生產(chǎn)過程的動態(tài)監(jiān)測：灰色預(yù)測模型的一種應(yīng)用外文翻譯

1、Li-Lin KuT ung-Chen HuangSequential monitoring of manufacturing processes: an application of grey forecasting modelsAbstractThis study used statistical control charts as an efficient tool for improving and monitoring the quality of manufacturing processes. Under the normality assumption, when a proc

2、ess variable is within control limits, the process is treated as being in-control. Sometimes, the process acts as an in-control process for short periods; however, once the data show that the production process is out-of-control, a lot of defective products will have already been produced, especiall

3、y when the process exhibits an apparent normal trend behavior or if the change is only slight. In this paper, we explore the application of grey forecasting models for predicting and monitoring production processes. The performance of control charts based on grey predictors for detecting process cha

4、nges is investigated. The average run length(ARL) is used to measure the effectiveness when a mean shift exists. When a mean shift occurs, the grey predictors are found to be superior to the sample mean, especially if the number of subgroups used to compute the grey predictors is small. The grey pre

5、dictor is also found to be very sensitive to the number of subgroups.Keywords Average run length Con trol chart Con trol limit Grey predictorIntroductionStatistical control charts have long been used as an efficient tool for improving and monitoring the quality of manufacturing processes. Traditiona

6、l statistical process control (SPC) methods assume that the process variable is distributed normally, and that the observed data are independent. Under the normality assumption, when the process variable is within the control limits, the process is treated as being in-control; otherwise, the process

7、 assumes that some changes have occurred, i.e., the process may be out-of-control.There are many situations in which processes act as in-control while in they are in fact out-of-control, such as tool-wear1 and when the raw material has been consumed. Sometimes the process acts as an in-control proce

8、ss for short periods; however, once the data show that the production process is out-of-control, a lot of defective products have already been produced, especially when the process exhibits an apparent normal trend behavior2 or if the change is only slight. Though these kinds of shifts in the proces

9、s are not easy to detect, the process is nevertheless predictable. If the process failure costs are very large, then detecting these shifts as soon as possible becomes very important.In this paper, we explore the application of grey forecasting models for predicting and monitoring production process

10、es. The performance of control charts based on grey predictors for detecting process changes is studied. The average run length(ARL) is used to measure the effectiveness when a mean shift exists. The ARL means that an average number of observations is required before an out-of-control signal is crea

11、ted indicating special circumstances. Small ARL values are desired. The performance of grey predictors is compared with sample means x . All procedures are studied via simulations. When a mean shift occurs, the grey predictors are found to be superior to the sample mean if the number of subgroups th

12、at are used to compute the grey predictors is small. The grey predictor is also found to be very sensitive to the number of subgroups. The advantage of the grey methods is that the grey predictor only needs a few samples in order to detect the process changes even when the process shifts are slight.

13、 The number of subgroups(samples) can be adjusted so that the performance of grey predictors can be changed according to the desired criteria.In the next section, the grey forecasting models are introduced and an overview of the proposed monitoring procedure will be given. The details of the numeric

14、al analytical results and conclusions are then given in Sect. 3, in which the results of the grey predictors are compared with sample means. The Type I error based on X-bar control charts for sample means and the grey predictors are also described. Finally, recommendations and suggestions based on t

15、he results are then discussed in Sect. 4.Grey forecasting models and procedural stepsThe grey system was proposed by Deng 3. The grey system theory has been successfully applied in many fields such as management, economy, engineering, finance 4 6, etc. There are three types of systems white, black,

16、and grey. A system is called a white system when its information is totally clear. When a system s information is totally unknown, it is called a black system. If a system s information is partially known, then it is called a grey system.In manufacturing processes, the operational conditions, facili

17、ty reliability and employee behaviors are all factors that are impossible to be totally known or be fully under control. In order to control the system behavior, a grey model is used to construct an ordinary differential equation, and then the differential equation of the grey model is solved. By us

18、ing scarce past data, the grey model can accurately predict the output. After the output is predicted, it can be checked if the process is under control or not.In this paper, we monitor and predict the process output by means of the GM(1,1) model 7. Sequential monitoring is a procedure in which a ne

19、w output point is chosen (usually it is a sample mean) and the cumulative results of the grey forecasts are analyzed before proceeding to the next new point. The procedure can be separated into five steps:Step 1: Collect original data and build a data sequence. The observed original data are defined

20、 as xi(0) , where i is i th sample mean. The raw sequence of k samples is defined as TOC o 1-5 h z x(0)x1(0),x2(0),x3(0), xk(0)(1)Step 2: Transform the original data sequence into a new sequence. A new sequence (1)x is generated by the accumulated generating operation(AGO), wherex(1)x1(1),x2(1),x3(1

