CN102929148A - Multiple product production mode statistical process control method based on T-K control chart - Google Patents

Abstract

The invention discloses a multiple product production mode statistical process control method based on a T-K control chart. The multiple product production mode statistical process control method includes the following steps: (1) building a T control chart for monitoring process parameter mean value; (2) building a K control chart for monitoring process parameter standard deviation; and (3) under a multiple product production mode, the T-K control chart is used for monitoring a device operation state, as long as each type of product sample data reaches more than 2 batches, if distribution parameters are known, one batch of data is enough, and sample capacity of each batch of products is guaranteed to be identical and larger than 1. Control limits are determined according to whether the mean value of the process parameter matrix or standard deviation is known, and the T-K control chart is built. Instance analysis and simulation verification prove that the process control method can detect abnormal factors causing an incontrollable phenomenon in a production process under the multiple product production mode timely and effectively, prompts operation staff to make response timely, and enables the production process to remain in a statistical controlled state so as to guarantee product quality.

Description

Many variety production pattern statistical process control method based on the T-K control chart

Technical field

The invention belongs to commercial production and manufacturing industry technical field, relate to a kind of statistical process control method, specifically, relate to a kind of many variety production pattern statistical process control method based on the T-K control chart.

Background technology

Statistical Process Control (Statistical Process Control, SPC) technology is the Main Means of estimating production run statistics slave mode and then guaranteeing product quality.Many scholars have proposed numerous dissimilar control chart technology, and these Charts technology are used widely in the batch commercial production, and improvement productivity effect, the aspect of improving the quality of products have been produced positive effect.But use Charts to require production run to meet some requirements.At first, in order to calculate the control limit, require the average of process and standard deviation known, or can be calculated by the data that gather.But the data that are based on collection are calculated control in limited time, in order to guarantee confidence level, need at least 20 to 25 lot sample notebook datas, and every batch of 5 sample points that is: need 100 to 125 data from the sample survey at least.In addition, the prerequisite of using Charts is that each batch sample data of gathering in the requirement production run is separate and obey same normal distribution, it is usually said same (Independent and IdenticallyNormally Distributed, IIND) condition that distributes of independent normal state.

Yet, in actual production, there are the characteristics of obvious many kinds, short run in some production run, and same procedure is processed several even tens kinds of products that technological specification is different possibly, and this is so that the Charts technology is difficult to the quality control of this class production run.The production run of at first this many kinds, short run can not be known average and the standard deviation of procedure parameter usually in advance, and because product category is various, the product of single variety is less in batches, is difficult to satisfy the data volume requirement of setting up control chart.As adopt less data construct control chart, with so that the monitoring capacity of control chart reduce.In addition, dissimilar products often owing to produce starting material, working condition or processing request difference, causes the technological parameter parent to disobey same distribution, and this has obviously run counter to the IIND condition.These problems have all proposed challenge to implementing SPC in the multiple variety and small batch production.

For solving the very few problem of data volume in the small serial production pattern, some scholars have proposed self-starting (self-starting) control chart technology.The advantage of Self-starting control chart is the sample data amount less demanding, need not undergoing analysis and can set up control chart with control chart, and along with the constant renewal estimator of parent distribution parameter of monitor procedure.Propose a series of self-starrting control charts based on the Q statistic in the prior art, be called as Xiu Hate Q control chart, for detection of the fluctuation of process mean parameter and standard deviation.Discussed average running length (Average Run Length, the ARL) characteristic of Q control chart in detail and the Q control chart has been optimized.CUSUM control chart and the MEWMA control chart of self-starting characteristics have been proposed to have.Recently, the people such as Zhang have proposed Xiu Hate t control chart, compare with traditional Xiu Hate mean chart, and the t control chart need not undergoing analysis with the control chart stage and more stable monitoring capacity is arranged.Choosing of t control chart control limit made the t control chart of revising and propose to be used for the small serial production pattern, and this technology possesses the advantage of Xiu Hate t control chart and is more suitable for quality monitoring in small batch production process.

