CN108445761A - Combine modeling method with maintenance strategy based on GERT network statistics process control - Google Patents

Abstract

The present invention provides a kind of statistical Process Controls based on GERT networks to combine modeling method with maintenance strategy, including：Manufacturing process is monitored online with X bar control figures and obtains production status；It calculates the sampling interval and monitors the probability that process makes I classes, II class mistakes；The GERT network architectures are built, and calculates the cost that joint modeling needs and it is expected it is expected with the time；Non-linear Optimal Model is established, algorithm for design solving model obtains Optimal Control graph parameter and maintenance strategy.

Description

Joint modeling method of statistical process control and maintenance strategy based on GERT network

technical field

The invention relates to a statistical process control and maintenance decision-making technology, in particular to a GERT network-based statistical process control and maintenance strategy joint modeling method.

Background technique

With the continuous progress of the global economy, technology and industry, new manufacturing technologies such as flexible manufacturing and agile manufacturing are developing rapidly, and the competition in the manufacturing industry is becoming increasingly fierce. How to control manufacturing costs and improve product quality is an important way for an enterprise to gain a competitive advantage in the market, and it is also an important magic weapon for an enterprise to last forever.

Online monitoring is an effective means to discover production problems in time and avoid unnecessary losses and waste. Control chart is a classic method of on-line monitoring, and its parameter design results directly affect product quality and manufacturing cost. The study of control chart parameter design from the economic perspective is called control chart economic design. Maintenance refers to the actions of updating, maintaining or replacing some parts of the equipment according to the production status, so as to reduce the number of defective products produced. The purpose of maintenance strategy optimization is to reasonably arrange maintenance frequency and maintenance level, and avoid over-maintenance or under-maintenance as much as possible.

The two important issues, the economical design of the control chart and the optimal design of the maintenance strategy, are interrelated and affect each other. In statistical process control, it is necessary to implement planned maintenance (preventive maintenance) to reduce the failure rate of equipment as much as possible, and to implement corrective maintenance actions to restore the out-of-control process to a controllable state. Similarly, maintenance strategies affect the failure mechanism of the production process, and maintenance makes Product quality drift is reduced. Therefore, integrating process control and maintenance strategies and realizing the joint optimization of the two can not only improve product quality but also improve the economic benefits of production.

The goal of joint modeling of statistical process control and maintenance strategy is usually to minimize the cost expectation in the production cycle or the cost expectation per unit time. From this, it can be seen that the core of the problem is the solution of the cost expectation and time expectation of the production cycle. With the continuous deepening of research on this problem, there are currently three main solutions:

(1) Recursive method

The research based on the recursive method first divides the production status into several categories, and then uses the recursive method to deduce the time expectation and cost expectation (including quality control cost, maintenance cost, etc.) in the production cycle according to the transition relationship between the states.

(2) Scenario Classification

The research based on the scenario classification method mainly divides the production process into several different production scenarios, calculates the time expectation and cost expectation of each type and the occurrence probability of each scenario, on this basis, calculates the time expectation and cost in the production cycle expect.

(3) Markov method

The research based on the Markov method mainly relies on the Markov process/chain, and calculates the probability of different states, time expectations and cost expectations according to the transition relationship between states.

The above three methods simplify the production process and maintenance parameters. With the increase of production state/scenario classification, the complexity of modeling based on the recursive method and the scenario classification method will increase sharply, resulting in very difficult calculations. Moreover, due to the limitations of the model itself, it is impossible to list all production scenarios in the scenario classification method. The Markov method also has the problem that it cannot consider all production states, but due to the advantages of the stochastic model itself, it can solve more complex production state problems, which is slightly better than the first two methods. Judging from the research trend in recent years, it is one of the mainstream methods to use Markov method to solve the joint modeling problem of statistical process control and maintenance strategy. However, the Markov method still has some flaws in solving this problem:

First, it is assumed that the failure time of equipment obeys a negative exponential distribution or a geometric distribution (when the failure time is expressed by the number of products before mass shift), so as to facilitate the modeling of the Markov method. However, in practice the failure times of equipment may follow an arbitrary distribution.

