CN113627970A - Economic design method for variable sampling interval VAR control chart - Google Patents

Economic design method for variable sampling interval VAR control chart Download PDF

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CN113627970A
CN113627970A CN202110690269.7A CN202110690269A CN113627970A CN 113627970 A CN113627970 A CN 113627970A CN 202110690269 A CN202110690269 A CN 202110690269A CN 113627970 A CN113627970 A CN 113627970A
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薛丽
李聪凯
邹妍
贾元忠
曹逗逗
郑含笑
吴昊辰
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Zhengzhou University of Aeronautics
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Abstract

本发明涉及一种可变抽样区间VAR控制图的经济设计方法，针对现有技术中，对于VAR控制图的设计，只考虑到了可变抽样区间的动态设计，虽然可以提高VAR控制图的监控效率，但是却没有考虑VAR控制图的运行成本问题，本发明通过分析数据、构建统计量、构建控制图、构建经济模型、构建经济函数、求取经济函数中的各项参数等步骤来求取、分析验证可变抽样区间VAR控制图的经济设计方法的最优性，本发明与用统计方法设计的控制图相比，可以取得更好的经济效益，即单位时间的运行成本更小。

Figure 202110690269

The invention relates to an economical design method for a VAR control chart with variable sampling interval. For the design of the VAR control chart in the prior art, only the dynamic design of the variable sampling interval is considered, although the monitoring efficiency of the VAR control chart can be improved. , but does not consider the operating cost of the VAR control chart. The present invention obtains, through the steps of analyzing data, constructing statistics, constructing control charts, constructing economic models, constructing economic functions, and obtaining various parameters in the economic functions. The optimality of the economic design method of the variable sampling interval VAR control chart is analyzed and verified. Compared with the control chart designed by the statistical method, the present invention can obtain better economic benefits, that is, the operation cost per unit time is smaller.

Figure 202110690269

Description

Economic design method for variable sampling interval VAR control chart
Technical Field
The invention relates to the technical field of economic design of control charts, in particular to an economic design method of a VAR control chart with a variable sampling interval.
Background
Control charts are a widely used quality control tool in the manufacturing of products. The traditional Huhart control chart usually monitors the process change of a single quality characteristic value, but because the production process is increasingly complicated, a plurality of quality characteristic values need to be monitored simultaneously in a plurality of practical processes, and if the conventional single-variable control chart is still used at the moment, a good monitoring effect is difficult to obtain; on the other hand, with the development of an automatic data acquisition technology and the smaller and smaller data acquisition intervals, the quality characteristic observed value often has an autocorrelation phenomenon in the production process of the industries such as chemical engineering, semiconductors and the like. At this time, monitoring the multivariate autocorrelation process using a conventional control map increases the probability of controlling a map false alarm. Aiming at the small dynamic production process of the multivariate autocorrelation process, some scholars design VAR control charts and obtain good monitoring effect, and then some scholars design VAR control charts with variable sampling intervals, thereby effectively improving the monitoring efficiency of control charts. In the related design of the VAR control chart at present, no method for economic design exists;
when there is little fluctuation in the multivariate autocorrelation process, the monitoring effect of the Vector Autoregressive (VAR) control chart is superior to the residual T2 control chart. However, in the existing design method for the VAR control chart, only a dynamic design method of a Variable Sampling Interval (VSI) is involved, and although the monitoring efficiency of the VAR control chart is improved, the economic benefit of the control chart is not considered. Based on the method, the economic design is carried out on the VAR control chart in the variable sampling interval, and the purpose is to further enable the VSI VAR control chart to have smaller economic cost through the design of the method on the basis that the process monitoring efficiency is improved through the dynamic control chart.
Disclosure of Invention
In view of the above situation, to overcome the defects of the prior art, the present invention aims to provide an economic design method for a VAR control chart between variable sampling areas, which aims to improve monitoring efficiency and achieve better economic effect.