21、),xk(1)(2)(1)The xiis derived as follows:xi(1)xn(0),i1,2, , k.(3)n1Step 3: Build a first-order differential equation of the GM(1,1) model. By transforming the original data sequence into a first-order differential equation, the time series can be approximated by an exponential function. The grey dif

22、ferential model is obtained asdx11ax b(4)dtwhere a is a developing coefficient and b represents the grey input. According to Eqs. 2 and 4, the parameters a and b can be estimated by the least-squares method.The parameters a? are represented as(5)(6)(7)a? a,bT (BTB) 1 BTYkwhere TOC o 1-5 h z 1(1)(1)(

23、x1x2 ) 12B12 (x2(1)x3(1)11 (xk(1)1xk(1) )12andYkx2(0),x3(0),xk(0)TStep 4: The grey forecasting predictor is obtained as follows: by substituting the estimated parameters obtained from Eqs. 5 to 7 into Eq. 4, we get x?k(1)1 (x1(0)b)eak b(8)aawhere x?1(1)x1(0) . After one order inverse-accumulated gen

24、erating operation (IAGO),the (k 1)th predicted data x?k(0)1 can be calculated byx?k(01) x?k(1)1 x?k(1)x1(0) b 1 ea e ak , k 1,2,3,(9)aStep 5: Check the process: after a new point (a sample mean) is obtained, Steps 1 to 4 are followed to predict the behavior of the process, until unusual conditions o

25、ccur.The control limits are based on X-bar control charts. The upper and lower limits will be x 3 x and x 3 x , respectively. Once a new predicted point is plotted beyondthe upper or lower limits, it means that the process may be out-of-control and that an investigation will be started; otherwise, t

26、he process continues to be monitored.3 Numerical analysis and conclusionsIn this section, the simulation results are given. We begin by making the assumption that the process variable has a normal probability distribution. The data points were generated from a normal distribution with mean 3 and var

27、iance 1. Samplesizes of 3, 5, and 7 were simulated. When the sample size equals 3, the central line of the X-bar control chart is assumed equal to mean 3, and the standard deviation of themean is equal to 1 .The upper and lower limits of the X-bar control charts are then 311calculated and equal to (

28、3 3,3 3) . The same procedures can be applied33for sample sizes of 5 and 7, respectively. The process can then be analyzed as per the data obtained and plotted. All programs were written in the MATLAB language and all samples were generated through MATLAB. All results are based on 1000 replications.

29、Once the upper and lower limits of the X-bar control charts are obtained, the performances of the grey predictors and sample means are compared by the calculated ARL. The processes are simulated as in-control for the first 10 samples and as out-of-control after the 11th sample. Once a mean shift is

30、detected by points outside the control limits, the ARL will be recorded.We will discuss the following three levels of mean shifts: (i) a mean shift of 0.1 standard deviations from a target. i.e., the mean shifts from the target to .01 s+target, where s is a standard deviation of the sample mean desc

31、ribed as above (i.e., s / n , where n is the sample size) in this paper, the target equals 3; (ii) a mean shift of 0.5 standard deviations from a target; (iii) a mean shift of 1.5 standard deviations from a target.The probability of a Type I error of sample means, and grey predictors under the same

32、control limits are also compared. To understand the sensitivity and influence of the number of subgroups that are used to compute the grey predictors (i.e., the k values are set at 4, 6, 7 and 8) the raw sequence ko fs amples is defined as(0) x(0)(0)(0)(0)x1,x2 , x3 , xk where k equals 4, 6, 7 or 8,

33、 respectively. The results of the observed probability of Type I errors, the ARL for sample means, and the grey predictors fork = 4, are given in Table 1. The other results in the cases ofk = 6, 7 and 8 are summarized in Tables 2 to 4.Table 1. Type I error and ARL for x and grey predictors when k 4A

34、RLType I errorSample0.1 s target0.5 s target1.5 s targetsizexgreyxgreyxgreyxgrey30.00270.0526333.42517.374151.28112.49215.5913.62750.00270.0513357.10518.543165.512.39915.0213.75370.00270.0508368.07719.641163.81713.71815.2213.63Table 2. Type I error and ARL for x and grey predictors when k 6Sample si

35、zeType I errorARL0.1 s target0.5 s target1.5 s targetxgreyxgreyxgreyxgrey30.00270.0064346.44145.016153.98965.73214.9728.27650.00270.0055361.911161.731154.99471.07215.0059.24770.00270.0052358.536172.506162.59480.5215.52210.389Table 3. Type I error and ARL for x and grey predictors when k 7Sample size