Second, maintenance parameters are usually ignored or treated as assumed constants. In fact, describing the maintenance time and maintenance cost of the system through random variables or functions, assuming that they satisfy a certain statistical function or law, this description is more in line with the actual manufacturing process.

Contents of the invention

The purpose of the present invention is to provide a GERT network-based statistical process control and maintenance strategy joint modeling method, which is more in line with the actual situation of the manufacturing process.

The technical scheme that realizes the object of the present invention is: a kind of joint modeling method based on GERT network statistical process control and maintenance strategy, comprises the following steps:

Step 1, use the X-bar control chart to monitor the manufacturing process online to obtain the production status;

Step 2, calculate the sampling interval and the probability of making Type I and Type II errors during the monitoring process;

Step 3, construct the GERT network architecture, and calculate the cost expectation and time expectation required for joint modeling;

Step 4, establish a nonlinear optimization model, design an algorithm to solve the model, and obtain the optimal control chart parameters and maintenance strategy.

Compared with the prior art, the present invention has the following advantages: (1) the present invention proposes to use the semi-Markov process method—GERT network to carry out the joint modeling of the statistical process control and maintenance strategy of the manufacturing process, and obtain the production cycle time expectation and The total cost expectation in the cycle; (2) Add statistical constraints, economic constraints and other practical constraints, take the minimization of the expected cost per unit time as the optimization goal, establish a nonlinear programming model, and obtain the best parameters by solving. This method not only considers the obedience Arbitrary distribution of equipment failure mechanisms, while considering the uncertainty of maintenance parameters, the maintenance parameters are set as random variables subject to arbitrary distribution, which is more in line with the actual situation of the manufacturing process; (3) the present invention not only enriches the knowledge in this field It not only provides a new overall solution for the joint decision-making of quality control and maintenance strategies, but also has very important practical significance for reducing manufacturing costs, ensuring product quality, and improving enterprise economic benefits.

The present invention will be further described below in conjunction with the accompanying drawings.

Description of drawings

Fig. 1 is the specific implementation flowchart of the present invention.

Figure 2 is a state transition diagram of the GERT network model.

Detailed ways

Combined with Figure 1, a joint modeling method of statistical process control and maintenance strategy based on GERT network, including the following steps:

Step S101, analyzing state nodes and state transition relationships, constructing a GERT network architecture;

Use the X-bar control chart to monitor the manufacturing process online to obtain the production status;

Step S102, calculating the sampling interval;

Step S103, calculating the probability of making Type I and Type II errors during the monitoring process;

Step S104, calculating the probability density function of state transition probability P _ij and parameters (cost, time);

Step S105, calculating the moment generating function M _ij of the parameter;

Step S106, calculating the parameter transfer function W _ij ;

Step S107, calculating the network equivalent transfer function W;

Step S108, converting the equivalent transfer function into an equivalent moment generating function M;

Step S107, deduce the time expectation E _t and the cost expectation E _c through the equivalent moment generating function;

Step S108, establishing a nonlinear optimization model, designing an algorithm to solve the model, and obtaining optimal control chart parameters and maintenance strategies.

In conjunction with Fig. 2, in step S101, a GERT network architecture is constructed by analyzing the relationship between state nodes and state transitions. The GERT network architecture of the present invention is as follows: the present invention assumes that the production process is a discrete-time random process, and the discrete time points correspond to sampling detection. Because each sampling considers four types of production status, that is, no alarm S _i1 is controlled, S _i2 is controlled and alarmed, S _i3 is not alarmed when out of control, and S _i4 is alarmed when out of control. Then the state space includes 4(m-1)+2 possible states, including the initial state point S ₀ , then, S ₁₁ , S ₁₂ , S ₁₃ , S ₁₄ ,…, S _(m-1)1 , S _{(m -1)2} , S _(m-1)3 , S _(m-1)4 , and finally, wait for the preventive maintenance point S _m . The production process is in a state of control when it starts running. Assuming that there have been i(i=1,2,...,m-1) sampling intervals since the last corrective maintenance or preventive maintenance, then if the process is controllable and there is no alarm, the process is in state S _i1 ; if the process is If the process _{is out of control but no alarm is issued, the process is in state S i3 ;} if the process is space-time and an alarm is issued, the process is in state S _i4 . If it has persisted for m sampling intervals, the process is in state S _m , ie awaiting preventive maintenance. Use the GERT network model to solve the semi-Markov problem. The state is the network node, and the state transition relationship obeys the XOR logic, that is, there is only one state at each time point. At the same time, the maintenance parameters that obey the arbitrary distribution are used as the network original parameters. Note that for the convenience of modeling, a virtual node S ₀ ′ is added, which has the same meaning as S ₀ .