Although above-mentioned technology has solved the very few problem of sample size that small batch production process brings well, still can not solve many variety problems.For many variety problems, conventional settling mode is first raw data to be changed, make result data obey same normal distribution, again the result data implementation process is controlled, as: with desired value deviation control figure (Deviations from Nominal, DNOM) and standardization DNOM control chart (Standardized DNOM).Yet, utilize these control charts, need to estimate dissimilar Product Process parameter and standard deviations, need all will gather abundant data to the product of each kind for this reason, otherwise standard deviation estimated value error is larger, to cause the control chart hydraulic performance decline, show as False Rate lifting or insensitive to abnormal cause.For the multi-varieties and small-batch situation, usually be difficult to satisfy the requirement of this data acquisition amount.

Summary of the invention

In order to solve the problems of the technologies described above, overcome the defective that exists in the prior art, the invention provides a kind of many variety production pattern statistical process control method based on the T-K control chart, the method has proposed the T-K control chart under many variety production pattern, for detection of the fluctuation of technological parameter Parent Mean and standard deviation.Its basic thought is to adopt a kind of method not high to sample size requirements, based on the every lot sample notebook data that gathers, calculates T, K statistic, so that T, K statistic are separate and obey same distribution separately.According to actual conditions, the method for two kinds of compute statistics has been proposed.A kind of known for the technological parameter average, the another kind of situation that is used for average the unknown.

Its technical scheme is as follows:

A kind of many variety production pattern statistical process control method based on the T-K control chart may further comprise the steps:

1) sets up the T control chart that is used for monitoring technological parameter average

Suppose that one produces procedure planning according to the different code requirement converted products of P kind, that is: produce P kind product.If X is key process parameter, every batch of product is randomly drawed n observation sample, { X ^(r)I, j, l ..., X ^(r)I, j, n} are i group sample, i=1, and 2 ...; J represents the product type sequence number that this batch sample is corresponding, j=1, and 2 ... P; N is sample size, and subscript r represents this sequence number in the same type product of this lot sample.If no special instructions, aiming symbol under related the having in the literary composition, the first subscript represents the sample batch serial number, and the second subscript represents kind sequence number corresponding to this sample, and the 3rd subscript shows the sequence number of these data in affiliated batch; Subscript in the bracket represents the sequence number of this group sample in the respective type product sample.Under the statistics slave mode, in same batch and batch between the technological parameter data separate, the sample data of same kind is obeyed same normal distribution, and the sample data of different cultivars is obeyed different normal distributions, that is: X _{I, j, k}～N (μ _j, σ _j), i=1,2 ..., j=1,2 ... P, k=1,2 ... n, wherein μ _jAnd σ _jAverage and standard deviation for j kind Product Process parameter distribution that parent is obeyed under the slave mode.The sample average of i group sample

And sample standard deviation S _{I, j}Be respectively:

${\stackrel{\‾}{X}}_{i,j}=\frac{1}{n}\underset{k=1}{\overset{n}{\mathrm{\Σ}}}{X}_{i,j,k}$ $S_{i,j}=\sqrt{\frac{1}{n-1}\underset{k=1}{\overset{n}{\mathrm{\Σ}}}{(X_{i,j,k}-{\stackrel{\‾}{X}}_{i,j})}^2}---\left(1\right)$

1.1) T adds up definition of quantity

In the actual production, the average μ of technological parameter estimator of parent distribution is generally unknown parameter.Little such as fruit product editing objective value and technological parameter actual distribution mean bias, the available targets value is as the estimator of parent distribution average, or can determine the value of μ according to practical production experience the time, μ can be considered as known parameters.When making up the T statistic, whether known according to all types of Product Process parameter Parent Mean, there is dual mode to set up statistic.

