CN108445761B - Joint modeling method for process control and maintenance strategy based on GERT network statistics - Google Patents

Joint modeling method for process control and maintenance strategy based on GERT network statistics Download PDF

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CN108445761B
CN108445761B CN201810212387.5A CN201810212387A CN108445761B CN 108445761 B CN108445761 B CN 108445761B CN 201810212387 A CN201810212387 A CN 201810212387A CN 108445761 B CN108445761 B CN 108445761B
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李亚平
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Nanjing Forestry University
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Abstract

The invention provides a statistical process control and maintenance strategy joint modeling method based on a GERT network, which comprises the following steps: monitoring the manufacturing process on line by using an X-bar control chart to obtain a production state; calculating the sampling interval and the probability of I and II errors in the monitoring process; constructing a GERT network architecture, and calculating a cost expectation and a time expectation required by joint modeling; and establishing a nonlinear optimization model, designing an algorithm solving model, and obtaining the optimal control chart parameters and a maintenance strategy.

Description

Joint modeling method for process control and maintenance strategy based on GERT network statistics
Technical Field
The invention relates to a statistical process control and maintenance decision making technology, in particular to a statistical process control and maintenance strategy combined modeling method based on a GERT network.
Background
With the continuous progress of global economy, science and technology and industry, novel manufacturing technologies such as flexible manufacturing and agile manufacturing are rapidly developed, and the manufacturing industry is increasingly competitive. How to control the manufacturing cost and improve the product quality is an important way for enterprises to obtain market competitive advantages and is also an important legal treasure for the enterprises to be durable.
On-line monitoring is an effective means for finding production problems in time and avoiding unnecessary loss and waste. The control chart is a classic method for on-line monitoring, and the parameter design result directly influences the product quality and the manufacturing cost. The control chart parameter design problem is researched under the economic view angle, namely the control chart is economically designed. Maintenance is to perform operations such as updating, maintaining or replacing parts of the equipment according to the production conditions, so as to reduce the number of defective products. The purpose of the maintenance strategy optimization is to reasonably arrange the maintenance frequency and the maintenance degree and avoid excessive maintenance or insufficient maintenance as much as possible.
The two important problems of control map economic design and maintenance strategy optimization design are related and mutually influenced. In statistical process control, it is necessary to perform scheduled maintenance (preventive maintenance) to reduce equipment failure rate as much as possible, and it is further necessary to perform corrective maintenance actions to restore the process out of control to a controlled state. Therefore, the process control and maintenance strategies are integrated, the combined optimization of the process control and the maintenance strategies is realized, the product quality can be improved, and the economic benefit of production can be improved.
The goal of statistical process control and maintenance strategy joint modeling is generally to minimize the cost expectancy in the production cycle or the cost per unit time expectancy, thus it is seen that the core of the problem is the solution of the cost expectancy and time expectancy of the production cycle. With the continuous and deep research of the problem, there are 3 main solutions at present:
(1) recursive method
Research based on a recursion method firstly divides production states into several types, and then deduces time expectation and cost expectation (including quality control cost, maintenance cost and the like) in a production period by using the recursion method according to a transfer relation between the states.
(2) Method for classifying scenes
The research based on the scene classification method mainly divides the production process into several different production scenes, respectively calculates various time expectations and cost expectations and the occurrence probability of various scenes, and on the basis, calculates the time expectations and the cost expectations in the production period.
(3) Markov method
Markov-method-based research relies primarily on markov processes/chains, computing probabilities for different states and time and cost expectations based on transition relationships between states.
The above 3 methods simplify the production process and maintenance parameters. As production state/scenario classification increases, the complexity of modeling based on recursion and scenario classification increases dramatically, making computation very difficult. Moreover, the situation classification method cannot enumerate all production situations due to the limitations of the model itself. The markov method also has a problem that all production states cannot be considered, but can solve a more complicated production state problem slightly better than the former two methods due to the advantages of the stochastic model itself. In recent years, the markov method has been used to solve the problem of joint modeling of statistical process control and maintenance strategies. However, the markov approach still has some drawbacks in solving this problem:
first, it is assumed that the plant failure times follow a negative exponential distribution or geometric distribution (when failure times are expressed in terms of pre-quality-shift product numbers) to facilitate modeling by the Markov method. However, in practice, the equipment failure times may follow an arbitrary distribution.