The technical scheme adopted by the invention for solving the technical problems is as follows: the economic design method for the VAR control chart with the variable sampling interval is characterized by comprising the following steps of:
step 1: and analyzing data, namely assuming that the p-dimensional quality characteristic of a product has autocorrelation, and aiming at the collected data, after the collected data is analyzed by SPSS software, the data obeys a stable q-order vector autoregressive model:
Figure BDA0003125897330000021
in the above formula, Xt＝(X1t,X2t,...,Xpt)/Representing the p-dimensional column vector, X, of the observed value at time ttIs given as the mean column vector of (mu) ═ mu1，μ2,...,μp)/,AiRepresenting a matrix of autocorrelation coefficients in the dimension p x p. In the stationary VAR (q) process, μ and AiAre independent of time t. Epsilont＝(ε1t,ε2t,...,εpt)/Is a p-dimensional white noise sequence with a mean value of 0t:Np(0,∑)。
Step 2: constructing statistics, namely assuming that the sample capacity is n, and the observed value vector of t time is xt＝(x1,t,x2,t,...,xp,t)TThen the n observations of the kth subgroup can be represented as xkn+j＝(x1,kn+j,x2,kn+j,...,xp,kn+j)TK is 0,1,2, a., and j is 1,2, akThe following were used:
Figure BDA0003125897330000022
in the above formula, the first and second carbon atoms are,
Figure BDA0003125897330000023
representing the estimated value of the k-th subgroup mean.
Figure BDA0003125897330000024
And is
Figure BDA0003125897330000025
And step 3: and (3) constructing a control chart, wherein an upper control limit H and an upper warning limit W of the VSI VAR control chart are given, and a monitoring mechanism of the VSI VAR control chart is as follows: two sampling intervals h are set1And h2And h is1＞h2. When statistic VkWhen falling in the safety domain (V is more than or equal to 0)kLess than or equal to W), the next sampling takes a long sampling interval h1(ii) a When statistic VkWhen falling in the warning domain (W < V)kLess than or equal to H), the next sampling adopts a short sampling interval H2(ii) a When the statistic exceeds the control limit (V)kH) to control the alarm.
And 4, step 4: constructing an economic model:
wherein the step 4 is divided into the following steps:
step 4-1: to simplify the complex real world scenario, four assumptions are proposed:
(1) the process of each period is in a controlled state when the process is started, the process starts to be out of control after a period of time, the process is recovered to be normal after detection and correction, and one period is finished.
(2) Process controlled duration obeys an average of
Figure BDA0003125897330000031
Is used as the index distribution of (1).
(3) After an anomaly in a process occurs, the process is left out of control until the cause of the anomaly is discovered and corrected.
(4) Only affected by a single abnormal factor in a sampling interval, and the abnormality does not occur in the sampling process.
Step 4-2: the expected value of a process cycle is divided into the following parts:
(1) desired value of process controlled duration:
Figure BDA0003125897330000032
where λ represents the frequency of occurrence of the cause of the abnormality.
(2) Expected time due to false alarm: t isb＝(1-r1)T0ANF
Wherein, γ1Represents: when gamma is1When the value is 1, the control chart is operated in the abnormal process, and when the value is gamma1When the control chart is 0, finding the control chart in the abnormal process is stopped;
T0represents the average time to look for a false alarm;
ANF indicates the number of false alarms, and
Figure BDA0003125897330000033
ANSS0indicating the average number of samples taken for the control chart alarm while the process is controlled, s is the average of the number of samples taken while the process is controlled, and
Figure BDA0003125897330000035
(3) time expectation for process runaway: t isc＝ATS1-τ+nE+T1+T2
Wherein, ATS1The average alarm time of the control chart is shown when the process is out of control;
τ represents the average time between two samples when the process is controlled and an anomaly occurs.
Figure BDA0003125897330000034
E represents the average time of one sampling and drawing;
T1an average time indicating the finding of the cause of the abnormality;
T2mean time to eliminate the cause of the abnormality;
thus, the expectation of the cycle length is: t ═ Ta+Tb+Tc。
Step 4-3: the expectation of the total cost within one process cycle is divided into the following parts:
(1) cost expectation of defective products when the process is controlled:
Figure BDA0003125897330000041
wherein, CicRepresenting the cost of producing defective products per unit time when the process is controlled.
