A new alternative Hotelling’s T2 control chart using MEWMA to monitor minimal mean deviation

Hotelling 2 T control chart not only reflects the correla-tions between different quality characteristics but also has good efficiency on monitoring multivariate quality characteristics in production process. A new alternative control chart was constructed after the original products data are processed by using multivariate exponentially weighted moving average for cumulating failure effects because 2 T control chart is ineffective on detecting minimal mean deviations. Exemplified by bivariate quality characteristics, we compared the monitoring effects of Hotelling 2 T control chart and new control chart which is called as 2 MEWMA T control chart. Paper showed the improved 2 MEWMA T control chart has smaller average run length than Hotelling 2 T control chart on monitoring minimal mean deviation and that also studied the relationships between 2 MEWMA T control chart’s forgetting factor, sample sizes N and type II error. It indicated the smaller forgetting factor is more sensitive to minimal mean value deviation and that average run length tended to become bigger gradually along with increase of sample sizes N when production process is out of control.


Introduction
Quality management is becoming increasingly important in manufacturing and other industries. Quality control is a key element of quality management. H. F. Dodge, H. G. Romig and W. A. Shewhart together created the statistical quality control which the statistics data mining techniques are used in industrial processes for sampling and quality control in the 1920s.
Quality control chart is a common statistical quality control theory and divided by the number of quality characteristics indicators into univariate and multivariate control charts. Although univariate control chart has matured, the actual production process usually accompanies two or more key quality characteristics, so the researches of creating multidimensional quality control charts are urgent. Currently multivariate control charts including MCUSUM, MEWMA, Hotelling 2 T and various optimized control charts based on all above researches [1][2][3][4][5][6][7][8][9][10][11][12]. Quality characteristic generally follow a given normal distribution in industrial production. Univariate control chart need to control the process mean and volatility simultaneously, multivariate control charts not only need to control both the mean and volatility of each variable but also need to control various correlations. Hotelling 2 T control chart takes full advantage of the correlations and has a good effect on monitoring multiple quality characteristics, but Lin(2005) studied that hotelling 2 T control chart does not make use of products historical data effectively and detect process mean minimal deviation. Li andQiu(2012,2014) found that Multivariate exponentially weighted moving average (MEWMA) detects minimal fault before the process mean shift, thereby leading to early prevention and trouble resolutions.
There are two types of errors of control charts. Type I error , the probability of which control charts judge process unsteady when the production process is steady. Type II error , the probability of which control charts judge process steady when the production process is unsteady. Generally, you want that Type I error is as small as possible when process is under control, Type II error β is as small as possible when uncontrolled. In fact, the two types of errors can't become small simultaneously. We usually fix a type error to compare the other in order to determine the efficiency of control charts.
In this paper we proposed an alternative Hotelling's using MEWMA to detect minimal mean deviation. In Section 2, we introduced the MEWMA process in detail. Afterwards we constructed a new control chart to detect process minimal deviation quickly, namely control charts. In Section 3, firstly, we compared two kinds of control chart's average run length by monte-carlo simulations when the process is out of control; secondly, we researched the relationships between and detection efficiency of control chart on mean minimal deviation; Finally, we studied the efficiency of control chart by changing on the sample sizes N. In last Section, We came to some conclusions and considered a further study to be done.

2
T control chart

Multivariate exponentially weighted moving average
Products quality characteristics follow P multiv-ariate normal distribution, ∼ ( , ), with the process mean 0 and the covariance 0 . The processed data is below according to multivariate exponentially weighted moving average (MEWMA), See formula (1).
, forgetting factor matrix λ is unknown. The larger j λ ,the smaller the influence whose historical data make on the current indicators.
( ) Let us consider the equation(1) can be simplified into under this condition if P quality characteristics priorities are equal value .At this moment, the average value and covariance of will depend on .
According to the property of independent normal random variable: (5)

Group samples ( > )
Hotelling 2 T control chart gradually developed into a main multivariable quality monitoring tool by several years of development which is mentioned by Wang et al(2008) Quality characteristics data extracted by the quality inspector may be a group data or single-valued data. Hotelling(1947) found that the two different data patterns have different distribution characteristics. Production quality characteristics X follow multivariate normal distribution, There are the statistics 2 T .
p m n mn n p p m n = .