Second, maintenance parameters are typically ignored or assumed to be constant processing. In fact, the maintenance time, cost, etc. of a system are described by random variables or functions, which description is more consistent with actual manufacturing process conditions, provided that it satisfies some statistical function or law.
Disclosure of Invention
The invention aims to provide a process control and maintenance strategy combined modeling method based on GERT network statistics, which is more consistent with the actual situation of the manufacturing process.
The technical scheme for realizing the purpose of the invention is as follows: a process control and maintenance strategy combined modeling method based on GERT network statistics comprises the following steps:
step 1, using an X-bar control chart to monitor the manufacturing process on line to obtain a production state;
step 2, calculating the sampling interval and the probability of I-type and II-type errors in the monitoring process;
step 3, constructing a GERT network architecture, and calculating a cost expectation and a time expectation required by joint modeling;
and 4, establishing a nonlinear optimization model, designing an algorithm solving model, and obtaining the optimal control chart parameters and a maintenance strategy.
Compared with the prior art, the invention has the following advantages: (1) the invention provides a method for carrying out combined modeling of statistical process control and maintenance strategies in the manufacturing process by using a semi-Markov process-GERT network to obtain a production cycle time expectation and a total cost expectation in a cycle; (2) adding statistical constraint, economic constraint and other practical constraints, minimizing the unit time cost expectation as an optimization target, establishing a nonlinear programming model, and solving to obtain an optimal parameter, wherein the method not only considers an equipment failure mechanism obeying any distribution, but also considers the uncertainty problem of the maintenance parameter, sets the maintenance parameter as a random variable obeying any distribution, and better accords with the practical situation of the manufacturing process; (3) the invention not only enriches the theory and method in the field, but also provides a new overall solution for the combined decision of quality control and maintenance strategy, and has very important practical significance for reducing the manufacturing cost, ensuring the product quality, improving the economic benefit of enterprises and the like.
The invention is further described below with reference to the accompanying drawings.
Drawings
FIG. 1 is a flow chart of an embodiment of the present invention.
Fig. 2 is a GERT network model state transition diagram.
Detailed Description
With reference to fig. 1, a joint modeling method for statistical process control and maintenance strategy based on GERT network includes the following steps:
step S101, analyzing a state node and a state transition relation, and constructing a GERT network architecture;
monitoring the manufacturing process on line by using an X-bar control chart to obtain a production state;
step S102, calculating a sampling interval;
step S103, calculating the probability of I and II errors in the monitoring process;
step S104, calculating the probability P of state transitionijProbability density function of parameters (cost, time);
step S105, calculating a moment mother function M of the parametersij；
Step S106, calculating a transfer function W of the parameterij；
Step S107, calculating a network equivalent transfer function W;
step S108, converting the equivalent transfer function into an equivalent moment mother function M;
step S107, deducing a time expectation E through an equivalent moment mother functiontAnd cost expectation Ec；
And S108, establishing a nonlinear optimization model, designing an algorithm solving model, and obtaining the optimal control chart parameters and a maintenance strategy.
In step S101, with reference to FIG. 2, the method comprisesAnalyzing the state node and state transfer relation, and constructing a GERT network architecture, wherein the GERT network architecture comprises the following steps: the invention assumes that the production process is a random process of discrete time, and the discrete time point corresponds to sampling detection. Because each sampling considers 4 types of production states, the S can be controlled without alarmingi1Controllable alarm Si2Out of control no alarm Si3Out of control alarm Si4. The state space then comprises 4(m-1) +2 possible states, including the initial state point S0Then, S11,S12,S13,S14,…,S(m-1)1,S(m-1)2,S(m-1)3,S(m-1)4And finally, waiting for preventive maintenance point Sm. The production process is in a controllable state when the production process starts to run. Assuming that i (i ═ 1,2, …, m-1) sampling intervals have continued since the last corrective or preventative maintenance, then if the process is controlled and there is no alarm, the process is in state Si1(ii) a If the process is controllable but gives a false alarm, the process is in state Si2(ii) a If the process is out of control but there is no alarm, the process is in state Si3(ii) a If the process is time-space and an alarm is raised, the process is in state Si4. If m sampling intervals have continued, the process is in state SmI.e. waiting for preventive maintenance. The semi-Markov problem is solved by applying a GERT network model, wherein the states are network nodes, the state transfer relationship obeys exclusive OR logic, namely, only one state occurs at each time point, and meanwhile, maintenance parameters obeying arbitrary distribution are used as original parameters of the network. Note that for convenient modeling, a virtual node S is added0', its meaning with S0And (5) the consistency is achieved.