(2) Cost expectation for defective products when the process is out of control: cb＝Coc[ATS1-τ+nE+r1T1+r2T2]
Wherein, CocRepresents the cost of defective products per unit time when the process is out of control;
γ2is expressed as gamma2When the value is 1, the control chart is operated in the abnormal process, and when the value is gamma2When 0, the control chart stops in the process of finding the abnormity
(3) Cost expectation for false alarm generation: cc＝afsANF
Wherein, afsRepresenting the cost of a false alarm.
(4) Cost expectation for discovery and elimination of anomalies: cd＝arp
(5) Cost expectation for sampling:
Figure BDA0003125897330000042
wherein, afixRepresents a fixed cost of one sampling and detection;
avarrepresents a variable cost of sampling and detection;
h0represents an average sampling interval, and
Figure BDA0003125897330000043
ATS0representing an average alarm time of a control map for the process under control;
thus, the overall cost expectation is: c ═ Ca+Cb+Cc+Cd+Cf。
And 5: and (5) constructing an economic function ETC.
Figure BDA0003125897330000044
Step 6: the economic design is to obtain the optimal parameter combination (n, H, W, H) when ETC is minimized1,h2,λ)。
Preferably, in step 5, in the economic function, the average alarm time ATS for the control chart and the average number of samples ANSS required for controlling the chart alarm are calculated by a monte carlo simulation method:
(1) and (4) setting. Setting an upper control limit H, an upper warning line W and a long sampling interval H1Short sampling interval h2；
(2) The simulation generates an observed value. In the j simulation experiment, for a P-dimensional autocorrelation process VAR (q) model, according to a formula
Figure BDA0003125897330000051
Generating a P-dimensional random number vector { XtWhen controlled, mean μ ═ μ1，μ2,...,μp)/When out of control, mu1μ + δ. Then according to the formula
Figure BDA0003125897330000052
Generating a statistic Vk；
(3) And simulating control chart monitoring. For the statistic generated in step (2), if t time statistic VkFalling in the safety domain (0 ≦ V)kW) or less, continuously monitoring the next statistic, wherein the sampling interval is h1(ii) a If statistic VkFalling in the warning domain (W < V)kH or less), continuously monitoring the next statistic, wherein the sampling interval is H2(ii) a Up to statistic VkOut of control limit (V)kAnd (H), controlling the chart to alarm and stop running, recording the value of the alarm time TS (j) and the value of the sampling times RL (j) of the experiment, and ending the experiment.
(4) The experiment was repeated. Repeating the processes (2) to (3), and setting the number of times of repeating the test to a larger number M.
(5) The value of the average alarm time ATS may be estimated using the average of M alarm times { ts (j) }, j ═ 1, 2.. M }; the value of the average number of samples ANSS can be estimated as the average of M sampling times rl (j), j 1, 2.
The invention has the beneficial effects that: compared with the control chart designed by a statistical method, the economic design method for the VAR control chart with the variable sampling interval can obtain better economic benefit, namely the running cost in unit time is lower.
Drawings
FIG. 1 is a flow diagram of the steps of the present invention.
Detailed Description
The embodiments of the present invention will be described in further detail with reference to the accompanying drawings.
Firstly, the invention relates to an economic design method of a VAR control chart with variable sampling intervals, which comprises the following steps:
step 1: and analyzing data, namely assuming that the p-dimensional quality characteristic of a product has autocorrelation, and aiming at the collected data, after the collected data is analyzed by SPSS software, the data obeys a stable q-order vector autoregressive model:
Figure BDA0003125897330000061
in the above formula, Xt＝(X1t,X2t,...,Xpt)/Representing the p-dimensional column vector, X, of the observed value at time ttIs given as the mean column vector of (mu) ═ mu1，μ2,...,μp)/,AiRepresenting a matrix of autocorrelation coefficients in the dimension p x p. In the stationary VAR (q) process, μ and AiAre independent of time t. Epsilont＝(ε1t,ε2t,...,εpt)/Is a p-dimensional white noise sequence with a mean value of 0t:Np(0,∑)。
Step 2: constructing statistics, namely assuming that the sample capacity is n, and the observed value vector of t time is xt＝(x1,t,x2,t,...,xp,t)TThen the n observations of the kth subgroup can be represented as xkn+j＝(x1,kn+j,x2,kn+j,...,xp,kn+j)TK is 0,1,2, a., and j is 1,2, akThe following were used:
Figure BDA0003125897330000062
in the above formula, the first and second carbon atoms are,
Figure BDA0003125897330000063
representing the estimated value of the k-th subgroup mean.