Single-value sample ( n 1 = )
When the sample sizes is 1, that is n=1, statistics 2 T is saw equation (4).
We usually consider formula (4) as a special example of equation (2). In fact, statistics 2 T follow 2 χ distribution with P degrees of freedom, that is If production process is statistical process control in the initial stages, that is, multivariate process mean and covariance matrix are unknown, we should estimate overall mean and covariance based on samples. Easy to understand, overall group average and group average covariance respectively are the uniformly minimum variance unbiased estimate of sample population and sample population .
is the sample mean value of the group ,Then, is the covariance array of the group i.
is an average value of all of groups mean value, and is an average covariance of all of groups average covariance.  2 T control chart is insensitive to process average minimum deviation which., we achieve inspiration from MEWMA and construct a new control chart, namely 2 MEWMA T control chart, which deals with samples and cumulates failure effects according to multivariate exponentially weighted moving average to detect process minimal deviation quickly. m n × samples are extracted from production process. First of all, we should have a judgement whether the process is in control by using hotelling 2 T control chart. If you are uncertain on process mean's minimal deviation, the following procedure can solve the problem.
a) m single-value datas are constructed by multivariate exponentially weighted moving average, see equation (1).
T control chart is constructed by using single-value data to decide whether process mean gets minimal deviation, see equation(9).(11). c) 0 µ and 0 Σ are estimated based on single-value data, see equations(13). (14).

d)
we should focus on the problem how to select the appropriate sample sizes N and forgetting factor matrix λ in the process of implementation of the 2

MEWMA
T control chart to make the control chart playing best efficiency.

Monte-Carlo simulation
Article compared two different control chart's average run length by monte-carlo simulation in order to confirm whether 2 T control chart is beneficial to detecting process mean slight shift when the process is out of control. In the end, we discussed the relationships between forgetting factor, sample sizes N and detection efficiency of mean minimal deviation about 2 MEWMA T control chart, the article took bivariate quality characteristics for an example. .     b)

A comparative analysis of
The average run length is 100, 200, 300, 370, 500 in turn when both control charts are in control. Minimal deviations of two quality indicators 1 X and 2 X appeared simultaneously, the offset K L = .

e)
We get a 1 ARL after every simulation, where, ARL which is an average run length when the production process is unsteady will be expected smaller to detect process deviation as soon as possible.  Legends in Figure 1 are match with Figure2,3,4. Table 1 shows that control chart has a smaller average run length than hotelling control chart on monitoring minimal mean shift when sample sizes N=10, forgetting factor =0.99, =100,200, 300, 370, 500. What's more, we find that control chart only has efficiency when the mean deviation gets big Figure 1 Figure 2 reflects that decreases smoothly with increase of the mean deviation regardless of . ger rather than smaller.
In contrast, this problem does not appear in control chart. Formula. (1) shows that influences of historical data on the new data will reduce when forgetting factor 0 λ gradually close to 1; In contrast, influences will increase when gradually close to 0. What are the influences of forgetting factor on the new control chart? Answers will be revealed by the simulation. The research is similar to the simulation method in section 3.1, the forgetting factor is the only difference, 0 λ =0.50, 0.80, 0.90 in turn. Table 2 show, average run length (out of control) will reduce with the decrease of when average run length (in control) and the mean deviation hold at the same value. Therefore, it indicates that the smaller forgetting factor is more sensitive to mean minimal deviation for control chart. Figure 3 reveals the above phenomenon, data from the 5th row of Table 1  Actually, Figure 1-3 also demonstrates the relationship between two types of error in all of control charts. The smaller is, the larger type 1 error is, because type 1 error and type 2 error cannot change toward the same direction at the same time, that is to say, the larger is, the smaller is; In addition, the production process and also change to the same direction simultaneously. Therefore, and change toward the same direction, the phenomenon in Figure1-3 that the whole curve strongly confirm the relationship between two types of error stay at the bottom when all average run length are 100.

Relationships between N and mean minimal deviation detection efficiency about control chart.
As previously described,  from left to right on the horizontal axis in Fig.4. nearly has no change with the N variation from 5 to 30 regardless of on table 3. It reveals that significant change of 1 ARL cannot be caused with the variation in sample sizes N. However, tends to become bigger gradually along with the increase of sample sizes N, these conclusions are depicted in Fig.4.

Conclusions
We proposed a new alternative multivariate control chart which is based on multivariate exponentially weighted moving average for monitoring process mean minimal deviation. The above study results showed: (1) control chart has a smaller average run length than Hotelling control chart on monitoring minimal mean deviation in statistical quality control. Hotelling control chart has lower efficiency when the mean deviation gets smaller rather than bigger.
(2) will reduce with the forgetting factor decreasing when and mean value deviation hold at own same value. Therefore, it indicates that the smaller forgetting factor is more sensitive to minimal mean value deviation about . (3) Significant changes in cannot be caused of the variation of sample sizes N, However, tends to become large gradually along with the increase in sample sizes N.

ARL
Meanwhile, it is worth noting that the study is based on two dimensional quality characteristics, a further study needs to be done on whether the research findings can be applied into high dimensional quality characteristics for multivariate statistical quality control. The selection of forgetting factor discussed with consumers and manufactures ought to be deter-mined to achieve optimal economic benefits if the control chart will be used in statistical quality control.