When the manufacturing process is in a controllable state, the quality characteristic value x randomly fluctuates around the mean value mu and follows a normal distribution, namely x to N (mu, sigma)2) Where μ and σ are known.
Production run-away taking into account mass property sample mean from μ only0(μ0μ) to
Figure BDA0001597577010000041
Regardless of the offset generated by the standard deviation, letIt remains constant.
The implementation of the maintenance plan is fused while the quality monitoring in step S101. And if the process is out of control, implementing correction maintenance, and if the process is controllable but the preventive maintenance time is up, implementing preventive maintenance. Setting the failure time of the equipment to comply with two-parameter Weibull distribution, wherein the failure density function is f (t) ═ gamma vtv-1e-γtT is more than 0, v is more than or equal to 1, and gamma is more than 0, wherein gamma is more than 0, and v is more than or equal to 1, which are a proportion parameter and a shape parameter respectively.
Preventive maintenance comprises the tasks of preparation, detection and diagnosis, part replacement, adjustment, inspection, original part repair and the like; the correction and maintenance comprises the tasks of preparation, abnormal reason troubleshooting, piece changing, adjustment, inspection, original part repair or replacement and the like; the compensation maintenance comprises the tasks of preparation, detection diagnosis and the like. The preventive maintenance and the correction maintenance belong to perfect maintenance, and the equipment is recovered as new after the maintenance. Maintenance time and maintenance cost follow an arbitrary probability distribution (constants can be considered as special cases of probability distributions).
In step S102, the specific process of the sampling interval calculation is as follows:
to equalize the risk of equipment failure within the sampling interval, i.e.
Figure BDA0001597577010000042
In the formula (1), tiDenotes a point in time at which the ith sampling is performed, and λ (t) is a failure rate function indicating a probability that a device which has not failed by the time t has operated and failed in the next unit time after the time t, that is
Figure BDA0001597577010000051
In the formula (2), f (t) is a probability density function of equipment failure,
Figure BDA0001597577010000052
are complementary cumulative probability distribution functions.
Suppose the sampling interval is h1,h2,…,hmThen, then
Figure BDA0001597577010000053
Formula (1) can be changed to formula (3),
Figure BDA0001597577010000054
since the failure time of the plant follows a two-parameter Weibull distribution, the failure rate function λ (t) is then, according to equation (2)
λ(t)＝γνtν-1 (4)
Bringing formula (4) into formula (3) to obtain formula (5),
h1 ν＝(ti-1+hi)ν-ti-1 ν,i＝1,…,m (5)
according to
Figure BDA0001597577010000055
With equation (5), all sampling intervals can be converted to a function h of the first sampling interval1I.e. by
Figure BDA0001597577010000056
In step S103, let α denote the probability of a class I error, β denote the probability of a class II error, and Φ denote the cumulative distribution function of the normal random variables, then equations (7) and (8) are obtained,
α＝1-[Φ(k)-Φ(-k)] (7)
Figure BDA0001597577010000057
in step S104, assume that the process is at time point ti-1I is 1,2, …, m is controllable, let piAt time intervals (t) for the processi-1,ti) The probability of internal runaway, then,
Figure BDA0001597577010000058
in the formula (9), F (. cndot.) is a cumulative function of Weibull distribution.