Situation one: the technological parameter Parent Mean is known

For each lot sample originally, i=1,2 ..., statistic T _{I, j}Expression formula as follows:

$T_{i,j}=\frac{{\stackrel{\‾}{X}}_{i,j}-{\mathrm{\μ}}_j}{S_{i,j}/\sqrt{n}},$ j＝1，2，..P (2)

Wherein n is every lot sample capacity originally, μ _jJ kind Product Process parameter Parent Mean during for controlled process.When production run is in statistics slave mode, T _{I, j}Separate and obedience degree of freedom is that the t of n-1 distributes.

Situation two: the technological parameter Parent Mean is unknown

Definition:

${\stackrel{\‾}{\stackrel{\‾}{X}}}_j^{(r-1)}=\frac{1}{r-1}\underset{h=1}{\overset{r-1}{\mathrm{\Σ}}}{\stackrel{\‾}{X}}_j^{\left(h\right)},$ j＝1，2，..P，r＝2，3... (3)

The expression product type is the average of the front r-1 lot sample notebook data of j, and the expression formula of T statistic is:

$\begin{array}{c}{T}_{i,j}^{\left(1\right)}=0\\ T_{i,j}^{\left(r\right)}=\frac{{\stackrel{\‾}{X}}_{i,j}^{\left(r\right)}-{\stackrel{\‾}{\stackrel{\‾}{X}}}_j^{(r-1)}}{S_{i,j}^{\left(r\right)}}\sqrt{\frac{n(r-1)}{r}}\end{array}$ j＝1，2，..P，r＞1 (4)

As seen first statistic of all types of products is constant 0 in the T control chart, and follow-up statistic is by its appearance position in the respective type product and in the past data decision.

1.2) the determining of T control chart control limit

The control of T control chart is limited to:

$\mathrm{UCL}=G_t^{-1}(1-\frac{\mathrm{\α}}{2}|n-1)$

CL＝0 (5)

LCL＝-UCL

Wherein

Be that degree of freedom is the inverse function of the t distribution cumulative distribution function of n-1, α is level of significance, and according to statistical process control, upper lower control limit is corresponding to ± 3 σ positions, and namely the α value is 0.0027.For same production run, if process is in slave mode, the technological parameter average is offset, variety classes product so, as long as the sample size of each subgroup is identical, the T statistic of then being calculated by each subgroup process parameter measurement value and getting is separate and obey same t distribution, and has identical control limit.These T statistics are painted in same the control chart, just can carry out quality monitoring to production run.Can find by above-mentioned analytic process, even if the initial period in the SPC analysis, the level of significance of T control chart is the identical fixed value 0.0027 that remains on traditional Shewhart control chart still, even if this shows that the T control chart still can normally be implemented quality monitoring to production run under the small serial production pattern.

Choosing of the level of significance of T control chart, control limit is all consistent with Shewhart control chart, and the T statistic is separate, so the judgment rule of Charts still is applicable to the T control chart.In the monitoring of technological process, whether have the point that exceeds the control limit except detecting, any point that shows nonrandom phenomenon in control chart also is considered as point out of control, in these criterions out of control, also need by control chart ± σ (probable value 0.3173) and ± position of 2 σ (probable value 0.0455).The fractile that the T control chart distributes according to t is determined the control limit, and its control limit is relevant with the sample size of every batch data, for making things convenient for practical application, listed in the table 1 that under Different Sample T control chart center line, control are limit and ± value of σ and ± 2 σ positions.

The control limit of table 1.T control chart reaches ± σ and ± 2 σ values

Sample size	CL	±σ	±2σ	UCL/LCL
					n＝2	0	±1.8373	±13.9677	+/-235.7837
n＝3	0	±1.3213	±4.5265	+/-19.2060
					n＝4	0	±1.1969	±3.3068	+/-9.2187
n＝5	0	±1.1416	±2.8693	+/-6.6201
					n＝6	0	±1.1105	±2.6486	+/-5.5070
n＝7	0	±1.0906	±2.5165	+/-4.9040
					n＝8	0	±1.0767	±2.4288	+/-4.5299
n＝9	0	±1.0665	±2.3664	+/-4.2766
					n＝10	0	±1.0587	±2.3198	+/-4.0942

2) for the K control chart of monitoring the technological parameter standard deviation

Similar with T control chart statistics definition of quantity, whether known according to the technological parameter standard deviation, the foundation of K control chart in two kinds of situation.

