part order time point;
2) a component failure occurs between the point in time when the spare part has issued an order and the spare part has not been delivered;
3) failure of the component occurs between the time when the spare part has been delivered and the optimal replacement time.
4) The order request has been issued by the spare part, but the spare part is not delivered until the optimal replacement time point.
5) Spare parts have been delivered and failure of the part occurs after the optimal replacement time.
The expected spare part shortage times in the five cases are respectively represented by ES1, ES2, ES3, ES4 and ES5, and the expected spare part shortage times in each condition are calculated as follows:
Figure RE-GDA0002980211480000093
Figure RE-GDA0002980211480000101
ES3＝0 (18)
Figure RE-GDA0002980211480000102
ES5＝0 (20)
wherein l is lead time; (l) is a probability density function of delivery time;
Figure RE-GDA0002980211480000103
is tkThe time of the order of the spare part at the moment,
Figure RE-GDA0002980211480000104
is tkTime of day component replacement.
The expected spare part shortage times in the five cases are respectively represented by EH1, EH2, EH3, EH4 and EH5, and the expected spare part holding time under each condition is calculated as follows:
EH1＝0 (21)
EH2＝0 (22)
Figure RE-GDA0002980211480000105
EH4＝0 (24)
Figure RE-GDA0002980211480000106
from the above analysis, the expected spare part shortage time ES in a life cycle is:
ES＝ES1+ES2+ES3+ES4+ES5 (26)
the expected spare part hold time EH for a life cycle is:
EH＝EH1+EH2+EH3+EH4+EH5 (27)
the expected total cost e (cc) for a life cycle is then:
Figure RE-GDA0002980211480000107
the expected length of time in a life cycle, e (cl), is:
Figure RE-GDA0002980211480000108
combining the above analysis, the replacement and spare part ordering joint optimization function based on the prediction information is:
Figure RE-GDA0002980211480000111
in the above equation, the value that minimizes the average cost rate of one cycle is the optimal replacement time
Figure RE-GDA0002980211480000112
And spare part order time
Figure RE-GDA0002980211480000113
The constraint indicates that the spare part order time should be between the current sampling time and the replacement time.
And if the optimized spare part ordering time is larger than the next sampling time, the next sampling is continued without adopting a spare part ordering strategy. And if the spare part ordering time is before the next sampling, taking maintenance measures according to the optimized replacement time and the spare part ordering time.
Although the present invention has been described with reference to the preferred embodiments, it is not intended to be limited thereto. Those skilled in the art can make various changes and modifications without departing from the spirit and scope of the invention. Therefore, the protection scope of the present invention should be determined by the appended claims.

Claims (5)

1. A method for jointly optimizing maintenance and spare part supply under partially observable information, comprising the steps of:
(1) selecting a healthy data part to establish a vector autoregressive model based on real-time multidimensional state monitoring data of a monitored mechanical system, and calculating a residual error of an integral data set to enable preprocessed data to meet normality and independence;
(2) establishing a hidden Markov model reflecting the system degradation process based on residual data, and estimating unknown state parameters in the hidden Markov model by using an expectation-maximization algorithm
Figure RE-FDA0002887068250000011
And observed parameters
Figure RE-FDA0002887068250000012
(3) Real-time updating system at each sampling moment t based on Bayesian theoremkProbability density function f (t | Π) of remaining useful life ofk)；
(4) The optimal part replacement time and the spare part ordering time are dynamically updated by considering the delivery time of the spare part as a random variable rather than a traditional constant with the aim of minimizing the average cost rate.