State S(i-1)1I-1, …, m-1 (specifically, S)01Is S0) Transition to State Sir(r is 1,2,3,4) each
Pi1＝(1-pi)(1-α) (10)
Pi2＝(1-pi)α (11)
Pi3＝piβ (12)
Pi4＝pi(1-β) (13)
State Si2I-1, 2, …, m-1 transitions to state Si1Has a probability of
Pi5＝1 (14)
State Si4I-1, 2, …, m-1 to S0′Has a probability of
Pi6＝1 (15)
State S(i-1)3I-2, …, m-1 to SirAnd the probability of r being 3,4 is
Pi7＝β (16)
Pi8＝1-β (17)
State S(m-1)r(r-1, 3) to SmHas a probability of
Pmj＝1,j＝1,2 (18)
State SmTransfer to S0A probability of
Pm3＝1 (19)
Let Q0For the loss of mass per unit time with controllable process, the calculation of the relevant parameters is divided into the following cases, which are analyzed as follows:
(1) state S(i-1)1I-1, …, m-1 (specifically, S)01Is S0) Transition to State Si1Time spent is constant hi. The process has no mass shift and the mass loss is Q0hiSampling cost of cf+cvn(cfFor a fixed cost of one sampling, cvIs a unit variable cost), then the total cost is a constant (c)f+cvn)+Q0hi。
(2) State S(i-1)1I-1, …, m-1 (specifically, S)01Is S0) Transition to State Si2In time, no deviation occurs in the process, and only a false alarm is given. Similarly to the case (1), the elapsed time is constant hiThe total cost is constant (c)f+cvn)+Q0hi。
(3) State S(i-1)1I-1, …, m-1 (specifically, S)01Is S0) Transition to State Si3At that time, although no alarm is issued, the offset has already occurred. Time is still constant hi. The cost is composed of the loss of quality when the process is controllable and the loss of quality when the process is not controllable, and the sampling cost. Let Q1Is the loss of mass per unit time when the process is out of control. Suppose the process is offset from ti-1Rear tauiAfter a unit of time, the total mass loss is then Q0τi+Q1(hi-τi). Plus the sampling cost, the total cost is cf+cvn+Q0τi+Q1(hi-τi). Here, τiIs a random variable with a probability density function of f taui(x)＝f(x+ti-1)/(F(ti)-F(ti-1)). On the basis of the above, the total cost probability density is derived as
Figure BDA0001597577010000071
(4) State S(i-1)1I-1, …, m-1 (specifically, S)01Is S0) Transition to State Si4The process is skewed and an alarm is raised. Similarly to case (3), the time is still constant hi. The total cost is a random variable and the probability density function is similar to equation (20).
(5) State Si2I-1, …, m-1 to Si1Only compensation maintenance is performed during this time interval, and the validation process does not actually occurAnd (4) offsetting and releasing the alarm. The consumed time and cost are the time and cost of one compensation maintenance, i.e. tCpM，cCpM. Both are random variables.
(6) State S(i-1)3, i-2, …, m-1 transfer to SirAnd when r is 3,4, the process keeps an uncontrolled state. The elapsed time is constant h whether or not an alarm is issuediThe cost is made up of the loss of mass when the process is out of control and the cost of sampling, i.e. (c)f+cvn)+Q1hi。
(7) State Si4I-1, 2, …, m-1 to S0′Only corrective maintenance is performed during this time interval, restoring the runaway process to, for example, the new state. The time and cost of the correction is the time and cost of one correction maintenance, i.e. tCM，cCM. Both are random variables.
(8) State S(m-1)1Transfer to SmTime spent is constant hm. At a time interval (t)m-1,tm) Inner, taumAfter a unit of time, the process may drift, with τi(i.ltoreq.m-1) analogously, τmIs a random variable whose probability density function is f taum(x)＝f(x+tm-1)/(F(tm)-F(tm-1)). Total cost of consumption in the interval of cm1＝Q0τm+Q1(hm-τm) Deriving a probability density function of
Figure BDA0001597577010000072
The m-th sampling time point is not sampled any more, but is directly subjected to preventive maintenance.
(9) State S(m-1)3Transition to State SmTime spent is constant hmTotal cost is the loss of mass Q of the process out of control1hm。
(10) State SmTransfer to S0In this period, only the replacement type preventive maintenance is performed, and the process is restored to the new state. The time and cost areTime and cost of one preventive maintenance, i.e. tPM，cPM. Both are random variables.