Figure BDA0003125897330000064
And is
Figure BDA0003125897330000065
And step 3: and (3) constructing a control chart, wherein an upper control limit H and an upper warning limit W of the VSI VAR control chart are given, and a monitoring mechanism of the VSI VAR control chart is as follows: two sampling intervals h are set1And h2And h is1＞h2. When statistic VkWhen falling in the safety domain (V is more than or equal to 0)kLess than or equal to W), the next sampling takes a long sampling interval h1(ii) a When statistic VkWhen falling in the warning domain (W < V)kLess than or equal to H), the next sampling adopts a short sampling interval H2(ii) a When the statistic exceeds the control limit (V)kH) to control the alarm.
And 4, step 4: constructing an economic model:
wherein the step 4 is divided into the following steps:
step 4-1: to simplify the complex real world scenario, four assumptions are proposed:
(1) the process of each period is in a controlled state when the process is started, the process starts to be out of control after a period of time, the process is recovered to be normal after detection and correction, and one period is finished.
(2) For treatingThe controlled duration obeys the mean value of
Figure BDA0003125897330000071
Is used as the index distribution of (1).
(3) After an anomaly in a process occurs, the process is left out of control until the cause of the anomaly is discovered and corrected.
(4) Only affected by a single abnormal factor in a sampling interval, and the abnormality does not occur in the sampling process.
Step 4-2: the expected value of a process cycle is divided into the following parts:
(1) desired value of process controlled duration:
Figure BDA0003125897330000072
where λ represents the frequency of occurrence of the cause of the abnormality.
(3) Expected time due to false alarm: t isb＝(1-r1)T0ANF
Wherein, γ1Represents: when gamma is1When the value is 1, the control chart is operated in the abnormal process, and when the value is gamma1When the control chart is 0, finding the control chart in the abnormal process is stopped;
T0represents the average time to look for a false alarm;
ANF indicates the number of false alarms, and
Figure BDA0003125897330000073
ANSS0indicating the average number of samples taken for the control chart alarm while the process is controlled, s is the average of the number of samples taken while the process is controlled, and
Figure BDA0003125897330000075
(3) time expectation for process runaway: t isc＝ATS1-τ+nE+T1+T2
Wherein, ATS1The average alarm time of the control chart is shown when the process is out of control;
τ represents the average time between two samples when the process is controlled and an anomaly occurs.
Figure BDA0003125897330000074
E represents the average time of one sampling and drawing;
T1an average time indicating the finding of the cause of the abnormality;
T2mean time to eliminate the cause of the abnormality;
thus, the expectation of the cycle length is: t ═ Ta+Tb+Tc。
Step 4-3: the expectation of the total cost within one process cycle is divided into the following parts:
(3) cost expectation of defective products when the process is controlled:
Figure BDA0003125897330000081
wherein, CicRepresenting the cost of producing defective products per unit time when the process is controlled.
(4) Cost expectation for defective products when the process is out of control: cb＝Coc[ATS1-τ+nE+r1T1+r2T2]
Wherein, CocRepresents the cost of defective products per unit time when the process is out of control;
γ2is expressed as gamma2When the value is 1, the control chart is operated in the abnormal process, and when the value is gamma2When 0, the control chart stops in the process of finding the abnormity
(3) Cost expectation for false alarm generation: cc＝afsANF
Wherein, afsRepresenting the cost of a false alarm.