2.1) technological parameter parent standard deviation is known

At this moment, the K of each batch sample _{I, j}Statistic is:

$K_{i,j}=\frac{(n-1)S_{i,\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000799509800011.png,HSA00000799509800012.png"\; state="\{\{state\}\}">j</figure-callout>}}^2}{{\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000799509800011.png,HSA00000799509800012.png"\; state="\{\{state\}\}">j</figure-callout>}}^2},$ j＝1，2，..P (6)

Wherein n is the subgroup sample size, σ _jJ series products technological parameter parent standard deviation during for controlled process.When production run is in statistics slave mode, K _{I, j}Obeying degree of freedom is card side's distribution of n-1, and the control of K control chart is limited to:

$\mathrm{LCL}=H_{\mathrm{\&chi;}}^{-1}\left(\frac{\mathrm{\&alpha;}}{2}\right|n-1)$

$\mathrm{CL}=H_{\mathrm{\&chi;}}^{-1}\left(0.5\right|n-1)---\left(7\right)$

$\mathrm{UCL}=H_{\mathrm{\&chi;}}^{-1}(1-\frac{\mathrm{\&alpha;}}{2}|n-1)$

Wherein That degree of freedom is the inverse function of card side's distribution cumulative distribution function of n-1, α is level of significance, value is 0.0027, and is similar with the T control chart, the up and down control line of K control chart in the time of can obtaining Different Sample equally, center line and ± value of σ and ± 2 σ positions.For same production run, slave mode, technological parameter standard deviation are offset as long as process is in, the sample size of each batch measurement data is identical, the K statistic of then being calculated by every batch data process parameter measurement value and getting is separate and obey the distribution of same card side, and has identical control limit.Therefore can carry out quality monitoring by same control chart from the K statistic of dissimilar products.

2.2) the unknown of technological parameter parent standard deviation

Definition:

${\stackrel{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="S\; j"\; filenames="HSA00000799509800011.png,HSA00000799509800012.png"\; state="\{\{state\}\}">S</figure-callout>}}_j^{}}}^{(r-1)}=\frac{1}{r-1}\underset{h=1}{\overset{r-1}{\mathrm{\&Sigma;}}}{S}_j^{2\left(h\right)},$ j＝1，2，..P，r＝2，3... (8)

The expression product type is the average that the front r-1 of j criticizes sample variance.

The definition intermediate variable:

${\mathrm{\&lambda;}}_{i,j}^{\left(r\right)}=\frac{S_{i,j}^{2\left(r\right)}}{{\stackrel{\&OverBar;}{{\mathrm{<figure-callout\; id="2"\; label="S\; j"\; filenames="HSA00000799509800011.png,HSA00000799509800012.png"\; state="\{\{state\}\}">S</figure-callout>}}_j^{}}}^{(r-1)}},$ j＝1，2，..P，r＝2，3，... (9)

Statistic K _{I, j}Expression formula be:

$\begin{array}{c}{K}_{i,j}^{\left(1\right)}=1\\ K_{i,j}^{\left(r\right)}={\mathrm{\&Phi;}}^{-1}\left[F_{n-1,(n-1)(r-1)}\left({\mathrm{\&lambda;}}_{i,j}^{\left(r\right)}\right)\right]\end{array}$ j＝1，2，..P，r＞1 (10)

Wherein

First freely to be v ₁The second degree of freedom is v ₂The cumulative distribution function that distributes of F.Suppose 0＜s＜t, then press the statistic of following formula definition

With

Separate, separate from the sample of dissimilar products because as j ≠ k, so both must be independent; If j=k is defined as follows variable:

$W_1=(n-1)\underset{h=1}{\overset{r_s-1}{\mathrm{\&Sigma;}}}{S}_{\&CenterDot;,j}^{2\left(h\right)},$ ${\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="HSA00000799509800011.png,HSA00000799509800012.png"\; state="\{\{state\}\}">W</figure-callout>}}_2=(n-1)\underset{h=r_s}{\overset{r_i-r_s}{\mathrm{\&Sigma;}}}{S}_{\&CenterDot;,j}^{2\left(h\right)},$ ${\mathrm{<figure-callout\; id="3"\; label="W"\; filenames="HSA00000799509800012.png"\; state="\{\{state\}\}">W</figure-callout>}}_3=(n-1)S_{i,j}^{2\left(r_i\right)}$

Obvious W ₁:

W ₂:

W ₃:

Wherein: v ₁=(n-1) (r _s-1), v ₂=(n-1) (r _t-r _s), v ₃=n-1 is because W ₁, W ₂With W ₃Independent in twos, so W ₂/ W ₁With W ₃/ (W ₁+ W ₂) separate.Can get by conversion:

${\mathrm{\&lambda;}}_{s,j}^{\left(r_s\right)}=\frac{(n-1)(r_s-1)S_{s,j}^{2\left(r_s\right)}}{W_1},$ ${\mathrm{\&lambda;}}_{i,j}^{\left(r_i\right)}=\frac{v_1+v_2}{v_3}\frac{W_3}{W_1+{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="HSA00000799509800011.png,HSA00000799509800012.png"\; state="\{\{state\}\}">W</figure-callout>}}_2}$

So

With

Separate.Because obeying the first degree of freedom, (9) formula distributes for (n-1) F (r-1) for (n-1), the second degree of freedom, so the K statistic is separate and obey N (0,1), that is: in the K control chart all statistics points separate and obey a standardized normal distribution.With reference to the selection mode of traditional Shewhart control chart control limit, the control of K control chart was limited to when standard deviation was unknown:

LCL＝-3

CL＝0 (11)

UCL＝3

In sum, under many variety production pattern, adopt the T-K control chart that equipment running status is monitored, as long as every type of product sample data reach more than 2 batches (if distribution parameter is known, 1 batch data gets final product) and guarantee that the sample size of every batch of product is identical and greater than 1, whether average or standard deviation according to the technological parameter estimator of parent distribution be known, by (2) formula or (4) formula, (6) formula or (10) formula compute statistics, according to (5) formula, (7) formula or (11) formula are determined the control limit, and set up the T-K control chart.

Beneficial effect of the present invention:

The T-K control chart that technical scheme of the present invention is applicable to the Multi-varieties and Small-batch Production environment has two aspect advantages.It to the monitoring of average and standard deviation, only need respectively a control chart to make correct evaluation to the running status of many variety production process, and this control chart algorithm is simple, is convenient to practical application.Its two, the T-K control chart has the self-starting characteristics, need not undergoing analysis and estimates that with the control chart stage process of setting up of distribution parameter, especially the T control chart of parent has nothing to do with the parent standard deviation.Even if in the situation of estimator of parent distribution unknown parameters, as long as the data of each kind reach 2 batches, the T-K control chart can be controlled the multiple variety and small batch production implementation quality.

Description of drawings

Fig. 1 is average T-K control chart analysis result when out of control;

Fig. 2 is standard deviation T-K control chart analysis result when out of control;

Fig. 3 is bonding technology T-K control chart.

Embodiment

Below in conjunction with the accompanying drawing embodiment method of the present invention is described in more detail.