In step S105, let S be any real number, and if the total cost of the state transition is constant, then
Figure BDA0001597577010000081
Otherwise
Figure BDA0001597577010000082
Where f (y) is the probability density function of the total cost.
Step S106, taking cost expectation as an example, gives the transfer function W in detailijAnd (3) calculating:
(1) if m.gtoreq.2
For 1. ltoreq. i.ltoreq.m-1, let Wij(j is 1,2,3,4) is S(i-1)1(specifically, S)01Is equivalent to S0) Transfer to Sir(r ═ 1,2,3,4) transfer function;
for i is 2. ltoreq. m-1, let Wij(j is 5,6,7, 8) is Si2Transfer to Si1，Si4Transfer to S0′，S(i-1)3Transfer to Si3，S(i-1)3Transfer to Si4The transfer function of (a);
let Wmj(j＝1，2，3)Wij(j ═ 1) is S(i-1)1Transfer to Si，S(i-1)3Transfer to Si，SiTransfer to S0' transfer function.
(2) If m is 1
Let W11Is S0Transfer to SmThe transfer function of (a) is selected,
let W13Is SmTransfer to S0' transfer function.
In step S107, after the cost transfer function is obtained in step S106, let Wcc(m, S) is transition S0To S0 ′The equivalent transfer function of (1), hereinafter, W is obtained by using the induction methodcc(m,s)：
If m is equal to 1, the ratio of m,
Wcc(1,s)＝W11W13 (22)
if m is equal to 2, the ratio of m,
Figure BDA0001597577010000083
if m is 3, the number of the atoms is 3,
Figure BDA0001597577010000091
if m is equal to 4, the ratio of m,
Figure BDA0001597577010000092
if m is more than or equal to 5,
Figure BDA0001597577010000093
in step S108, let Eq _ Mcc(m, s) is Wcc(m, s) corresponding to an equivalent moment mother function, then
Figure BDA0001597577010000094
Let Ec(m) cost expectation over period, according to the nature of the intalox function: the value of the nth derivative of the moment mother function at s ═ 0, i.e. the nth origin moment of the random variable, yields equation (28),
Figure BDA0001597577010000095
in steps S105 to S107, if the parameter is time, let Wtt(m, S) is S0Transfer to S0' equivalent transfer function, Eq _ Mtt(m, s) is WttEquivalent moment mother function corresponding to (m, s), Et(m) for time expectation, adding Wtt(m, s) instead of Wcc(m,s)、Eq_Mtt(M, s) instead of Eq _ Mcc(m,s)、Et(m) in place of Ec(m), using the same method as above, a specific expression can be obtained.
And S108, establishing a nonlinear optimization model, designing an algorithm solving model, and obtaining the optimal control chart parameters and a maintenance strategy.
According to the time expectation and the cost expectation obtained in step 3, the cost per unit time is expected to be,
Figure BDA0001597577010000101
adding constraint conditions, establishing a nonlinear optimization model,
Figure BDA0001597577010000102
in the formula (30), ARL0≥C1The average run length representing the controllable state is not less than C1，ARL1≤C2Mean run length indicating runaway conditions not greater than C2，ai,biI-1, 2,3,4 respectively denote constraints on the decision variables, C1、C2Are all known parameters.
For the nonlinear optimization problem, model optimal solution can be obtained by solving through pattern search, genetic algorithm and the like, and the optimal combination scheme of control chart and maintenance can be obtained.