(4) Cost expectation for discovery and elimination of anomalies: cd＝arp
(5) Cost expectation for sampling:
Figure BDA0003125897330000082
wherein, afixRepresents a fixed cost of one sampling and detection;
avarrepresents a variable cost of sampling and detection;
h0represents an average sampling interval, and
Figure BDA0003125897330000083
ATS0representing an average alarm time of a control map for the process under control;
thus, the overall cost expectation is: c ═ Ca+Cb+Cc+Cd+Cf。
And 5: and (5) constructing an economic function ETC.
Figure BDA0003125897330000084
In the economic function, the average alarm time ATS of the control chart and the average sample number ANSS required by the control chart alarm are calculated by adopting a Monte Carlo simulation method:
(1) and (4) setting. Setting an upper control limit H, an upper warning line W and a long sampling interval H1Short sampling interval h2；
(2) The simulation generates an observed value. In the j simulation experiment, for a P-dimensional autocorrelation process VAR (q) model, according to a formula
Figure BDA0003125897330000091
Generating a P-dimensional random number vector { XtWhen controlled, mean μ ═ μ1，μ2,...,μp)/When out of control, mu1μ + δ. Then according to the formula
Figure BDA0003125897330000092
Generating a statistic Vk；
(3) And simulating control chart monitoring. For the statistic generated in step (2), if t time statistic VkFalling in the safety domain (0 ≦ V)kW) or less, continuously monitoring the next statistic, wherein the sampling interval is h1(ii) a If statistic VkFalling in the warning domain (W < V)kH or less), continuously monitoring the next statistic, wherein the sampling interval is H2(ii) a Up to statistic VkOut of control limit (V)kAnd (H), controlling the chart to alarm and stop running, recording the value of the alarm time TS (j) and the value of the sampling times RL (j) of the experiment, and ending the experiment.
(4) The experiment was repeated. Repeating the processes (2) to (3), and setting the number of times of repeating the test to a larger number M.
(5) The value of the average alarm time ATS may be estimated using the average of M alarm times { ts (j) }, j ═ 1, 2.. M }; the value of the average number of samples ANSS can be estimated as the mean of M sampling times rl (j), j =1,2,. M.
Step 6: the economic design is to obtain the optimal parameter combination (n, H, W, H) when ETC is minimized1,h2,λ)。
Example analysis
Chemical building materials company Limited mainly produces aircraft glass fiber cloth. To improve production quality, the width, length and diameter of the circular pattern on a batch of aircraft glass fiber cloth are monitored.
Step 1: the data is analyzed. Analyzing the collected data to obtain the three-dimensional quality characteristic value obeying VAR (2) autocorrelation process Xt＝A1Xt-1+A2Xt-2+εt,
Figure BDA0003125897330000093
Figure BDA0003125897330000094
Without loss of generality, the fluctuation vector is [ delta, delta ] assuming that the three quality characteristic values fluctuate equally]。
Step 2: and (5) constructing a monitoring model. The solution vector of the model is an unknown number vector (n, H, W, H)1,h2,λ)。
Suppose the observed vector at time t is xt＝(x1,t,x2,t,...,xp,t)TThen the n observations of the kth subgroup can be represented as xkn+j＝(x1,kn+j,x2,kn+j,...,xp,kn+j)TK is 0,1,2, a., and j is 1,2, akThe method comprises the following steps:
Figure BDA0003125897330000101
in the above formula, the first and second carbon atoms are,
Figure BDA0003125897330000102
representing the estimated value of the k-th subgroup mean.
Figure BDA0003125897330000103
Monitoring by using a variable sampling interval control chart when the statistic VkWhen falling in the safety domain (V is more than or equal to 0)kLess than or equal to W), the next sampling takes a long sampling interval h1(ii) a When statistic VkWhen falling in the warning domain (W < V)kLess than or equal to H), the next sampling adopts a short sampling interval H2(ii) a When the statistic exceeds the control limit (V)kH) to control the alarm.
And step 3: and (5) constructing an economic model.