36、Type I errorARL0.1 s target0.5 s target1.5 s targetxgreyxgreyxgreyxgrey30.00270.0023343.727368.188160.249150.27215.57213.83950.00270.0019360.713425.558154.797158.2315.00914.07370.00270.0017360.338494.867155.729181.09215.28615.464Table 4. Type I error and ARL for x and grey predictors whenSampl e siz

37、eType I errorARL0.1 s target0.5 s target1.5 s targetxgreyxgreyxgreyxgrey30.00270.000842360.09984.227144.174307.57315.71121.20250.00270.000676343.0191245.5152.785386.415.422.55470.00270.000597367.6121401.1158.62396.78416.24521.726The Type I error of sample means is virtually fixed on 0.0027, i.e., 0

38、.27%. This is because the X-bar control charts are based on x 3 x , where x and x are assumed values in Sect. 3. If the process variable has a normal distribution, then the probability of the population mean will fall within 3 standard deviations of the sample mean, and will be about 99 .74%. From o

39、ur simulation results, the grey predictor is very sensitive to the number of subgroups, i.e., the k values. From Tables 1 to 4, the Type I error of grey predictors decreases rapidly when thek value increases.Once the mean shift levels become larger, the ARLs become smaller for both methods; i.e., it

40、 becomes easier to detect out-of-control conditions. When k equals 4 or 6, the performance of the grey predictors dominates the sample means, but the Type I error of the grey predictors is larger than that of the sample means. When the value of k becomes larger (k = 7), the differences of the ARL of

41、 the sample means and the grey predictors are not significant, but the Type I error of grey predictors is smaller than x s. This means that the probability of a false indication of th eprocess change is small if grey predictors are used. When k is increased to 8, the ARL of the grey predictors doesn

42、 t pweerflol,r bmut their Type I error is very small.From our simulation results, when k equals 4 or 6, the capabilities of grey predictors of detecting out-of-control situations are outstanding, but their Type I error is relatively larger than that of the sample means. Once thke values are increase

43、d to 7, the capabilities of detecting unusual conditions are similar for the sample means and the grey predictors, but the grey predictors will have a smaller Type I error.4 Suggestions and recommendationsIn conclusion, if the failure costs of a process are very large, i.e., the costs of recovering

44、from or repairing a defect are substantial and important, the number of subgroups used to compute grey predictors should be small, because that way grey predictors are more sensitive to process changes. Because, when k = 4, the Type I error of the grey predictors is too large, k = 6 is suggested for

45、 monitoring the process. If the costs of the process interruptions can be ignoredk, = 4 is suggested. Whenk = 6, the Type I error of grey predictors is larger than that of the sample means. The importance is that when k =6, the grey predictors can detect the process shifts very quickly, even when th

46、e shift level is small, so there is a saving on repair costs. Also, the process chaos that is created by out-of-control situations can be predicted and reduced quickly. The pros and cons should be explored according to each individual process.If failure costs are low, but the costs of interrupting t

47、he process are high,k = 7 is suggested. Whenk = 7, the performance of x and grey predictors is competitive, but the Type I error of the grey predictors is smaller than that of the sample means; that is, grey predictors have small probabilities of detecting an out-of-control signal when in fact the p

48、rocess is in-control. It is not suggested for the k value to be increased to 8, because the ARL of the grey predictors is too large, unless the false interrupting costs of process changes are immense. If the costs of failure and interrupting the process are both high, then the sample means and the g

49、rey predictors can be used to monitor the process behavior simultaneously. Once the grey predictors or the sample means have pointed out that the process may be out-of-control, an advanced investigation should be made before the process is actually interrupted.生產(chǎn)過程的動態(tài)監(jiān)測：灰色預(yù)測模型的一種應(yīng)用摘要本文運(yùn)用統(tǒng)計(jì)控制圖作為有效工具來