When the manufacturing process is in a controllable state, the quality characteristic value x fluctuates randomly around the mean value μ, and obeys the normal distribution, that is, x～N(μ,σ ² ), where μ and σ are known.

The production process is out of control and only considers that the sample mean of the quality characteristic shifts from μ ₀ (μ ₀ = μ) to Disregard the offset from the standard deviation and keep it constant.

Simultaneously with the quality monitoring in step S101, the implementation of the maintenance plan is integrated. If the process is out of control, corrective maintenance will be implemented. If the process is controllable but the preventive maintenance time is reached, preventive maintenance will be implemented. It is assumed that the failure time of the equipment obeys the two-parameter Weibull distribution, and its failure density function is f(t)=γvt ^v-1 e ^-γt , t＞0, v≥1, γ＞0, where γ＞0 and ν≥1 are the scale parameter and the shape parameter, respectively.

Preventive maintenance includes tasks such as preparation, detection and diagnosis, replacement, adjustment, inspection, and repair of original parts; corrective maintenance includes tasks such as preparation, investigation of abnormal causes, replacement, adjustment, inspection, and repair or replacement of original parts; compensatory maintenance includes tasks such as preparation, tasks such as detection and diagnosis. Both preventive maintenance and corrective maintenance are perfect maintenance, and the equipment will be restored as new after maintenance. The maintenance time and maintenance cost obey any probability distribution (the constant can be regarded as a special case of the probability distribution).

In step S102, the specific process of sampling interval calculation is as follows:

Make the risk of equipment failure in the sampling interval equal, that is,

In formula (1), t _i represents the time point when the i-th sampling is performed, and λ(t) is the failure rate function, which means that the equipment that has not failed until the time t will fail in the next unit time after the time t probability, i.e.

In formula (2), f(t) is the probability density function of equipment failure, is the complementary cumulative probability distribution function.

Suppose the sampling interval is h ₁ ,h ₂ ,…,h _m , then Formula (1) can be changed into formula (3),

Since the failure time of equipment obeys the Weibull distribution of two parameters, according to formula (2), the failure rate function λ(t) is

λ(t)=γνt ^ν-1 (4)

Put formula (4) into formula (3), get formula (5),

h ₁ ^ν ＝(t _i-1 +h _i ) ^ν -t _i-1 ^ν ,i=1,…,m (5)

according to With formula (5), all sampling intervals can be transformed into the function h ₁ of the first sampling interval, namely

In step S103, let α represent the probability of type I error, β represent the probability of type II error, and Ф represent the cumulative distribution function of a standard normal random variable, then formula (7) and formula (8) are obtained,

α=1-[Φ(k)-Φ(-k)] (7)

In step S104, assuming that the process is controllable at the time point t _i-1 , i=1, 2,...,m, let p _i be the probability that the process is out of control within the time interval (t _i-1 , t _i ), then,

In formula (9), F(·) is the cumulative function of Weibull distribution.

