Abstracts
In practice, sometimes the process data did not come from a known population distribution. So the commonly used Shewhart variables control charts are not suitable since their performance could not be properly evaluated. In this paper, we propose a new EWMA Control Chart based on a simple statistic to monitor the small mean shifts in the process with nonnormal or unknown distributions. The sampling properties of the new monitoring statistic are explored and the average run lengths of the proposed chart are examined. Furthermore, an Arcsine EWMA Chart is proposed since the average run lengths of the Arcsine EWMA Chart are more reasonable than those of the new EWMA Chart. The Arcsine EWMA Chart is recommended if we are concerned with the proper values of the average run length.
EWMA chart; Process mean; Binomial distribution; Arcsine transformation
Na prática a distribuição de probabilidade de muitas variáveis não é conhecida e sabese que não é proveniente de uma distribuição normal. Segue que o uso dos gráficos de controle Shewhart não é conveniente e daí há necessidade de procurar outros gráficos de controle alternativos. Neste artigo um novo gráfico de controle do tipo EWMA é proposto. Ele utiliza uma estatística não paramétrica para monitorar a média de um processo e observouse que é ágil para detectar pequenos desvios da média. Propriedades amostrais da estatística são exploradas e um exemplo ilustra a nova proposta. Além disto, outro gráfico do tipo EWMA é apresentado utilizando como estatística o arcoseno da estatística não paramétrica. Os valores de ARL’s deste gráfico apresentaram melhor desempenho do que a proposta anterior. Desta forma o gráfico ArcoSeno EWMA é recomendado se o critério do ARL for empregado.
Gráfico de controle EWMA; Monitoramento de média de processo; Distribuição binomial; Transformada arcoseno
Monitoring process mean with a new EWMA control chart
Um novo gráfico de controle EWMA para monitoramento da média de processo
SuFen Yang^{I,*}; WenChi Tsai^{II}; TzeeMing Huang^{III}; ChiChin Yang^{IV}; Smiley Cheng^{V}
^{I}yang@nccu.edu.tw, National Chengchi University, Taiwan
^{II}wctsai@nccu.edu.tw, National Chengchi University, Taiwan
^{III}tmhuang@nccu.edu.tw, National Chengchi University, Taiwan
^{IV}96354502@nccu.edu.tw, National Chengchi University, Taiwan
^{V}smiley_cheng@umanitoba.ca, University of Manitoba, Canada
ABSTRACT
In practice, sometimes the process data did not come from a known population distribution. So the commonly used Shewhart variables control charts are not suitable since their performance could not be properly evaluated. In this paper, we propose a new EWMA Control Chart based on a simple statistic to monitor the small mean shifts in the process with nonnormal or unknown distributions. The sampling properties of the new monitoring statistic are explored and the average run lengths of the proposed chart are examined. Furthermore, an Arcsine EWMA Chart is proposed since the average run lengths of the Arcsine EWMA Chart are more reasonable than those of the new EWMA Chart. The Arcsine EWMA Chart is recommended if we are concerned with the proper values of the average run length.
Keywords: EWMA chart. Process mean. Binomial distribution. Arcsine transformation.
RESUMO
Na prática a distribuição de probabilidade de muitas variáveis não é conhecida e sabese que não é proveniente de uma distribuição normal. Segue que o uso dos gráficos de controle Shewhart não é conveniente e daí há necessidade de procurar outros gráficos de controle alternativos. Neste artigo um novo gráfico de controle do tipo EWMA é proposto. Ele utiliza uma estatística não paramétrica para monitorar a média de um processo e observouse que é ágil para detectar pequenos desvios da média. Propriedades amostrais da estatística são exploradas e um exemplo ilustra a nova proposta. Além disto, outro gráfico do tipo EWMA é apresentado utilizando como estatística o arcoseno da estatística não paramétrica. Os valores de ARL’s deste gráfico apresentaram melhor desempenho do que a proposta anterior. Desta forma o gráfico ArcoSeno EWMA é recomendado se o critério do ARL for empregado.
Palavraschave: Gráfico de controle EWMA. Monitoramento de média de processo. Distribuição binomial. Transformada arcoseno.