The cost parameters and settings are as follows:
Cic＝$8，Coc＝$100，λ＝0.01，afix＝$5，avar＝$0.5，E＝0.1hr，arp＝$50，afs＝$20， γ1＝1，γ2＝1，T0＝1.5hr，T1＝5hr，T2＝5hr，δ＝0.05。
the economic model is as follows:
Figure BDA0003125897330000104
calculating the average alarm time ATS of the control chart and the average sample number ANSS required by the alarm of the control chart by adopting a Monte Carlo simulation method:
(1) and (4) setting. Control chart design parameters are (n, H, W, H)1,h2,λ)。
(2) The simulation generates an observed value. In the j simulation experiment, for a P-dimensional autocorrelation process VAR (q) model, according to a formula
Figure RE-GDA0003257706020000103
Generating a P-dimensional random number vector { XtWhen controlled, mu ═ mu (mu)1，μ2,...,μp)/When out of control, mu1μ + δ. Then according to the formula
Figure RE-GDA0003257706020000106
Generating a statistic Vk；
(3) And simulating control chart monitoring. For the statistic generated in step (2), if t time statistic VkFalling in the safety domain (0 ≦ V)kW) or less, continuously monitoring the next statistic, wherein the sampling interval is h1(ii) a If statistic VkFalling in the warning domain (W < V)kH or less), continuously monitoring the next statistic, wherein the sampling interval is H2(ii) a Up to statistic VkOut of control limit (V)kAnd (H), controlling the chart to alarm and stop running, recording the value of the alarm time TS (j) and the value of the sampling times RL (j) of the experiment, and ending the experiment.
(4) The experiment was repeated. Repeating the processes (2) to (3), and setting the number of times of repeating the test to a larger number M.
(5) The value of the average alarm time ATS may be estimated using the average of M alarm times { ts (j) }, j ═ 1, 2.. M }; the value of the average number of samples ANSS can be estimated as the average of M sampling times rl (j), j 1, 2.
The economic design is to obtain the optimal parameter combination (n, H, W, H) when ETC is minimized1,h2,λ)。
And 4, step 4: and (6) solving the economic model. Solving the model by utilizing a Matlab genetic algorithm tool box, wherein the parameters of the genetic algorithm are set as follows:
nvars＝5；
Aineq＝[0 -1 1 0 0]；
bineq＝-2；
lb＝[50 4 1 1 0.1]；
ub＝[100 15 10 2 1]；
PopulationSize_Data＝30；
MaxGenerations_Data＝50；
other parameters are set to default values. The solution results for this test are: ETC 47.6792, solution vector: (53,6.1810,3.4071,1.9024,0.7831).
And 5: and (5) sensitivity analysis. The step is to research the model parameters
(Cic,Coc,λ,afix,avar,E,arp,afs,T1,T2δ) control chart design parameters (n, H, W, H)1,h2λ), and ETC.
Setting an orthogonal test, and setting the high level and the low level of model parameters as follows:
Figure BDA0003125897330000121
the values of other three parameters in the economic model parameters are as follows: gamma ray1＝1，γ2＝1，T0＝1。
The orthogonal test table is set as follows:
Figure BDA0003125897330000122
the results of the orthogonal test are:
Figure BDA0003125897330000131
using SPSS software, regression analysis was performed on the 16 results to reach the following conclusions:
(1) model parameter versus sample volume n, short sampling interval h2The effect of (a) was not significant.
(2) Variable cost a of control limit H with one samplevarAnd an increase in the offset δ.
(3) Variable cost a of alarm limit W with one samplingvarThe offset delta and the average time E of the first sampling and plotting increases.
(4) Long sample interval h1Increases as the frequency λ at which the cause of the abnormality occurs increases.
(5) Frequency lambda of cost function ECT occurring along with abnormality cause, variable cost a of one-time samplingvarCost C of defective products per unit time when the process is out of controlocAverage time E of primary sampling and drawing and average time T of finding abnormal cause1Increase in the cost C of defective products per unit time as the process is out of controlicIs increased and decreased.
Step 6: and (5) carrying out optimality analysis. The purpose of this step is to verify that the economic benefits achieved by the present invention are optimal.
Step 6-1: designing VSI VAR by using a statistical method: average alarm time ATS when VAR control chart of fixed VSI multivariable autocorrelation process is controlled0Control chart design parameter combination (n, h) satisfying this condition1,h2K, w) selecting average alarm time ATS under the condition of out-of-control chart1And combining the minimum set of parameters as the design parameters of the VAR control chart of the VSI multivariable autocorrelation process.
