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

Abstract

The invention relates to an economic design method of a VAR control chart in a variable sampling interval, which aims at the problems that in the prior art, the design of the VAR control chart only considers the dynamic design of the variable sampling interval, the monitoring efficiency of the VAR control chart can be improved, but the operation cost of the VAR control chart is not considered.

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:

in the above formula, X_t＝(X_1t,X_2t,...,X_pt)^/Representing the p-dimensional column vector, X, of the observed value at time t_tIs given as the mean column vector of (mu) ═ mu₁，μ₂,...,μ_p)^/,A_iRepresenting a matrix of autocorrelation coefficients in the dimension p x p. In the stationary VAR (q) process, μ and A_iAre independent of time t. Epsilon_t＝(ε_1t,ε_2t,...,ε_pt)^/Is a p-dimensional white noise sequence with a mean value of 0_t:N_p(0,∑)。

Step 2: constructing statistics, namely assuming that the sample capacity is n, and the observed value vector of t time is x_t＝(x_1,t,x_2,t,...,x_p,t)^TThen the n observations of the kth subgroup can be represented as x_kn+j＝(x_1,kn+j,x_2,kn+j,...,x_p,kn+j)^TK is 0,1,2, a., and j is 1,2, a_kThe following were used:

in the above formula, the first and second carbon atoms are,

representing the estimated value of the k-th subgroup mean.

And is

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 V_kWhen 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 V_kWhen 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

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:

where λ represents the frequency of occurrence of the cause of the abnormality.

(2) Expected time due to false alarm: t is_b＝(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

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

(3) time expectation for process runaway: t is_c＝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.

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 ═ T_a+T_b+T_c。

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:

wherein, C_icRepresenting 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: c_b＝C_oc[ATS₁-τ+nE+r₁T₁+r₂T₂]

Wherein, C_ocRepresents 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: c_c＝a_fsANF

Wherein, a_fsRepresenting the cost of a false alarm.

(4) Cost expectation for discovery and elimination of anomalies: c_d＝a_rp

(5) Cost expectation for sampling:

wherein, a_fixRepresents a fixed cost of one sampling and detection;

a_varrepresents a variable cost of sampling and detection;

h₀represents an average sampling interval, and

ATS₀representing an average alarm time of a control map for the process under control;

thus, the overall cost expectation is: c ═ C_a+C_b+C_c+C_d+C_f。

And 5: and (5) constructing an economic function ETC.

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

Generating a P-dimensional random number vector { X_tWhen controlled, mean μ ═ μ₁，μ₂,...,μ_p)^/When out of control, mu₁μ + δ. Then according to the formula

Generating a statistic V_k；

(3) And simulating control chart monitoring. For the statistic generated in step (2), if t time statistic V_kFalling in the safety domain (0 ≦ V)_kW) or less, continuously monitoring the next statistic, wherein the sampling interval is h₁(ii) a If statistic V_kFalling 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 V_kOut 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:

in the above formula, X_t＝(X_1t,X_2t,...,X_pt)^/Representing the p-dimensional column vector, X, of the observed value at time t_tIs given as the mean column vector of (mu) ═ mu₁，μ₂,...,μ_p)^/,A_iRepresenting a matrix of autocorrelation coefficients in the dimension p x p. In the stationary VAR (q) process, μ and A_iAre independent of time t. Epsilon_t＝(ε_1t,ε_2t,...,ε_pt)^/Is a p-dimensional white noise sequence with a mean value of 0_t:N_p(0,∑)。

Step 2: constructing statistics, namely assuming that the sample capacity is n, and the observed value vector of t time is x_t＝(x_1,t,x_2,t,...,x_p,t)^TThen the n observations of the kth subgroup can be represented as x_kn+j＝(x_1,kn+j,x_2,kn+j,...,x_p,kn+j)^TK is 0,1,2, a., and j is 1,2, a_kThe following were used:

in the above formula, the first and second carbon atoms are,

representing the estimated value of the k-th subgroup mean.

And is

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 V_kWhen 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 V_kWhen 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

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:

where λ represents the frequency of occurrence of the cause of the abnormality.

(3) Expected time due to false alarm: t is_b＝(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

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

(3) time expectation for process runaway: t is_c＝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.

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 ═ T_a+T_b+T_c。

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:

wherein, C_icRepresenting 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: c_b＝C_oc[ATS₁-τ+nE+r₁T₁+r₂T₂]

Wherein, C_ocRepresents 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: c_c＝a_fsANF

Wherein, a_fsRepresenting the cost of a false alarm.

(4) Cost expectation for discovery and elimination of anomalies: c_d＝a_rp

(5) Cost expectation for sampling:

wherein, a_fixRepresents a fixed cost of one sampling and detection;

a_varrepresents a variable cost of sampling and detection;

h₀represents an average sampling interval, and

ATS₀representing an average alarm time of a control map for the process under control;

thus, the overall cost expectation is: c ═ C_a+C_b+C_c+C_d+C_f。

And 5: and (5) constructing an economic function ETC.