1. Introduction
Control charts are commonly used tools in quality improvement, such as the Xbar and R charts, EWMA and CUSUM charts for variables data; p and C charts for attributes data. However in order to properly construct the chart, we need to know the sampling properties of the monitoring statistic, study the chart’s behavior and compare its performance with other existing charts. In most cases, normality is assumed for variables data, some with distributions of known form. Hence the question is – what if we had no knowledge of the underlying distribution or the known distribution doesn’t help us derive the necessary sampling properties? Using a nonparametric approach seems to be a good choice. Only limited researches were done in this area, like Ferrell (1953); Bakir and Reynolds Junior (1979); Amin, Reynolds and Baker (1995); Chakraborti, Van der Lann and Van de Wiel (2001); Altukife (2003a, 2003b); Bakir (2004, 2006); Chakraborti and Eryilmaz (2007); Chakraborti and Graham (2007); Chakraborti and Van der Wiel (2008) and Das and Bhattacharya (2008).
A major drawback of the Shewhart variables charts is that they are not effective in detecting small shifts, so EWMA and/or CUSUM charts are used to achieve this purpose. In this paper, we propose a new chart for variables data to monitor small shifts of the process mean.
2. The new EWMA chart
A random sample of size n, X1, X2, ..., Xn is taken from a process and let m be the process mean. Define
and
Let S be the total number of Yj> 0, then
would follow a Binomial distribution with parameters (n, p) for an incontrol process, where p = P (Yj > 0).
Note that, although the resulting chart is a np chart, this is a new chart in that the binomial variable is not the count of nonconforming units in the sample but rather the number of Xj values in a sample that are above the incontrol process mean.
To monitor the small shifts of the process mean quickly and effectively, we will apply a New EWMA chart. Hence we define the EWMA statistic as:
where S_{i} represents the value of S (according to (3)) in the i^{th} sample (or time). Adopt the starting value, EWMA_{S0}, as the mean of S; that is EWMA_{S0}= np, the mean and variance are then E(EWMA_{Si}) = np and
If time is infinite then
The control limits of the EWMA chart are often based on the asymptotic standard deviation of the control statistic. Hence we could construct the New EWMA chart as follows:
and plot EWMA_{Si}.
The two parameters, k and λ, are chosen to satisfy certain required average run length (ARL).
3. Incontrol average run lengths of the new EWMA chart
We will use the ARL as a measure of performance of the proposed chart. Following Lucas and Saccucci (1990), the ARL_{s} of the New EWMA chart are evaluated by Markov chain approach. The procedure to calculate ARL is described below.
Step 1. Divide the interval (LCL, UCL) into (N – 1) subintervals each with an equal width.
Step 2. Denote the 1^{st} interval (or State) as (∞, LCL]; the (N + 1)^{th} interval (State) as (UCL, ∞); the (N – 1) subintervals of the interval (LCL, UCL) are denoted as State 2, State 3, …, State N.
Step 3. Since State 1 and (N + 1) are the action regions of the New EWMA chart, it is considered the absorption states for the Markov chain. States 2, 3, …, and N are transient states.
Step 4. Let P_{ij} be the transition probability that the statistic EWMA_{St} reaches State j at time t, given that EWMA_{St–1} was in State i at time (t – 1), i, j = 2,…, N.
Step 5. Let b be a (N – 1)vector of the initial probabilities that the process started in State 2, …, N. We have in our study that
Step 6. Let P = P_{ij} be a (N – 1) × (N – 1) transition probability matrix, i, j = 2,…,N. The ARL is thus computed by
ARL = b´(I  P)–11,
where 1´ = (1, 1, 1, ..., 1)is a (N – 1)vector with element 1 in State 2, 3,…, N.
Step 7. To calculate the incontrol and outofcontrol ARL_{s}, let p be incontrol proportion, then ARL = ARL_{0}, the incontrol ARL. If p is the outofcontrol proportion, then ARL = ARL_{1}, the outofcontrol ARL.
Table 1 and 2 list the ARL_{s} under various combinations of (n, p), for n ranging from 9 to 20 and for a number of values of p between 0.25 and 0.75, by adopting the respective combination (λ0.2, k = 2.84) and (λ = 0.05, k = 2.84) in the New EWMA chart for the incontrol process, i.e. ARL_{0}.