State S _(i-1)1 , i=1,...,m-1 (specially, S ₀₁ is S ₀ ) is transferred to state S _ir (r=1, 2, 3, 4) as

P _i1 ＝(1-p _i )(1-α) (10)

P _i2 =(1-p _i )α (11)

P _i3 = p _i β (12)

P _i4 = p _i (1-β) (13)

The probability of state S _i2 , i=1,2,...,m-1 transferring to state S _i1 is

P _i5 =1 (14)

The probability of state S _i4 , i=1,2,...,m-1 transitioning to S _0′ is

P _i6 =1 (15)

State S _(i-1)3 , i=2,..., m-1 transfers to S _ir , the probability of r=3,4 is

P _i7 =β (16)

P _i8 ＝1-β (17)

The probability that the state S _(m-1)r (r=1,3) transitions to S _m is

P _mj =1,j=1,2 (18)

The probability that state S _m transitions to S ₀ ′ is

P _m3 = 1 (19)

Let Q ₀ be the mass loss per unit time when the process is controllable, the calculation of relevant parameters is divided into the following situations, and the analysis is as follows:

₍ 1) When the state S _(i-1)1 , _i = _{1 ,} . There is no quality shift in the process, the quality loss is Q ₀ h _i , and the sampling cost is c _f +c _v n (c _f is the fixed cost of one sampling, and c _v is the unit variable cost), then the total cost is constant (c _f +c _v n)+Q ₀ h _i .

(2) When the state S _(i-1)1 , i=1,...,m-1 (specially, S ₀₁ is S ₀ ) transfers to the state S _i2 , the process does not deviate, only a false alarm is issued . Similar to case (1), the consumption time is a constant h _i , and the total cost is a constant (c _f +c _v n)+Q ₀ h _i .

(3) When the state S _(i-1)1 , i=1,..., m-1 (specially, S ₀₁ is S ₀ ) transitions to the state S _i3 , although no alarm is issued, the deviation has occurred. Time remains constant h _i . The cost consists of quality loss when the process is controllable, quality loss when the process is uncontrollable, and sampling cost. Let _Q1 be the mass loss per unit time when the process is out of control. Assuming that the process shift occurs after τ _i time units from ti _-1 , then the total mass loss is Q ₀ τ _i +Q ₁ (h _i -τ _i ). Adding the sampling cost, the total cost is c _f +c _v n+Q ₀ τ _i +Q ₁ (h _i -τ _i ). Here, τ _i is a random variable, and its probability density function is fτ _i (x)=f(x+t _i-1 )/(F(t _i )-F(t _i-1 )). On this basis, the total cost probability density is deduced as

(4) When the state S _(i-1)1 , i=1,...,m-1 (specially, S ₀₁ is S ₀ ) transitions to the state S _i4 , the process deviates and an alarm is issued. Similar to case (3), the time is still constant h _i . The total cost is a random variable, and the probability density function is similar to formula (20).

(5) When the state S _i2 , i=1,...,m-1 transitions to S _i1 , only compensation maintenance is performed within this time interval, and the confirmation process does not actually deviate, and the alarm is released. The consumption time and cost are the time and cost of one compensation maintenance, namely t _CpM and c _CpM . Both are random variables.

(6 ₎ When the state S _(i −1)3, i=2, . No matter whether the alarm is issued or not, the consumption time is constant h _i , and the cost consists of quality loss and sampling cost when the process is out of control, namely (c _f +c _v n)+Q ₁ h _i .

(7) When the state S _i4 , i=1, 2, ..., m-1 is transferred to S _0′ , only corrective maintenance is performed in this time interval, and the out-of-control process is restored to the as-new state. The consumed time and cost are the time and cost of a corrective maintenance, ie t _CM , c _CM . Both are random variables.

(8) When the state S _(m-1)1 transitions to S _m , the elapsed time is constant h _m . In the time interval (t _m-1 , t _m ), after τ _m time units, the process may deviate, similar to τ _i (i≤m-1), τ _m is a random variable, and its probability density function is fτ _m (x)=f(x+t _m-1 )/(F(t _m )-F(t _m-1 )). The total consumption cost in this interval is c _m1 = Q ₀ τ _m +Q ₁ (h _m -τ _m ), and its probability density function is derived as

At the mth sampling time point, no sampling is performed, but preventive maintenance is performed directly.

(9) When the state S _(m-1)3 is transferred to the state S _m , the consumption time is a constant h _m , and the total cost is the quality loss Q ₁ h _m out of control of the process.

(10) When the state S _m is transferred to S ₀ , only replacement preventive maintenance is performed during this period, and the process is restored to the as-new state. The consumption time and cost are the time and cost of a preventive maintenance, ie t _PM , c _PM . Both are random variables.