Step 6-2: the parameter combination selected in the step 6-1 is brought into 16 groups of orthogonal tests, and the cost function ETC is calculated1The size of (2).
Step 6-3: when ATS0Cost function value ETC of 16 test sets of control chart statistically designed in step 6-2 at 1001Compared with a control chart cost function value ETC designed by an economic method, the method verifies the advantages and the disadvantages of the control chart cost function value ETC and the cost function value ETC.
The results of the comparison of the two design methods are:
Figure BDA0003125897330000141
as can be seen from the table, the cost function values based on the economic models were all smaller than the control charts designed based on the statistical method in the 16 sets of experiments. That is, the control chart designed based on the economic model of the invention is superior to the control chart designed based on the statistical method.

Claims (2)

1. An economic design method of a variable sampling interval VAR control chart is characterized by comprising the following steps:
step 1: and analyzing data, namely assuming that the p-dimensional quality characteristic of a product has autocorrelation, and analyzing the acquired data by SPSS software, and then, obeying a stable q-order vector autoregressive model:
Figure FDA0003125897320000011
in the above formula, Xt＝(X1t,X2t,...,Xpt)/Representing the p-dimensional column vector, X, of the observed value at time ttIs given as the mean column vector of (mu) ═ mu1，μ2,...,μp)/,AiRepresenting a matrix of autocorrelation coefficients in the dimension p x p. In the stationary VAR (q) process, μ and AiAre independent of time t. Epsilont＝(ε1t,ε2t,...,εpt)/Is a p-dimensional white noise sequence with a mean value of 0t:Np(0,∑)。
Step 2: constructing statistics, namely assuming that the sample capacity is n, and the observed value vector of t time is xt＝(x1,t,x2,t,...,xp,t)TThen the n observations of the kth subgroup can be represented as xkn+j＝(x1,kn+j,x2,kn+j,...,xp,kn+j)TK is 0,1,2, a., and j is 1,2, akThe following were used:
Figure FDA0003125897320000012
in the above formula, the first and second carbon atoms are,
Figure FDA0003125897320000013
representing the estimated value of the k-th subgroup mean.
Figure FDA0003125897320000014
And is
Figure FDA0003125897320000015
And step 3: and (3) constructing a control chart, wherein an upper control limit H and an upper warning limit W of the VSI VAR control chart are given, and a monitoring mechanism of the VSI VAR control chart is as follows: two sampling intervals h are set1And h2And h is1＞h2. When statistic VkWhen falling in the safety domain (V is more than or equal to 0)kLess than or equal to W), the next sampling takes a long sampling interval h1(ii) a When statistic VkWhen falling in the warning domain (W < V)kLess than or equal to H), the next sampling adopts a short sampling interval H2(ii) a When the statistic exceeds the control limit (V)kH) to control the alarm.
And 4, step 4: constructing an economic model:
wherein the step 4 is divided into the following steps:
step 4-1: to simplify the complex real world scenario, four assumptions are proposed:
(1) the process of each period is in a controlled state when the process is started, the process starts to be out of control after a period of time, the process is recovered to be normal after detection and correction, and one period is finished.
(2) Process controlled duration obeys an average of
Figure FDA0003125897320000021
Is used as the index distribution of (1).
(3) After an anomaly in a process occurs, the process is left out of control until the cause of the anomaly is discovered and corrected.
(4) Only affected by a single abnormal factor in a sampling interval, and the abnormality does not occur in the sampling process.
Step 4-2: the expected value of a process cycle is divided into the following parts:
(1) desired value of process controlled duration:
Figure FDA0003125897320000022
where λ represents the frequency of occurrence of the cause of the abnormality.