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

Generating a P-dimensional random number vector { X_tWhen controlled, mean μ ═ μ₁，μ₂,...,μ_p)^/When out of control, mu₁μ + δ. Then according to the formula

Generating a statistic V_k；

(3) And simulating control chart monitoring. For the statistic generated in step (2), if t time statistic V_kFalling in the safety domain (0 ≦ V)_kW) or less, continuously monitoring the next statistic, wherein the sampling interval is h₁(ii) a If statistic V_kFalling 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 V_kOut 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 X_t＝A₁X_t-1+A₂X_t-2+ε_t,

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 x_t＝(x_1,t,x_2,t,...,x_p,t)^TThen the n observations of the kth subgroup can be represented as x_kn+j＝(x_1,kn+j,x_2,kn+j,...,x_p,kn+j)^TK is 0,1,2, a., and j is 1,2, a_kThe method comprises the following steps:

in the above formula, the first and second carbon atoms are,

representing the estimated value of the k-th subgroup mean.

Monitoring by using a variable sampling interval control chart when the statistic V_kWhen 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 V_kWhen 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:

C_ic＝$8，C_oc＝$100，λ＝0.01，a_fix＝$5，a_var＝$0.5，E＝0.1hr，a_rp＝$50，a_fs＝$20， γ₁＝1，γ₂＝1，T₀＝1.5hr，T₁＝5hr，T₂＝5hr，δ＝0.05。

the economic model is as follows:

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

Generating a P-dimensional random number vector { X_tWhen controlled, mu ═ mu (mu)₁，μ₂,...,μ_p)^/When out of control, mu₁μ + δ. Then according to the formula

Generating a statistic V_k；

(3) And simulating control chart monitoring. For the statistic generated in step (2), if t time statistic V_kFalling in the safety domain (0 ≦ V)_kW) or less, continuously monitoring the next statistic, wherein the sampling interval is h₁(ii) a If statistic V_kFalling 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 V_kOut 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

(C_ic,C_oc,λ,a_fix,a_var,E,a_rp,a_fs,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:

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:

the results of the orthogonal test are:

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 sample_varAnd an increase in the offset δ.

(3) Variable cost a of alarm limit W with one sampling_varThe 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 sampling_varCost C of defective products per unit time when the process is out of control_ocAverage 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 control_icIs 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:

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:

in the above formula, X_t＝(X_1t,X_2t,...,X_pt)^/Representing the p-dimensional column vector, X, of the observed value at time t_tIs given as the mean column vector of (mu) ═ mu₁，μ₂,...,μ_p)^/,A_iRepresenting a matrix of autocorrelation coefficients in the dimension p x p. In the stationary VAR (q) process, μ and A_iAre independent of time t. Epsilon_t＝(ε_1t,ε_2t,...,ε_pt)^/Is a p-dimensional white noise sequence with a mean value of 0_t:N_p(0,∑)。

Step 2: constructing statistics, namely assuming that the sample capacity is n, and the observed value vector of t time is x_t＝(x_1,t,x_2,t,...,x_p,t)^TThen the n observations of the kth subgroup can be represented as x_kn+j＝(x_1,kn+j,x_2,kn+j,...,x_p,kn+j)^TK is 0,1,2, a., and j is 1,2, a_kThe following were used:

in the above formula, the first and second carbon atoms are,

representing the estimated value of the k-th subgroup mean.

And is

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 V_kWhen 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 V_kWhen 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

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:

where λ represents the frequency of occurrence of the cause of the abnormality.

(2) Expected time due to false alarm: t is_b＝(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

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

(3) time expectation for process runaway: t is_c＝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.

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 ═ T_a+T_b+T_c。

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:

wherein, C_icRepresenting 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: c_b＝C_oc[ATS₁-τ+nE+r₁T₁+r₂T₂]Wherein, C_ocRepresents 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: c_c＝a_fsANF

Wherein, a_fsRepresenting the cost of a false alarm.

(4) Cost expectation for discovery and elimination of anomalies: c_d＝a_rp

(5) Cost expectation for sampling:

wherein, a_fixRepresents a fixed cost of one sampling and detection;

a_varrepresents a variable cost of sampling and detection;

h₀represents an average sampling interval, and

ATS₀representing an average alarm time of a control map for the process under control;

thus, the overall cost expectation is: c ═ C_a+C_b+C_c+C_d+C_f。

And 5: and (5) constructing an economic function ETC.

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

Generating a P-dimensional random number vector { X_tWhen controlled, mean μ ═ μ₁，μ₂,...,μ_p)^/When out of control, mu₁μ + δ. Then according to the formula

Generating a statistic V_k；

(3) And simulating control chart monitoring. For the statistic generated in step (2), if t time statistic V_kFalling in the safety domain (0 ≦ V)_kW) or less, continuously monitoring the next statistic, wherein the sampling interval is h₁(ii) a If statistic V_kFalling 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 V_kOut 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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