50、改善和監(jiān)測制造過程的質(zhì)量。在正態(tài)假設(shè)下，當(dāng)一個過程變量在控制范圍內(nèi)，就認(rèn)為這個過程處于被控制狀態(tài)。有時(shí)，這個過程作為一個短期的控制過程；但是，一旦數(shù)據(jù)顯示生產(chǎn)失去控制，有缺陷的產(chǎn)品將已大量生產(chǎn)，特別是當(dāng)過程表現(xiàn)出一個貌似正常的趨勢行為或發(fā)生細(xì)微的變化。在論文中，我們探索了預(yù)測和監(jiān)控生產(chǎn)過程中灰色預(yù)測模型的應(yīng)用。我們研究了基于檢測過程變化的灰色預(yù)測控制圖的性能。平均運(yùn)行長度（ARL）是用來衡量當(dāng)均值漂移存在時(shí)的效力。當(dāng)均值漂移時(shí)，特別是當(dāng)用于計(jì)算的灰色預(yù)測樣本數(shù)很小時(shí)，我們發(fā)現(xiàn)灰色預(yù)測要優(yōu)于樣本均值。我們還發(fā)現(xiàn)灰色預(yù)測對樣本數(shù)非常敏感。關(guān)鍵字平均運(yùn)行長度控制圖 控制極限灰色預(yù)測引言統(tǒng)計(jì)控制圖長期

51、以來作為一種改善和監(jiān)測制造過程質(zhì)量的有效工具。傳統(tǒng)統(tǒng)計(jì)過程控制（SPC）方法假設(shè)過程變量服從正態(tài)分布，且研究數(shù)據(jù)相互獨(dú)立。在正態(tài)假設(shè)下，當(dāng)過程變量在控制范圍內(nèi)，就認(rèn)為這個過程處于被控制狀態(tài)；否則，過程假設(shè)發(fā)生變化，例如過程失去控制。在許多情況下，過程顯示在控制狀態(tài)，而事實(shí)上過程已失去控制，如刀具磨損或原料耗盡。有時(shí)過程作為一個短期的控制過程；但是， 一旦數(shù)據(jù)顯示生產(chǎn)失去控制， 有缺陷的產(chǎn)品將已大量生產(chǎn)，特別是當(dāng)過程表現(xiàn)出一個貌似正常的趨勢行為或發(fā)生細(xì)微的變化。雖然過程中這樣的變動不易發(fā)現(xiàn)，但過程仍然是可預(yù)測的。如果過程失敗成本非常大，那么盡快發(fā)現(xiàn)這些變動就變得十分重要。在論文中，我們探索了預(yù)測

52、和監(jiān)控生產(chǎn)過程中灰色預(yù)測模型的應(yīng)用。我們研究了基于檢測過程變化的灰色預(yù)測控制圖的性能。平均運(yùn)行長度（ARL ）是用來衡量當(dāng)均值漂移存在時(shí)的效力。（ ARL ）意味著在失去控制信號產(chǎn)生以指示特殊情況之前，觀測值的均值是必要的。我們希望（ARL ）值較小，灰色預(yù)測值要與樣本均值x 相比較，所有步驟都進(jìn)行模擬。當(dāng)均值漂移時(shí)，且當(dāng)用于計(jì)算的灰色預(yù)測樣本數(shù)很小時(shí)，我們發(fā)現(xiàn)灰色預(yù)測要優(yōu)于樣本均值。我們還發(fā)現(xiàn)灰色預(yù)測對樣本數(shù)非常敏感。灰色預(yù)測僅需少量樣本以檢測過程變化甚至變化很細(xì)微，這正是灰色預(yù)測的優(yōu)勢所在。我們可以調(diào)整樣本數(shù)以便灰色預(yù)測值可以根據(jù)所需標(biāo)準(zhǔn)改變。在下一章節(jié)，我們引出灰色預(yù)測模型并給出所提出監(jiān)

53、測步驟的概述。數(shù)字分析具體結(jié)果以及結(jié)論在第三部分給出，同時(shí)也給出了灰色預(yù)測值與樣本值的比較。我們還作出了基于樣本均值及灰色預(yù)測值的X 條形控制圖類誤差。最后，在第四部分我們給出了基于結(jié)果的建議?；疑A(yù)測模型和程序步驟灰色系統(tǒng)是由鄧聚龍教授提出的，現(xiàn)在灰色系統(tǒng)理論已成功地運(yùn)用于許多領(lǐng)域如管理、經(jīng)濟(jì)、工程、金融等。系統(tǒng)分為三種白色、黑色和灰色。人們把信息了解得清清楚楚的系統(tǒng)稱為白色系統(tǒng)。與之相反，黑色系統(tǒng)表示人們對系統(tǒng)信息全然不知。如果系統(tǒng)信息部分已知，部分未知則稱為灰色系統(tǒng)。在制造流程中，操作環(huán)境、設(shè)施可靠性及雇員行為都是不可能完全得知或完全掌控的因素。為了控制系統(tǒng)行為，灰色模型用來構(gòu)建一個常微