Embodiment 1. is by emulation mode checking T-K control chart testing process average ability out of control.Utilize randomizer to produce 25 batch datas, every batch of 5 samples, suppose that this 25 batch data is by two types product A, B consists of, difference Normal Distribution N (10,0.1) and N (20,0.5), every lot sample product type is originally determined at random with equiprobability when producing random number, supposes that the 1st to 15 batch data is in slave mode, since the 16th batch data, the skew of 3 times of standard deviations occurs in the average of two kinds of products, and standard deviation does not change, that is: category-A type product parameters is obeyed N (10.3,0.1), category-B type product parameters is obeyed N (21.5,0.5).Adopt the T-K control chart this 25 batch data to be analyzed result such as Fig. 1.

The the 1st to the 15th batch data is in slave mode in theory, no matter is category-A type product or category-B type product, and average and standard deviation all do not fluctuate, and therefore point out of control should not appear in front 15 batch datas in the T-K control chart.Since the 16th batch data, because " appearance of abnormal factors ", cause mean parameter that skew occurs and standard deviation remains unchanged.Therefore, the point out of control of upper control line appears surpassing in the T control chart from the 16th batch data, and all statistics points of K control chart are in slave mode always.

Embodiment 2. utilizes emulation mode checking T-K control chart testing process standard deviation ability out of control.Proof procedure and example 2 are similar.Utilize randomizer to produce 25 batch datas, every batch of 5 samples, this 25 batch data is made of two types product A, B, difference Normal Distribution N (10,0.1) and N (20,0.5), suppose that the 1st to 15 batch data is in slave mode, since the 16th batch data, the standard deviation of two kinds of products occurs unusually expanding as original 3 times, and average does not change, that is: category-A type product parameters is obeyed N (10,0.3), category-B type product parameters is obeyed N (20,1.5).Adopt the T-K control chart to analyze this 25 batch data, result such as Fig. 2.This 25 batch data average is offset in theory, therefore should not occur point out of control on the T control chart; And since the 16th batch, " abnormal cause appears in production run " causes the technological parameter standard deviation to be offset, so point out of control occurs on the K control chart.

Embodiment 3. utilizes the T-K control chart to estimate the running status of actual process process.In the bond sequence that microcircuit is produced, the bonding wire of employing has 2 kinds, and model is respectively F30 and F50.In normal productive process, monitored altogether 25 batch datas, every batch of 5 samples adopt the T-K control chart to carry out SPC and analyze, and the result is as shown in Figure 3.Analysis result shows that the bonding technology process is in the statistics slave mode.

Conclusion

When many variety production process is carried out quality control, often because data volume is not enough and dissimilar Product Process parameters and disobey same normal distribution, so that the Charts technology can't be directly used in many variety production environment.This paper has proposed to be applicable to the T-K control chart technology of many variety production pattern, comprise determining of T, K statistics definition of quantity and account form and T-K control chart control limit, the method only needs respectively a control chart just can monitor technological parameter average and standard deviation, and many variety production pattern is carried out quality control.Simultaneously, the T-K control chart possesses the self-starting characteristics.Theoretical analysis shows that the monitoring capacity of T-K control chart is consistent with traditional Shewhart control chart, even if the T-K control chart still can keep stable monitoring performance in the less situation of sample size.Instance analysis and simulating, verifying show that the T-K control chart can be used for the quality monitoring of many variety production process effectively, can make correct judgement to the running status of production run.

The above; only be the better embodiment of the present invention; protection scope of the present invention is not limited to this; anyly be familiar with those skilled in the art in the technical scope that the present invention discloses, the simple change of the technical scheme that can obtain apparently or equivalence are replaced and are all fallen within the scope of protection of the present invention.