Examples
The following is a case of a cotton machine for making cotton. The known two-parameter weibull distribution γ for machine failure times is 0.05 and ν 2. The production process is monitored by using an X-bar control chart, and the sampling fixed cost is cf$2.0, variable cost is cv$ 0.5. The loss of mass per unit time when the process is controllable is Q0$50, the loss of mass per unit time when the process is controlled is Q1$ 950. The average deviation of process runaway is 1. Assuming that the compensatory maintenance parameters obey an exponential distribution, cCpM～Exp(0.002),tCpMExp (3), correctionThe positive maintenance parameter follows a normal distribution, cCM～N(1100,502),tCM～N(1,0.22) The preventive maintenance parameter is constant, cPM＝250,tPM＝0.1。
At the same time, the ARL is limited according to the production requirement0≥370,ARL1Less than or equal to 5. Considering 4 hours of average failure time under two-parameter Weibull distribution, the 1 st sampling time interval h is defined1Less than or equal to 4, and simultaneously, 0.1 is considered to meet the precision requirement of time. According to literature experience, the k value variation range is limited to [0.25,4 ]]The variation step length is 0.25; n has a value in the range of [1,7 ]],10,15,30。
According to the modeling steps and the actual constraint requirements, a statistical process control and maintenance strategy combined model is constructed, and the optimal solution k is 3.25, m is 7, n is 7, h is obtained10.9, then E (c)min＝$272.69。
The results show that: under the existing production conditions and requirements, the optimal control limit of a control chart is 3.25 sigma, the first sampling interval is 0.9 hour, the sample capacity is 7, and the sampling frequency is 7, namely even if the machine does not have a fault, the equipment is required to be replaced for production after 7 times of sampling. Under this integrated scheme, the cost per unit time is expected to be $ 272.69.

Claims (2)

1. A process control and maintenance strategy combined modeling method based on GERT network statistics is characterized by comprising the following steps:
step 1, using an X-bar control chart to monitor the manufacturing process on line to obtain a production state;
step 2, calculating the sampling interval and the probability of I-type and II-type errors in the monitoring process;
step 3, constructing a GERT network architecture, and calculating a cost expectation and a time expectation required by joint modeling;
step 4, establishing a nonlinear optimization model, designing an algorithm solving model, and obtaining optimal control chart parameters and a maintenance strategy;
step 1 obtains controllable non-alarm Si1Controllable alarm Si2Out-of-control no-alarm Si3Out-of-control alarm Si4Four production statesState; and is
When the production process is in a controllable state, the quality characteristic value x randomly fluctuates around the mean value mu and is subjected to normal distribution;
considering mass characteristic sample mean from mu only when the production process is out of control0，μ0Shift to μ ═ μ0±ησ0，
Figure FDA0002821641410000011
Keeping the standard deviation constant regardless of the offset generated by the standard deviation;
during online monitoring, fusion and maintenance are carried out:
if the control is out of control, the correction maintenance is carried out,
the process is controllable but the preventive maintenance time is up and the preventive maintenance is carried out,
the process is controllable, and compensation maintenance is carried out if the preventive maintenance time is not reached;
setting the failure time of the equipment to comply with two-parameter Weibull distribution, wherein the failure density function is f (t) ═ gamma vtv-1e-γtT is more than 0, v is more than or equal to 1, gamma is more than 0, wherein gamma is more than 0, and v is more than or equal to 1, which are respectively a proportion parameter and a shape parameter,
the correction maintenance comprises the tasks of preparing, checking abnormal reasons, replacing, adjusting, checking and original repair or replacement;
the preventive maintenance comprises the tasks of preparation, detection and diagnosis, part replacement, adjustment, inspection and original repair;
the compensation maintenance comprises preparation and detection diagnosis tasks;
the sampling interval calculation process in the step 2 specifically comprises the following steps:
step 2.1, equalize the risk of equipment failure within the sampling interval
Figure FDA0002821641410000012
In the formula (1), tiIndicating a time point at which the ith sampling is performed;
Figure FDA0002821641410000013
is a failure rate function which represents the probability of failure of the equipment which has not failed before the time t, occurring in the next unit time after the time t, f (t) is a probability density function of the failure of the equipment,
Figure FDA0002821641410000021
is a complementary cumulative probability distribution function;
step 2.2, assume a sampling interval of h1,h2,…,hmThen, then
Figure FDA0002821641410000022
The formula (1) is converted into the formula (3),
Figure FDA0002821641410000023
and 2.3, setting the failure time of the equipment to obey two-parameter Weibull distribution, wherein the failure density function is f (t) -gamma vtv- 1e-γtT is more than 0, v is more than or equal to 1, and gamma is more than 0, wherein gamma is more than 0 and v is more than or equal to 1, and the gamma is respectively a proportion parameter, a shape parameter and a complementary cumulative probability distribution function
Figure FDA0002821641410000024
Then
λ(t)＝γνtν-1 (4)；
Step 2.4, bringing formula (4) into formula (3) to obtain formula (5),
h1 ν＝(ti-1+hi)ν-ti-1 ν,i＝1,…,m (5)
step 2.5, according to
Figure FDA0002821641410000025
And equation (5), converting all sampling intervals to a function h of the first sampling interval1
Figure FDA0002821641410000026
The probability alpha of the type I error and the probability beta of the type II error in the monitoring process in the step 2 are respectively as follows:
α＝1-[Φ(k)-Φ(-k)] (7)
Figure FDA0002821641410000027
wherein k represents a control chart boundary coefficient, η represents an offset, n represents a sample size, and Φ represents a cumulative distribution function of a standard normal random variable;
the specific process of the step 3 is as follows:
step 3.1, constructing a GERT network model, wherein the states are network nodes, and the state transfer relationship obeys exclusive OR logic, namely, only one state is generated at each time point and maintenance parameters obeying arbitrary distribution are used as original parameters of the network; wherein, the production process is a random process of discrete time, and the discrete time point corresponds to sampling detection; each sample considers 4 types of production states, and the state space then includes 4(m-1) +2 possible states, including the initial state point S0，S11,S12,S13,S14,…,S(m-1)1,S(m-1)2,S(m-1)3,S(m-1)4And wait for preventive maintenance Point Sm，S01Is S0；
Step 3.2, assume that the process is at time ti-1I is 1,2, …, m is controllable, let piAt time intervals (t) for the processi-1,ti) The probability of internal runaway, then,
Figure FDA0002821641410000031
in formula (9), F (. cndot.) is a cumulative function of the Weibull distribution;
step 3.3, State S(i-1)1I-1, …, m-1 transitions to state SirR is 1,2,3,4 respectively
Pi1＝(1-pi)(1-α) (10)
Pi2＝(1-pi)α (11)
Pi3＝piβ (12)
Pi4＝pi(1-β) (13)
State Si2I-1, 2, …, m-1 transitions to state Si1Has a probability of
Pi5＝1 (14)
State Si4I-1, 2, …, m-1 to S0′Has a probability of
Pi6＝1 (15)
State S(i-1)3I-2, …, m-1 to SirAnd the probability of r being 3,4 is
Pi7＝β (16)
Pi8＝1-β (17)
State S(m-1)rR 1,3 to SmHas a probability of
Pmj＝1,j＝1,2 (18)
State SmTransfer to S0A probability of
Pm3＝1 (19)
And 3.4, calculating the related parameters according to the following situations:
(1) state S(i-1)1I-1, …, m-1 transitions to state Si1Time spent is constant hiMass loss of Q0hiSampling cost of cf+cvn，cfIs fixed into by sampling onceThis, cvIs a unit variable cost with a total cost of constant (c)f+cvn)+Q0hi，Q0The loss of mass per unit time when the process is controllable;
(2) state S(i-1)1I-1, …, m-1 transitions to state Si2Time spent is constant hiMass loss of Q0hiSampling cost of cf+cvn, total cost is constant (c)f+cvn)+Q0hi；
(3) State S(i-1)1I-1, …, m-1 transitions to state Si3While the time is still constant hiSampling cost of cf+cvn, process shift from ti-1Rear tauiThe total mass loss occurring after a unit of time is Q0τi+Q1(hi-τi)，Q1The total cost is c for the loss of mass per unit time when the process is out of controlf+cvn+Q0τi+Q1(hi-τi)，τiIs a random variable, τiProbability density function f taui(x)＝f(x+ti-1)/(F(ti)-F(ti-1))，
To obtain a total cost probability density of