In step S105, let s be any real number, if the total cost of the above state transition is constant, then otherwise where f(y) is the probability density function of the total cost.

In step S106, the calculation process of the transfer function W _ij is given in detail below taking the cost expectation as an example:

(1) If m≥2

For 1≤i≤m-1, let W _ij (j=1,2,3,4) be S _(i-1)1 (specially, S ₀₁ is equivalent to S ₀ ) transferred to S _ir (r =1,2,3,4) transfer function;

For 2≤i≤m-1, let W _ij (j=5, 6, 7, 8) be S _i2 transferred to S _i1 , S _i4 transferred to S ₀ ′, S _(i-1)3 transferred to S _i3 , the transfer function of S _(i-1)3 transferred to S _i4 ;

Let W _mj (j=1, 2, 3)W _ij (j=1) respectively transfer S _(i-1)1 to S _i , S _(i-1)3 to S _i , and S _i to S ₀ ' transfer function.

(2) If m=1

Let _W11 be the transfer function from _S0 to _Sm ,

Let W ₁₃ be the transfer function of S _m to S ₀ '.

In step S107, after obtaining the transfer function of the cost in step S106, let W _cc (m, s) be the equivalent transfer function of the transition state S ₀ to S ₀ ^′ . Next, use the induction method to obtain W _cc (m, s) s):

If m=1,

W _cc (1,s) = W ₁₁ W ₁₃ (22)

If m=2,

If m=3,

If m=4,

If m≥5,

In step S108, let Eq_M _cc (m, s) be the equivalent moment generator function corresponding to W _cc (m, s), then

Let E _c (m) be the cost expectation in the cycle. According to the properties of the moment generating function: the value of the nth order derivative of the moment generating function at s=0 is the nth order origin moment of the random variable, and the formula (28) can be obtained ,

In step S105 to step S107, if the parameter is time, let W _tt (m, s) be the equivalent transfer function from S ₀ to S ₀ ′, and Eq_M _tt (m, s) be W _tt (m, s) The corresponding equivalent moment generating function, E _t (m) is the time expectation, replace W _cc (m, s) with W _tt (m, s), replace Eq_M _cc (m, s) with Eq_M _tt (m, s), E _t (m) replaces E _c (m), and the specific expression can be obtained by using the same method as above.

Step S108, establishing a nonlinear optimization model, designing an algorithm to solve the model, and obtaining optimal control chart parameters and maintenance strategies.

According to the time expectation and cost expectation obtained in step 3, the unit time cost expectation is,

Add constraints, establish a nonlinear optimization model,

In formula (30), ARL ₀ ≥ C ₁ means that the average running length of the controllable state is not less than C ₁ , and ARL ₁ ≤ C ₂ means that the average running length of the out-of-control state is not greater than C ₂ , a _i , b _i , i=1 , 2, 3, 4 represent constraints on decision variables respectively, and C ₁ and C ₂ are known parameters.

For nonlinear optimization problems, the optimal solution of the model can be obtained by using pattern search, genetic algorithm, etc., that is, the best combination scheme of control chart and maintenance can be obtained.

Example

The following is a case of a cotton yarn machine making cotton yarn. It is known that the failure time of the machine obeys the Weibull distribution of two parameters γ=0.05, ν=2. The production process is monitored using an X-bar control chart, the sampling fixed cost is c _f =$2.0, and the variable cost is c _v =$0.5. The mass loss per unit time when the process is controllable is Q ₀ =$50, and the mass loss per unit time when the process is controllable is Q ₁ =$950. The average excursion of the process out of control is η=1. Assuming that compensation maintenance parameters obey exponential distribution, c _CpM ～Exp(0.002), t _CpM ～Exp(3), correction maintenance parameters obey normal distribution, c _CM ～N(1100,50 ² ), t _CM ～N(1, 0.2 ² ), preventive maintenance parameters are constant, c _PM =250, t _PM =0.1.