2. The joint optimization method for repair and spare part supply under partially observable information according to claim 1, wherein the step (1) is implemented as follows:
for multidimensional state monitoring data collected by the sensor, it is expressed as
Figure RE-FDA0002887068250000017
d is the dimension of the data; the health data part is assumed to follow a stationary vector autoregressive process:
Figure RE-FDA0002887068250000013
wherein epsilonnAre independently and identically distributed, obeyNd(0, Σ); p belongs to the order of the model; phir∈d×dIs an autocorrelation matrix; delta0∈dIs an average value; Σ ∈d×dIs a covariance;
using estimated VAR model parameters
Figure RE-FDA0002887068250000014
Calculating residual Y of overall monitoring datan：
Figure RE-FDA0002887068250000015
Wherein the content of the first and second substances,
Figure RE-FDA0002887068250000016
3. the joint optimization method for repair and spare part supply under partially observable information as claimed in claim 1, wherein the step (2) is implemented as follows:
establishing 3 states capable of reflecting the degradation of the monitored mechanical system, including a hidden Markov model of a state 0, a state 1 and a state 2, wherein the state 0 represents a healthy state, the state 1 represents an unhealthy state, and the state 2 represents a failed state; estimating state parameters in hidden Markov models using expectation-maximization algorithms
Figure RE-FDA0002887068250000021
And observed parameters
Figure RE-FDA0002887068250000022
By ΠkRepresents the sampling time t at the k-th timekGiven observation data yΔ,y2Δ,...,ykΔ∈dPosterior probability of system in state 1 under the conditions:
Πk＝Pr(Xk＝1|ξ＞kΔ,yΔ,y2Δ,...,ykΔ)。 (3)
4. the joint optimization method for repair and spare part supply under partially observable information as claimed in claim 1, wherein the step (3) is implemented as follows:
the posterior probability pi of each sampling point is determined by Bayes' theoremkThe update can be iteratively updated by:
Figure RE-FDA0002933339480000024
at sampling time tkThe conditional reliability function for the remaining life of the system is:
Figure RE-FDA0002933339480000025
the probability density function is:
Figure RE-FDA0002933339480000026
at sampling time tk(i.e., k-th sample, total time k Δ), the joint optimization function for repair and spare parts ordering under part of observable information is:
Figure RE-FDA0002933339480000031
wherein E (CC) is the expected total cost in a life cycle, E (CL) is the expected time length of a life cycle, and the value of minimizing the average cost rate of a cycle is the corresponding optimal replacement time
Figure RE-FDA0002933339480000032
And spare part order time
Figure RE-FDA0002933339480000033
Constraint conditions
Figure RE-FDA0002933339480000034
Indicating that the spare part ordering time should be before the current sampling time and the replacement time; if the optimized spare part ordering time is after the next sampling time, the next sampling is continued without adopting a spare part ordering strategy, and the optimal replacement time and the optimal spare part ordering time are updated; and if the spare part ordering time is before the next sampling, taking maintenance measures according to the optimized replacement time and the spare part ordering time.
5. The joint optimization method for repair and spare part supply under partially observable information according to claim 1, wherein the step (4) is implemented as follows:
at sampling time tkThere are five possible scenarios between the spare part order time, the spare part arrival time, the replacement time, and the time to failure: a component failure before the spare part order time point; a component failure occurs between the point in time when the spare part has issued an order and the spare part has not been delivered; failure of the component between the time the spare part has been delivered and the optimal replacement time; the spare part has issued an order request but not delivered before the optimal replacement time point; spare parts have been delivered, and component failure occurs after optimal replacement time;
the expected spare part shortage times in the five cases are respectively represented by ES1, ES2, ES3, ES4 and ES5, and the expected spare part shortage times in each condition are calculated as follows:
Figure RE-FDA0002887068250000036
Figure RE-FDA0002887068250000037
ES3＝0 (14)
Figure RE-FDA0002887068250000038
ES5＝0 (16)
the expected spare part shortage times in the five cases are respectively represented by EH1, EH2, EH3, EH4 and EH5, and the expected spare part holding time under each condition is calculated as follows:
EH1＝0 (17)
EH2＝0 (18)
Figure RE-FDA0002887068250000041
EH4＝0 (20)
Figure RE-FDA0002887068250000042
the expected spare part shortage time ES for a life cycle is:
ES＝ES1+ES2+ES3+ES4+ES5 (22)
the expected spare part hold time EH for a life cycle is:
EH＝EH1+EH2+EH3+EH4+EH5 (23)
the expected total cost e (cc) for a life cycle is then:
Figure RE-FDA0002887068250000043
the expected length of time in a life cycle, e (cl), is:
Figure RE-FDA0002887068250000044
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