Claims (4)

1. the many variety production pattern statistical process control method based on the T-K control chart is characterized in that, may further comprise the steps:

1) sets up the T control chart that is used for monitoring technological parameter average

1.1) T adds up definition of quantity

When making up the T statistic, whether known according to all types of Product Process parameter Parent Mean, have dual mode to set up statistic:

Situation one: the technological parameter Parent Mean is known

For each lot sample originally, i=1,2 ..., statistic T _{I, j}Expression formula as follows:

$T_{i,j}=\frac{{\stackrel{\&OverBar;}{X}}_{i,j}-{\mathrm{\&mu;}}_j}{S_{i,j}/\sqrt{n}},$ j＝1，2，..P

Wherein n is every lot sample capacity originally, μ _jJ kind Product Process parameter Parent Mean during for controlled process.When production run is in statistics slave mode, T _{I, j}Separate and obedience degree of freedom is that the t of n-1 distributes;

Situation two: the technological parameter Parent Mean is unknown

${\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{X}}}_j^{(r-1)}=\frac{1}{r-1}\underset{h=1}{\overset{r-1}{\mathrm{\&Sigma;}}}{\stackrel{\&OverBar;}{X}}_j^{\left(h\right)},$ j＝1，2，..P，r＝2，3...

The expression product type is the average of the front r-1 lot sample notebook data of j, and the expression formula of T statistic is:

$\begin{array}{c}{T}_{i,j}^{\left(1\right)}=0\\ T_{i,j}^{\left(r\right)}=\frac{{\stackrel{\&OverBar;}{X}}_{i,j}^{\left(r\right)}-{\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{X}}}_j^{(r-1)}}{S_{i,j}^{\left(r\right)}}\sqrt{\frac{n(r-1)}{r}}\end{array}$ j＝1，2，..P，r＞1

First statistic of all types of products is constant 0 in the T control chart, and follow-up statistic is by its appearance position in the respective type product and in the past data decision.The statistic of setting up by following formula is separate and obey same t and distribute;

1.2) the determining of T control chart control limit

The control of T control chart is limited to:

$\mathrm{UCL}=G_t^{-1}(1-\frac{\mathrm{\&alpha;}}{2}|n-1)$

CL＝0

LCL＝-UCL

Wherein

Be that degree of freedom is the inverse function of the t distribution cumulative distribution function of n-1, α is level of significance, and according to statistical process control, upper lower control limit is corresponding to ± 3 σ positions, and namely the α value is 0.0027;

2) set up the K control chart that is used for monitoring technological parameter standard deviation

Whether known according to the technological parameter standard deviation, the foundation of K control chart in two kinds of situation:

2.1) technological parameter parent standard deviation is known

At this moment, the K of each batch sample _I,The j statistic is:

$K_{i,j}=\frac{(n-1)S_{i,j}^2}{{\mathrm{\&sigma;}}_j^2},$ j＝1，2，..P

Wherein n is the subgroup sample size, σ _jJ series products technological parameter parent standard deviation during for controlled process.When production run is in statistics slave mode, K _{I, j}Obeying degree of freedom is card side's distribution of n-1, and the control of K control chart is limited to:

$\mathrm{LCL}=H_{\mathrm{\&chi;}}^{-1}\left(\frac{\mathrm{\&alpha;}}{2}\right|n-1)$

$\mathrm{CL}=H_{\mathrm{\&chi;}}^{-1}\left(0.5\right|n-1)$

$\mathrm{UCL}=H_{\mathrm{\&chi;}}^{-1}(1-\frac{\mathrm{\&alpha;}}{2}|n-1)$

Wherein

That degree of freedom is the inverse function of card side's distribution cumulative distribution function of n-1, α is level of significance, value is 0.0027, the up and down control line of K control chart when obtaining Different Sample, center line and ± value of σ and ± 2 σ positions, for same production run, as long as process is in slave mode, the technological parameter standard deviation is offset, the sample size of each batch measurement data is identical, then calculated by every batch data process parameter measurement value and the K statistic separate and obey the distribution of same card side, and have identical control limit, can carry out quality monitoring by same control chart from the K statistic of dissimilar products.

2.2) the unknown of technological parameter parent standard deviation

Definition:

${\stackrel{\&OverBar;}{S_j^2}}^{(r-1)}=\frac{1}{r-1}\underset{h=1}{\overset{r-1}{\mathrm{\&Sigma;}}}{S}_j^{2\left(h\right)},$ j＝1，2，..P，r＝2，3...