At the same time, according to production requirements, limit ARL ₀ ≥ 370, ARL ₁ ≤ 5. At the same time, considering that the average failure time under the two-parameter Weibull distribution is 4 hours, it is stipulated that the first sampling time interval h ₁ ≤ 4, and it is considered that 0.1 meets the time accuracy requirement. According to literature experience, the variation range of k value is limited to [0.25,4], and the variation step size is 0.25; the value range of n is [1,7],10,15,30.

According to the above-mentioned modeling steps and actual constraint requirements, a joint model of statistical process control and maintenance strategy is constructed, and the optimal solution is obtained as k=3.25, m=7, n=7, h ₁ =0.9, at this time E(c) _min = $272.69.

The results show that: under the existing production conditions and requirements, the optimal control limit of the control chart is 3.25σ, the first sampling interval is 0.9 hours, the sample size is 7, and the number of sampling is 7, that is, even if the machine does not fail, 7 times After sampling, the production equipment should also be replaced. Under this integration scheme, the cost per unit time is expected to be $272.69.

Claims (8)

1. one kind combining modeling method based on GERT network statistics process control with maintenance strategy, which is characterized in that including following Step：

Step 1, manufacturing process is monitored online with X-bar control figures and obtains production status；

Step 2, it calculates the sampling interval and monitors the probability that process makes I classes, II class mistakes；

Step 3, the GERT network architectures are built, and calculates the cost that joint modeling needs and it is expected it is expected with the time；

Step 4, Non-linear Optimal Model is established, algorithm for design solving model obtains Optimal Control graph parameter and maintenance strategy.

2. according to the method described in claim 1, it is characterized in that, step 1 obtains the controllable S that do not alarm_i1, controllable warning S_i2、 The S out of control that do not alarm_i3, alarm S out of control_i4Four kinds of production statuses；And

When production process is in controllable state, quality characteristic value x surrounds mean μ random fluctuation, and Normal Distribution；

When production process is in runaway condition, only consider mass property sample average from μ₀(μ₀=μ) it is displaced to The offset that standard deviation generates is not considered, and standard deviation is enabled to keep constant.

3. according to the method described in claim 2, it is characterized in that, when being monitored online, fusion is safeguarded：

It is out of control, implement correction and safeguards,

Process control but preventive maintenance time, which reach, then implements preventive maintenance,

The process control but preventive maintenance time does not reach is implemented compensation and is safeguarded；

The set device out-of-service time obeys two parameters of Weibull, and failure dense function is f (t)=γ vt^v-1e^-γt, t ＞ 0, v >=1, γ ＞ 0, wherein γ ＞ 0 and ν >=1 are scale parameter and form parameter respectively,

The correction maintenance includes preparing, investigating abnormal cause, changing part, adjustment, inspection and original paper reparation or displacement task；

The preventive maintenance include preparations, checkout and diagnosis, change part, adjustment, inspection and original paper reparation task；

The compensation maintenance includes preparation, checkout and diagnosis task.

4. according to the method described in claim 3, it is characterized in that, sampling interval calculating process is specially in step 2：

Step 2.1, enable the equipment failure risk in the sampling interval equal

In formula (1), t_iIt indicates to execute the time point that ith is sampled；It is failure rate estimation, indicates work to t moment The equipment not yet to fail, the probability to fail within next unit interval after t moment, f (t) are the general of equipment failure Rate density function,It is complementary cumulative distribution function；

Step 2.2, it is assumed that sampling interval h₁,h₂,…,h_m, thenFormula (1) is converted to formula (3),

Step 2.3, the set device out-of-service time obeys two parameters of Weibull, and failure dense function is f (t)=γ vt^{v- 1}e^-γt, t ＞ 0, v >=1, γ ＞ 0, wherein γ ＞ 0 and ν >=1 are scale parameter and form parameter, complementary cumulative probability distribution respectively FunctionThen

λ (t)=γ ν t^ν-1(4)；

Step 2.4, it brings formula (4) into formula (3), obtains formula (5),

h₁ ^ν=(t_i-1+h_i)^ν-t_i-1 ^ν, i=1 ..., m (5)

Step 2.5, according toWith formula (5), it converts all sampling intervals to first sampling The function h at interval₁

5. according to the method described in claim 4, it is characterized in that, monitoring probability α and II that process makes I class mistakes in step 2 The probability β of class mistake is respectively：

α=1- [Φ (k)-Φ (- k)] (7)

Wherein, k indicates that control figure marginal coefficient, η indicate that offset, n indicate that sample size, Ф indicate standard normal random variable Cumulative distribution function.

