Monitoring process mean with a new EWMA control chart

 

Um novo gráfico de controle EWMA para monitoramento da média de processo

 

 

Su-Fen Yang^I,*; Wen-Chi Tsai^II; Tzee-Ming Huang^III; Chi-Chin Yang^IV; Smiley Cheng^V

^Iyang@nccu.edu.tw, National Chengchi University, Taiwan
^IIwctsai@nccu.edu.tw, National Chengchi University, Taiwan
^IIItmhuang@nccu.edu.tw, National Chengchi University, Taiwan
^IV96354502@nccu.edu.tw, National Chengchi University, Taiwan
^Vsmiley_cheng@umanitoba.ca, University of Manitoba, Canada

 

 

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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 non-normal 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.

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RESUMO

Na prática a distribuição de probabilidade de muitas variáveis não é conhecida e sabe-se 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 observou-se 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 arco-seno da estatística não paramétrica. Os valores de ARL’s deste gráfico apresentaram melhor desempenho do que a proposta anterior. Desta forma o gráfico Arco-Seno EWMA é recomendado se o critério do ARL for empregado.

Palavras-chave: Gráfico de controle EWMA. Monitoramento de média de processo. Distribuição binomial. Transformada arco-seno.

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1. Introduction

Control charts are commonly used tools in quality improvement, such as the X-bar 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 in-control 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 in-control 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. In-control 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 in-control and out-of-control ARL_s, let p be in-control proportion, then ARL = ARL₀, the in-control ARL. If p is the out-of-control proportion, then ARL = ARL₁, the out-of-control 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 in-control process, i.e. ARL₀.

 

4. Out-of-control average run lengths of the new EWMA chart

The ARL_s of the New EWMA chart are function of (n, k,λ). Adopting in-control process proportion p = 0.613, n = 10, ARL₀ = 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 out-of-control, i.e. ARL₁, are listed in Table 3.

 

5. Example

We will use an example from Montgomery (2009) to illustrate the new chart.

The fill volume of soft-drink 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.

 

Table 2

 

Here, sample size = 10, number of samples = 15, = –0.0033, = 0.613, adopting ARL₀ = 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 in-control.

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 ARL₀s of EWMA chart with (λ = 0.2, k = 2.84) when the in-control process proportion p is between 0.25 and 0.75, but ARL₀ 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 in-control 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 in-control, 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 ARL₀s of the Arcsine EWMA chart with (λ = 0.2, k = 2.84) gave the same value 350 for all in-control values of process proportion p between 0.25 and 0.75.

Adopting in-control process proportion p = 0.613, ARL₀ = 350 with λ = 0.2 and k = 2.84, the ARL₁s 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 out-of-control are listed in Table 6.

Now the ARL₀s are all 350 but not those of the EWMA chart. However the values of ARL₁s 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 ARL₀ 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 non-normal. 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 ARL₀ 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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