Figure FDA0002821641410000041
(4) State S(i-1)1I-1, …, m-1 transitions to state Si4Time is constant hiThe total cost is a random variable, and the probability density function is an expression (20);
(5) state Si2I-1, …, m-1 to Si1The time interval executes compensation maintenance, and the consumed time and the cost are the time t of one compensation maintenanceCpMAnd cost cCpMBoth are random variables;
(6) state S(i-1)3I-2, …, m-1 to SirWhen r is 3,4, the elapsed time is constant hiThe cost is formed by the loss of mass and sampling cost when the process is out of control (c)f+cvn)+Q1hi；
(7) State Si4I-1, 2, …, m-1 to S0When S is0' is a virtual node and its meaning is with S0In accordance, the correction maintenance is executed in the time interval, the out-of-control process is recovered to the initial state, and the consumed time and the cost are the time t of one correction maintenanceCMAnd cost cCMBoth are random variables;
(8) state S(m-1)1Transfer to SmTime spent is constant hmAt a time interval (t)m-1,tm) Inner, taumAfter a unit of time, the probability density function is f τ m (x) f (x + t)m-1)/(F(tm)-F(tm-1) Total cost consumed in the interval of c)m1＝Q0τm+Q1(hm-τm) Deriving a probability density function of
Figure FDA0002821641410000042
The mth sampling time point does not sample any more, but directly carries out preventive maintenance;
(9) state S(m-1)3Transition to State SmTime spent is constant hmTotal cost is the loss of mass Q of the process out of control1hm；
(10) State SmTransfer to S0In the period, only the preventive maintenance of the replacement is carried out, the process is recovered to a new state, and the consumed time and the cost are the time t of one preventive maintenancePMAnd cost cPMBoth are random variables;
the specific method for acquiring the cost expectation and the time expectation in the step 3 comprises the following steps:
step 3.1, construct the transfer function Wij＝PijMijWherein M isijIs a function of the moment mother;
step 3.2, if m is more than or equal to 2,
for 1. ltoreq. i.ltoreq.m-1, let WijAre respectively S(i-1)1Transfer to Sir, r ═ 1,2,3,4 transfer function, j ═ 1,2,3, 4;
for i is 2. ltoreq. m-1, let WijAre respectively Si2Transfer to Si1And Si4Transfer to S0' and S(i-1)3Transfer to Si3And S(i-1)3Transfer to Si4J-5, 6,7, 8;
let WmjJ is 1,2,3 is S respectively(i-1)1Transfer to Si，S(i-1)3Transfer to Si，SiTransfer to S0' a transfer function;
if m is equal to 1, the process is repeated,
let W11Is S0Transfer to SmThe transfer function of (a) is selected,
let W13Is SmTransfer to S0' a transfer function;
step 3.3, if the parameter is cost, let Wcc(m, S) is transition S0To S0The equivalent transfer function of the' is,
if m is equal to 1, the ratio of m,
Wcc(1,s)＝W11W13 (22)
if m is equal to 2, the ratio of m,
Figure FDA0002821641410000051
if m is 3, the number of the atoms is 3,
Figure FDA0002821641410000052
if m is equal to 4, the ratio of m,
Figure FDA0002821641410000061
if m is more than or equal to 5,
Figure FDA0002821641410000062
step 3.4, let Eq _ Mcc(m, s) is Wcc(m, s) corresponding to an equivalent moment mother function, then
Figure FDA0002821641410000063
Let Ec(m) cost expectation over period, according to the nature of the intalox: the value of the nth derivative of the moment mother function at s ═ 0 is the nth origin moment of the random variable, and the equation (28) is obtained
Figure FDA0002821641410000064
Step 3.5, if the parameter is time, let Wtt(m, S) is S0Transfer to S0' equivalent transfer function, Eq _ Mtt(m, s) is WttEquivalent moment mother function corresponding to (m, s), Et(m) for time expectation, adding Wtt(m, s) instead of Wcc(m,s)、Eq_Mtt(M, s) instead of Eq _ Mcc(m,s)、Et(m) in place of Ec(m) performing step 3.3 and step 3.4.
2. The method according to claim 1, wherein the specific process of step 4 is as follows:
step 4.1, obtaining the unit time cost expectation E (c)
Figure FDA0002821641410000071
Step 4.2, adding constraint conditions, establishing a nonlinear optimization model,
Figure FDA0002821641410000072
in the formula (30), ARL0≥C1The average run length representing the controllable state is not less than C1，ARL1≤C2Mean run length indicating runaway conditions not greater than C2，ai,biI-1, 2,3,4 respectively represent constraints on the decision variables;
and 4.3, solving the nonlinear optimization model by using a pattern search method to obtain the optimal solution of the model.
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