6. according to the method described in claim 5, it is characterized in that, the detailed process of step 3 is：

Step 3.1, GERT network models are built, state is network node, and state transfer relationship obeys XOR logic, i.e., when each Between put initial parameter of the maintenance parameters as network that one and only one state occurs and obeys Arbitrary distribution；Where it is assumed that Production process is the random process of a discrete time, and discrete time point corresponds to sampling Detection；4 class of each sampling consideration produces shape State, then state space includes 4 (m-1)+2 possible states, including original state point S₀, S₁₁,S₁₂,S₁₃,S₁₄,…,S_(m-1)1, S_(m-1)2,S_(m-1)3,S_(m-1)4With waiting preventive maintenance point S_m, S₀₁For S₀；

Step 3.2, it is assumed that process is in time point t_i-1, i=1,2 ..., m is controllable, enables p_iIt is process in time interval (t_i-1,t_i) in Probability out of control, then,

In formula (9), F () is the cumulative function of Weibull distribution；

Step 3.3, state S_(i-1)1, i=1 ..., m-1 is transferred to state S_ir(r=1,2,3,4) it is respectively

P_i1=(1-p_i)(1-α) (10)

P_i2=(1-p_i)α (11)

P_i3=p_iβ (12)

P_i4=p_i(1-β) (13)

State S_i2, i=1,2 ..., m-1 is transferred to state S_i1Probability be

P_i5=1 (14)

State S_i4, i=1,2 ..., m-1 is transferred to S_0′Probability be

P_i6=1 (15)

State S_(i-1)3, i=2 ..., m-1 is transferred to S_ir, the probability of r=3,4 is

P_i7=β (16)

P_i8=1- β (17)

State S_(m-1)r(r=1,3) it is transferred to S_mProbability be

P_mj=1, j=1,2 (18)

State S_mIt is transferred to S₀' probability be

P_m3=1 (19)

Step 3.4, relevant parameter calculating divides following scenario described：

(1) state S_(i-1)1, i=1 ..., m-1 is transferred to state S_i1When, elapsed time is constant h_i, mass loss Q₀h_i, take out Sample cost is c_f+c_vN, c_fFor single sample fixed cost, c_vFor unit variable cost, totle drilling cost is constant (c_f+c_vn)+Q₀h_i, Q₀For process control when unit interval mass loss；

(2) state S_(i-1)1, i=1 ..., m-1 is transferred to state S_i2When, elapsed time is constant h_i, mass loss Q₀h_i, take out Sample cost is c_f+c_vN, totle drilling cost are constant (c_f+c_vn)+Q₀h_i；

(3) state S_(i-1)1, i=1 ..., m-1 is transferred to state S_i3When, the time is still constant h_i, cost of sampling c_f+c_vN, mistake Journey offset is from t_i-1τ afterwards_iThe total mass loss occurred after a chronomere is Q₀τ_i+Q₁(h_i-τ_i), Q₁When out of control for process Unit interval mass loss, totle drilling cost c_f+c_vn+Q₀τ_i+Q₁(h_i-τ_i), τ_iIt is stochastic variable, τ_iProbability density function f τ_i (x)=f (x+t_i-1)/(F(t_i)-F(t_i-1)),

Obtaining totle drilling cost probability density is

(4) state S_(i-1)1, i=1 ..., m-1 is transferred to state S_i4When, the time is constant h_i, totle drilling cost is stochastic variable, general Rate density function is formula (20)；

(5) state S_i2, i=1 ..., m-1 is transferred to S_i1When, which executes compensation and safeguards, elapsed time is one with cost The time t that secondary compensation is safeguarded_CpMWith cost c_CpM, both for stochastic variable；