(2) Expected time due to false alarm: t isb＝(1-r1)T0ANF
Wherein, γ1Represents: when gamma is1When the value is 1, the control chart is operated in the abnormal process, and when the value is gamma1When the control chart is 0, stopping searching the control chart in the abnormal process;
T0represents the average time to look for a false alarm;
ANF indicates the number of false alarms, and
Figure FDA0003125897320000023
ANSS0indicating the average number of samples taken for the control chart alarm while the process is controlled, s is the average of the number of samples taken while the process is controlled, and
Figure FDA0003125897320000024
(3) time expectation for process runaway: t isc＝ATS1-τ+nE+T1+T2
Wherein, ATS1The average alarm time of the control chart is shown when the process is out of control;
τ represents the average time between two samples when the process is controlled and an anomaly occurs.
Figure FDA0003125897320000025
E represents the average time of one sampling and drawing;
T1an average time indicating the finding of the cause of the abnormality;
T2mean time to eliminate the cause of the abnormality;
thus, the expectation of the cycle length is: t ═ Ta+Tb+Tc。
Step 4-3: the expectation of the total cost within one process cycle is divided into the following parts:
(1) cost expectation of defective products when the process is controlled:
Figure FDA0003125897320000031
wherein, CicRepresenting the cost of producing defective products per unit time when the process is controlled.
(2) Cost expectation for defective products when the process is out of control: cb＝Coc[ATS1-τ+nE+r1T1+r2T2]Wherein, CocRepresents the cost of defective products per unit time when the process is out of control;
γ2is expressed as gamma2When the value is 1, the control chart is operated in the abnormal process, and when the value is gamma2When 0, the control chart stops in the process of finding the abnormity
(3) Cost expectation for false alarm generation: cc＝afsANF
Wherein, afsRepresenting the cost of a false alarm.
(4) Cost expectation for discovery and elimination of anomalies: cd＝arp
(5) Cost expectation for sampling:
Figure FDA0003125897320000032
wherein, afixRepresents a fixed cost of one sampling and detection;
avarrepresents a variable cost of sampling and detection;
h0represents an average sampling interval, and
Figure FDA0003125897320000033
ATS0representing an average alarm time of a control map for the process under control;
thus, the overall cost expectation is: c ═ Ca+Cb+Cc+Cd+Cf。
And 5: and (5) constructing an economic function ETC.
Figure FDA0003125897320000034
Step 6: the economic design is to obtain the optimal parameter combination (n, H, W, H) when ETC is minimized1,h2,λ)。
2. The economic design method for the variable sampling interval VAR control chart according to claim 1, characterized in that in step 5, in the economic function, the average alarm time ATS of the control chart and the average sample number ANSS required for the control chart alarm are calculated by adopting a Monte Carlo simulation method:
(1) and (4) setting. Setting an upper control limit H, an upper warning line W and a long sampling interval H1Short sampling interval h2；
(2) The simulation generates an observed value. In the j simulation experiment, for a P-dimensional autocorrelation process VAR (q) model, according to a formula
Figure FDA0003125897320000041
Generating a P-dimensional random number vector { XtWhen controlled, mean μ ═ μ1，μ2,...,μp)/When out of control, mu1μ + δ. Then according to the formula
Figure FDA0003125897320000042
Generating a statistic Vk；
(3) And simulating control chart monitoring. For the statistic generated in step (2), if t time statistic VkFalling in the safety domain (0 ≦ V)kW) or less, continuously monitoring the next statistic, wherein the sampling interval is h1(ii) a If statistic VkFalling in the warning domain (W < V)kH or less), continuously monitoring the next statistic, wherein the sampling interval is H2(ii) a Up to statistic VkOut of control limit (V)kAnd (H), controlling the chart to alarm and stop running, recording the value of the alarm time TS (j) and the value of the sampling times RL (j) of the experiment, and ending the experiment.
(4) The experiment was repeated. The processes (2) to (3) are repeated, and the number of times of repeating the test is set to a larger number M.
(5) The value of the average alarm time ATS may be estimated using the average of M alarm times { ts (j) }, j ═ 1, 2.. M }; the value of the average number of samples ANSS can be estimated as the average of M sampling times rl (j), j 1, 2.
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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115167315A (en) * 2022-08-01 2022-10-11 武汉理工大学 A method and device for generating process control based on control chart

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115167315A (en) * 2022-08-01 2022-10-11 武汉理工大学 A method and device for generating process control based on control chart

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