生产过程的动态监测：灰色预测模型的一种应用外文翻译

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.生产过程的动态监测：灰色预测模型的一种应用摘要本文运用统计控制图作为有效工具来

50、改善和监测制造过程的质量。在正态假设下，当一个过程变量在控制范围内，就认为这个过程处于被控制状态。有时，这个过程作为一个短期的控制过程；但是，一旦数据显示生产失去控制，有缺陷的产品将已大量生产，特别是当过程表现出一个貌似正常的趋势行为或发生细微的变化。在论文中，我们探索了预测和监控生产过程中灰色预测模型的应用。我们研究了基于检测过程变化的灰色预测控制图的性能。平均运行长度（ARL）是用来衡量当均值漂移存在时的效力。当均值漂移时，特别是当用于计算的灰色预测样本数很小时，我们发现灰色预测要优于样本均值。我们还发现灰色预测对样本数非常敏感。关键字平均运行长度控制图 控制极限灰色预测引言统计控制图长期

51、以来作为一种改善和监测制造过程质量的有效工具。传统统计过程控制（SPC）方法假设过程变量服从正态分布，且研究数据相互独立。在正态假设下，当过程变量在控制范围内，就认为这个过程处于被控制状态；否则，过程假设发生变化，例如过程失去控制。在许多情况下，过程显示在控制状态，而事实上过程已失去控制，如刀具磨损或原料耗尽。有时过程作为一个短期的控制过程；但是， 一旦数据显示生产失去控制， 有缺陷的产品将已大量生产，特别是当过程表现出一个貌似正常的趋势行为或发生细微的变化。虽然过程中这样的变动不易发现，但过程仍然是可预测的。如果过程失败成本非常大，那么尽快发现这些变动就变得十分重要。在论文中，我们探索了预测

52、和监控生产过程中灰色预测模型的应用。我们研究了基于检测过程变化的灰色预测控制图的性能。平均运行长度（ARL ）是用来衡量当均值漂移存在时的效力。（ ARL ）意味着在失去控制信号产生以指示特殊情况之前，观测值的均值是必要的。我们希望（ARL ）值较小，灰色预测值要与样本均值x 相比较，所有步骤都进行模拟。当均值漂移时，且当用于计算的灰色预测样本数很小时，我们发现灰色预测要优于样本均值。我们还发现灰色预测对样本数非常敏感。灰色预测仅需少量样本以检测过程变化甚至变化很细微，这正是灰色预测的优势所在。我们可以调整样本数以便灰色预测值可以根据所需标准改变。在下一章节，我们引出灰色预测模型并给出所提出监

53、测步骤的概述。数字分析具体结果以及结论在第三部分给出，同时也给出了灰色预测值与样本值的比较。我们还作出了基于样本均值及灰色预测值的X 条形控制图类误差。最后，在第四部分我们给出了基于结果的建议。灰色预测模型和程序步骤灰色系统是由邓聚龙教授提出的，现在灰色系统理论已成功地运用于许多领域如管理、经济、工程、金融等。系统分为三种白色、黑色和灰色。人们把信息了解得清清楚楚的系统称为白色系统。与之相反，黑色系统表示人们对系统信息全然不知。如果系统信息部分已知，部分未知则称为灰色系统。在制造流程中，操作环境、设施可靠性及雇员行为都是不可能完全得知或完全掌控的因素。为了控制系统行为，灰色模型用来构建一个常微

54、分方程，然后求解灰色模型微分方程。运用稀缺的过去数据，灰色模型可以精确预测输出结果。得到预测结果，我们就可以检查过程是否失去控制。本文我们通过GM(1,1)模型监控及预测输出结果。顺序监控这个过程是选择输出节点(通常是样本值)，并在到达新节点之前分析灰色预测累积结果。这个过程可分为五步：步骤 1： 收集原始数据建立数据序列观察到的原始数据定义为xi(0)， i 是第 i 个样本值。则原序列k 个样本值定义为 TOC o 1-5 h z x(0)x1(0),x2(0),x3(0),xk(0)(1)步骤2：把原始数据序列转换成新序列。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)定义xi(1)如下：xi(1)i xn(0),i 1,2, ,k.(3)n1步骤3：建立GM(1,1)模型的一阶微分方程。通过把原始数据序列转化成一阶微分方程，时间序列近似为指数函数，得到灰色微分方程模型：dx11(4)axdt其中 a是发展系数，b代表灰色输入。根据方程2 和方程4，参数a和 b可用最小二乘法估计。参数a 的估计值a?为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 步骤4： 灰色预测

56、结果如下取得：把方程 5 至 7 得到的参数估计代入方程4，得到?(1)(0) b ak bx?k 1(x1)e(8)aa其中x?1(1)x1(0)。通过作一阶累减可生成第(k 1)个预测数据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步骤5：检查过程：得到新节点(一个样本值)后，依次完成步骤1 至步骤4 来预测过程直到异常情况出现。控制范围是基于X 条形控制图，最高限和最低限将分别是x 3 x 和 x 3 x 。一旦新预测节点超出或低于限制就意味着过程失去控制就要开始调查；否则，过程继续受监控。3 数值分析及结论在这

57、部分，我们给出模拟结果。我们假设过程变量服从正态分布，数据点产生于均值为3 方程为 1 的正态分布。样本大小是3,5及 7 分别进行模拟。当样本量等于 3 时， X 条形控制图的中线等于均值3，标准差等于1 。 X 条形控制图3上下限为(3 31 ,3 31 )。 同样的步骤可分别用于样本量为5 和 7 的情况。33得到每个数据后就可以分析过程。所有程序用MATLAB 语言写出，所有样本通过 MATLAB 生成。所有结果基于1000个重复抽样。一旦得到X 条形控制图的上下限，通过计算ARL 来比较灰色预测值和样本值。 模拟过程前10 个样本受控制而第11 个样本失去控制。一旦发现均值漂移不再控制范围，ARL 会有记录。我们来讨论均值漂移的下面三种水平：标准差改变0.1 的均值漂移。例如，均值偏离目标到0.1 s 目标，其中s是上面所描述的样本值的标准差(即s / n，其中n是样本量)在本文中，目标为3；标准差改变0.5的均值漂移；标准差改变1.5的均值漂移。我们还比较了样本值的类误差概率和灰色预测值在同样控制范围内的概k 值设为率。为了了解对用来计算灰色预测的