The expression product type is the average that the front r-1 of j criticizes sample variance,

The definition intermediate variable:

${\mathrm{\&lambda;}}_{i,j}^{\left(r\right)}=\frac{S_{i,j}^{2\left(r\right)}}{{\stackrel{\&OverBar;}{S_j^2}}^{(r-1)}},$ j＝1，2，..P，r＝2，3，...

Statistic K _{I, j}Expression formula be:

$\begin{array}{c}{K}_{i,j}^{\left(1\right)}=1\\ K_{i,j}^{\left(r\right)}={\mathrm{\&Phi;}}^{-1}\left[F_{n-1,(n-1)(r-1)}\left({\mathrm{\&lambda;}}_{i,j}^{\left(r\right)}\right)\right]\end{array}$ j＝1，2，..P，r＞1

Wherein

First freely to be v ₁The second degree of freedom is v ₂The cumulative distribution function that distributes of F.Suppose 0＜s＜t, then press the statistic of following formula definition

With Separate, separate from the sample of dissimilar products because as j ≠ k, so both must be independent; If j=k is defined as follows variable:

$W_1=(n-1)\underset{h=1}{\overset{r_s-1}{\mathrm{\&Sigma;}}}{S}_{\&CenterDot;,j}^{2\left(h\right)},$ $W_2=(n-1)\underset{h=r_s}{\overset{r_i-r_s}{\mathrm{\&Sigma;}}}{S}_{\&CenterDot;,j}^{2\left(h\right)},$ $W_3=(n-1)S_{i,j}^{2\left(r_i\right)}$

Obvious W ₁: W ₂:

W ₃:

Wherein: v ₁=(n-1) (r _s-1), v ₂=(n-1) (r _t-r _s), v ₃=n-1 is because W ₁, W ₂With W ₃Independent in twos, so W ₂/ W ₁With W ₃/ (W ₁+ W ₂) separate ^[9]Can get by conversion:

${\mathrm{\&lambda;}}_{s,j}^{\left(r_s\right)}=\frac{(n-1)(r_s-1)S_{s,j}^{2\left(r_s\right)}}{W_1},$ ${\mathrm{\&lambda;}}_{i,j}^{\left(r_i\right)}=\frac{v_1+v_2}{v_3}\frac{W_3}{W_1+W_2}$

So

With

Separate.Because obeying the first degree of freedom, (9) formula distributes for (n-1) F (r-1) for (n-1), the second degree of freedom, so the K statistic is separate and obey N (0,1), that is: in the K control chart all statistics points separate and obey a standardized normal distribution.With reference to the selection mode of traditional Shewhart control chart control limit, the control of K control chart was limited to when standard deviation was unknown:

LCL＝-3

CL＝0

UCL＝3

3) under many variety production pattern, adopt the T-K control chart that equipment running status is monitored, as long as every type of product sample data reach more than 2 batches, if distribution parameter is known, whether 1 batch data gets final product, and guarantees that the sample size of every batch of product is identical and greater than 1, known according to average or the standard deviation of technological parameter estimator of parent distribution, determine the control limit, and set up the T-K control chart.

2. the many variety production pattern statistical process control method based on the T-K control chart according to claim 1, it is characterized in that, choosing of the level of significance of the control chart of T step 1), control limit is all consistent with Shewhart control chart, and the T statistic is separate.

3. the many variety production pattern statistical process control method based on the T-K control chart according to claim 1, it is characterized in that, step 1) in the monitoring of technological process, whether have the point that exceeds the control limit except detecting, any point that shows nonrandom phenomenon in control chart also is considered as point out of control, in these criterions out of control, also need by control chart ± σ (probable value 0.3173) and ± position of 2 σ (probable value 0.0455).

4. the many variety production pattern statistical process control method based on the T-K control chart according to claim 1 is characterized in that step 1) in the fractile that distributes according to t of T control chart determine the control limit, its control limit is relevant with the sample size of every batch data.

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