54、分方程，然后求解灰色模型微分方程。運(yùn)用稀缺的過去數(shù)據(jù)，灰色模型可以精確預(yù)測輸出結(jié)果。得到預(yù)測結(jié)果，我們就可以檢查過程是否失去控制。本文我們通過GM(1,1)模型監(jiān)控及預(yù)測輸出結(jié)果。順序監(jiān)控這個過程是選擇輸出節(jié)點(diǎn)(通常是樣本值)，并在到達(dá)新節(jié)點(diǎn)之前分析灰色預(yù)測累積結(jié)果。這個過程可分為五步：步驟 1： 收集原始數(shù)據(jù)建立數(shù)據(jù)序列觀察到的原始數(shù)據(jù)定義為xi(0)， i 是第 i 個樣本值。則原序列k 個樣本值定義為 TOC o 1-5 h z x(0)x1(0),x2(0),x3(0),xk(0)(1)步驟2：把原始數(shù)據(jù)序列轉(zhuǎn)換成新序列。新序列x(1)是由一次累加操作生成的，即x(1)x1(1),x2

55、(1),x3(1),xk(1)(2)定義xi(1)如下：xi(1)i xn(0),i 1,2, ,k.(3)n1步驟3：建立GM(1,1)模型的一階微分方程。通過把原始數(shù)據(jù)序列轉(zhuǎn)化成一階微分方程，時(shí)間序列近似為指數(shù)函數(shù)，得到灰色微分方程模型：dx11(4)axdt其中 a是發(fā)展系數(shù)，b代表灰色輸入。根據(jù)方程2 和方程4，參數(shù)a和 b可用最小二乘法估計(jì)。參數(shù)a 的估計(jì)值a?為a? a,bT (BTB) 1 BTYk(5)1(1)(1)(x1x2 ) 11(1)(1)其中 B2 (x2x3 ) 1Ykx2(0),x3(0),xk(0)T2(6)(7) TOC o 1-5 h z 步驟4： 灰色預(yù)測

56、結(jié)果如下取得：把方程 5 至 7 得到的參數(shù)估計(jì)代入方程4，得到?(1)(0) b ak bx?k 1(x1)e(8)aa其中x?1(1)x1(0)。通過作一階累減可生成第(k 1)個預(yù)測數(shù)據(jù)x?k(0)1，即x?k(01)x?k(1)1x?k(1)x1(0)b 1 ea e ak , k 1,2,3,(9)a步驟5：檢查過程：得到新節(jié)點(diǎn)(一個樣本值)后，依次完成步驟1 至步驟4 來預(yù)測過程直到異常情況出現(xiàn)?？刂品秶腔赬 條形控制圖，最高限和最低限將分別是x 3 x 和 x 3 x 。一旦新預(yù)測節(jié)點(diǎn)超出或低于限制就意味著過程失去控制就要開始調(diào)查；否則，過程繼續(xù)受監(jiān)控。3 數(shù)值分析及結(jié)論在這

57、部分，我們給出模擬結(jié)果。我們假設(shè)過程變量服從正態(tài)分布，數(shù)據(jù)點(diǎn)產(chǎn)生于均值為3 方程為 1 的正態(tài)分布。樣本大小是3,5及 7 分別進(jìn)行模擬。當(dāng)樣本量等于 3 時(shí)， X 條形控制圖的中線等于均值3，標(biāo)準(zhǔn)差等于1 。 X 條形控制圖3上下限為(3 31 ,3 31 )。 同樣的步驟可分別用于樣本量為5 和 7 的情況。33得到每個數(shù)據(jù)后就可以分析過程。所有程序用MATLAB 語言寫出，所有樣本通過 MATLAB 生成。所有結(jié)果基于1000個重復(fù)抽樣。一旦得到X 條形控制圖的上下限，通過計(jì)算ARL 來比較灰色預(yù)測值和樣本值。 模擬過程前10 個樣本受控制而第11 個樣本失去控制。一旦發(fā)現(xiàn)均值漂移不再控制范圍，ARL 會有記錄。我們來討論均值漂移的下面三種水平：標(biāo)準(zhǔn)差改變0.1 的均值漂移。例如，均值偏離目標(biāo)到0.1 s 目標(biāo)，其中s是上面所描述的樣本值的標(biāo)準(zhǔn)差(即s / n，其中n是樣本量)在本文中，目標(biāo)為3；標(biāo)準(zhǔn)差改變0.5的均值漂移；標(biāo)準(zhǔn)差改變1.5的均值漂移。我們還比較了樣本值的類誤差概率和灰色預(yù)測值在同樣控制范圍內(nèi)的概k 值設(shè)為率。為了了解對用來計(jì)算灰色預(yù)測的