4. Outofcontrol average run lengths of the new EWMA chart
The ARL_{s} of the New EWMA chart are function of (n, k,λ). Adopting incontrol process proportion p = 0.613, n = 10, ARL_{0} = 374.0 with λ = 0.2 and k = 2.84, the ARL_{s} of the EWMA chart for n ranging from 9 to 20 and the values of p between 0.25 and 0.95 when the process is outofcontrol, i.e. ARL_{1}, are listed in Table 3.
5. Example
We will use an example from Montgomery (2009) to illustrate the new chart.
The fill volume of softdrink beverage bottles is an important quality characteristic. The volume is measured (approximately) by placing a gauge over the crown and comparing the height of the liquid in the neck of the bottle against a coded scale. On this scale, a reading of zero corresponds to the correct fill height. Fifteen samples of size n = 10 have been analyzed, and the fill heights are shown in Table 4.
Here, sample size = 10, number of samples = 15, = –0.0033, = 0.613, adopting ARL_{0} = 374.0 with λ = 0.2 and k = 2.84 based on Table 1. S = Sum of positive differences (X_{j} + 0.0033), i = 1, 2, ..., 15.
The New EWMA chart is plotted in Figure 1.
The chart shows that the process seems to be incontrol.
The chart (MONTGOMERY, 2009) shown in Figure 2 gave the same conclusion.
6. The ARL of the EWMA_{Y} chart
Table 1 shows that the ARL_{0}s of EWMA chart with (λ = 0.2, k = 2.84) when the incontrol process proportion p is between 0.25 and 0.75, but ARL_{0} values vary irregularly. The reason for this is that the binomial distribution is asymmetric when p ≠ 0.5 for small and moderate values of n. To rectify this problem, we would apply the arcsine transformation (MOSTELLER, YOUTZ, 1961). That is, let and then the distribution of Y would be approximately normal with mean and variance 1/(4n). A revised EWMA_{Y} chart, the Arcsine EWMA chart is thus constructed as follows:
Let the mean of Y be the starting value, EWMA_{Y0}; i.e. EWMA_{Y0} for incontrol process. The mean and variance of EWMA_{Yi} are E (EWMA_{Yi})= and If time gets sufficiently large then
The control limits and the center line of the Arcsine EWMA_{Y} chart are:
and plot EWMA_{Yi}.
Let us use the same example in section 5 and plot the Arcsine EWMA_{Y} chart in Figure 3.
This chart also showed that the process is incontrol, just as the EWMA_{S} chart.
To evaluate the performance of this new chart, we calculated the chart’s ARL_{s}. Table 5 shows that the ARL_{0}s of the Arcsine EWMA chart with (λ = 0.2, k = 2.84) gave the same value 350 for all incontrol values of process proportion p between 0.25 and 0.75.
Adopting incontrol process proportion p = 0.613, ARL_{0} = 350 with λ = 0.2 and k = 2.84, the ARL_{1}s of the EWMA_{Y} chart for n from 9 to 20 and the values of p between 0.25 to 0.95 when the process is outofcontrol are listed in Table 6.
Now the ARL_{0}s are all 350 but not those of the EWMA chart. However the values of ARL_{1}s are smaller for EWMA chart when the values of p are small, but it is reversed for large values of p. Overall we feel that the detection ability of the Arcsine EWMA_{Y} chart is better than that of the EWMA chart. Hence, we would recommend the Arcsine EWMA_{Y} chart if we were concerned with the proper ARL_{0} values.
7. Conclusion
A new chart, the EWMA chart, to monitor the process mean for variables data is proposed. It provides an alternative when the underlying distribution is unknown or nonnormal. We have shown that it performs quite well. However, we would recommend a modified version, the Arcsine EWMA_{Y} chart if we are concerned with attaining the proper ARL_{0} values.
Acknowledgements
This research was supported by Mathematics Research Promotion Center, National Science Council, Taiwan; Center for Service Innovation, National Chengchi University, Taiwan; Commercial College, National Chengchi University, Taiwan and Quality Control Research and Applications Group, Department of Statistics, University of Manitoba, Canada.
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Publication Dates

Publication in this collection
27 May 2011 
Date of issue
June 2011