(6) state S_(i-1)3, i=2 ..., m-1 is transferred to S_ir, r=3, when 4, elapsed time is constant h_i, cost is out of control by process When mass loss and cost of sampling constitute (c_f+c_vn)+Q₁h_i；

(7) state S_i4, i=1,2 ..., m-1 is transferred to S₀' when, S₀' it is dummy node and its meaning and S₀Unanimously, between the time It corrects and safeguards every interior execution, runaway event restores to original state, elapsed time and the time t that cost is that primary correction is safeguarded_CM With cost c_CM, both for stochastic variable；

(8) state S_(m-1)1It is transferred to S_mWhen, elapsed time is constant h_m, in time interval (t_m-1,t_m) in, τ_mAfter chronomere, Probability density function is f τ_m(x)=f (x+t_m-1)/(F(t_m)-F(t_m-1)), consumption totle drilling cost is c in the interval_m1=Q₀τ_m+Q₁ (h_m-τ_m), deriving its probability density function is

M-th of sample time point not resampling, but directly carry out preventive maintenance；

(9) state S_(m-1)3It is transferred to state S_mWhen, elapsed time is constant h_m, totle drilling cost is process mass loss Q out of control₁h_m；

(10) state S_mIt is transferred to S₀When, the preventive maintenance for changing part is only performed during this period, process is restored to such as new state, is disappeared Time-consuming and the time t that cost is a preventive maintenance_PMWith cost c_PM, both for stochastic variable.

7. according to the method described in claim 6, it is characterized in that, cost it is expected and time desired specific acquisition in step 3 Method is：

Step 3.1, structure transmission function W_ij=P_ijM_ij, wherein M_ijFor moment generating function.

Step 3.2, if m >=2,

For 1≤i≤m-1, W is enabled_ijRespectively S_(i-1)1It is transferred to S_ir(r=1,2,3,4) transmission function, j=1,2,3,4；

For 2≤i≤m-1, W is enabled_ijRespectively it is S_i2It is transferred to S_i1And S_i4It is transferred to S₀' and S_(i-1)3It is transferred to S_i3And S_(i-1)3 It is transferred to S_i4Transmission function, j=5,6,7,8；

Enable W_mj(j=1,2,3) is respectively S_(i-1)1It is transferred to S_i, S_(i-1)3It is transferred to S_i, S_iIt is transferred to S₀' transmission function；

If m=1,

Enable W₁₁For S₀It is transferred to S_mTransmission function,

Enable W₁₃For S_mIt is transferred to S₀' transmission function；

Step 3.3, if parameter is cost, W is enabled_cc(m, s) is transition S₀To S₀' transfer function of equal value,

If m=1,

W_cc(1, s)=W₁₁W₁₃ (22)

If m=2,

If m=3,

If m=4,

If m >=5,

Step 3.4, Eq_M is enabled_cc(m, s) is W_cc(m, s) corresponding moment generating function of equal value, then

Enable E_c(m) it is expected for the cost in the period, according to the property of moment generating function：Number of the n order derivatives of moment generating function at s=0 Value, as the n rank moment of the origns of stochastic variable, obtain formula (28)

Step 3.5, if parameter is the time, W is enabled_tt(m, s) is S₀It is transferred to S₀' transmission function of equal value, Eq_M_tt(m, s) is W_tt (m, s) corresponding moment generating function of equal value, E_t(m) it is it is expected the time, by W_tt(m, s) substitutes W_cc(m,s)、Eq_M_tt(m, s) is substituted Eq_M_cc(m,s)、E_t(m) E is substituted_c(m) step 3.3 and step 3.4 are carried out.

8. the method according to the description of claim 7 is characterized in that the detailed process of step 4 is：

Step 4.1, it obtains long-run cost rate and it is expected E (c)

Step 4.2, increase constraints, establish Non-linear Optimal Model,

In formula (30), ARL₀≥C₁Indicate that the average run-length of controllable state is not less than C₁, ARL₁≤C₂Indicate runaway condition Average run-length is not more than C₂, a_i,b_i, i=1,2,3,4 indicates the constraint to decision variable respectively；

Step 4.3, application mode search method solves Non-linear Optimal Model and obtains model optimal solution.