Keywords

5.1 Introduction

Control chart introduced by Walter A. Shewhart in 1920’s, is the most important Statistical Process Control (SPC) tool used to monitor reliability and performance of industrial processes. The basic purpose of implementing control chart procedures is to detect abnormal variations in the process (location & scale) parameters. Although first proposed for manufacturing industry, control charts have recently been applied in a wide variety of disciplines, such as in nuclear engineering [7], health care [16], education [15], analytical laboratories [10] etc.

Monitoring process dispersion is an important component of SPC. Dispersion control charts are a well known tool used for improving process capability and productivity by reducing variability in the process. \(R\) and \(S\) charts are the two most widely used control charts for monitoring changes in process dispersion [11]. The design of these charts is based on estimating the process standard deviation \(\sigma \) using sample range and sample standard deviation respectively. These charts perform better under the ideal assumptions but are well known to be very inefficient when the assumption of normality is violated. In this study we propose a new dispersion control chart, namely the \(D\) chart, based on Downton’s based estimate of process standard deviation. The design of \(D\) chart is established and is shown to be more efficient as compared to both \(R\) and \(S\) charts particularly for non-normal processes.

Assume \(X\) be a normally distributed quality characteristic with in-control mean \(\mu \) and standard deviation \(\sigma \) (i.e. \(X\sim N(\mu ,\sigma ^2)\)) . Let \(X_1,X_2,\ldots ,X_n\) represents a random sample of size \(n\) and the corresponding order statistics are represented by \(X_{(1)}, X_{(2)},\ldots ,X_{(n)}\). The Downton’s estimator is defined as (see [2, 5] and [1]):

$$\begin{aligned} D = \frac{2\sqrt{\pi }}{n(n-1)}\sum _{i=1}^{n}{\left[i-\frac{1}{2}(n+1)\right]}X_{(i)} \end{aligned}$$
(5.1)

For normally distributed quality characteristic, \(D\) is an unbiased estimator of \(\sigma \) [3] and it has been shown in the past that \(D\) is not much affected by non-normality. The purpose of this study is to develop a variability chart based on \(D\) that performs better than existing variability charts, such as \(R\) or \(S\) charts, under the existence and violation of normality assumption. The rest of study is organized as follows: In the next section the widely used 3-sigma and probability limit structure of \(D\) chart is established following [14] and [6]. The following section compares the performance of \(D\), \(R\) and \(S\) charts assuming normality of quality characteristics. The comparison is made using probability to signal as a performance measure. Fourth section presents comparison of these charts when the assumption of normality is violated and quality characteristic is assumed to follow non-normal (heavy tailed symmetric and skewed) distributions following [14] and [12]. Finally conclusions have been made in the last section.

5.2 Design of D Control Chart

Suppose the relationship between \(D\) and \(\sigma \) be defined by a random variable \(Z\) as \( Z = D/\sigma \) (similar to \(W = R/\sigma \) for R chart; [11]). For setting up control limits of the proposed \(D\) chart, estimates of \(\sigma \) and \(\sigma _D\) are required. By taking expectations on both sides of \(Z\), we obtain:

$$\begin{aligned} E(Z) = E(D/\sigma ) = E(D)/\sigma \end{aligned}$$
(5.2)

\(E(D)\) can be replaced with average of sample \(D^{\prime }s\) (\(\overline{D}\)), computed from an appropriate number of random samples obtained from a process during normal operating conditions (similar to \(\overline{R}\) and \(\overline{S}\) used in the construction of \(R\) and \(S\) charts). Let \(E(Z) = z_2\), as \(D\) is an unbiased estimator of \(\sigma \) hence we have \(z_2 = 1\) (for every value of \(n\)). Thus under normality, an unbiased estimator of \(\sigma \) based on Downton’s estimator is given as \(\widehat{\sigma } = \overline{D}\).

Similarly for an estimate of \(\sigma _D\) we have \(\sigma _Z = \sigma _D/\sigma \). Let \(\sigma _Z = z_3\), hence we have

$$\begin{aligned} \sigma _D =z_3 \sigma \end{aligned}$$
(5.3)

Barnett et al. [3] showed that

$$\begin{aligned} var(D) = \frac{\sigma ^2}{n(n-1)}\left\{ n\left(\frac{1}{3}\pi +2\sqrt{3}-4\right)+\left(6-4\sqrt{3}+\frac{1}{3}\pi \right)\right\} \end{aligned}$$
(5.4)

From Eqs. (5.3) and (5.4) we have

$$\begin{aligned} z_3 = \frac{1}{\sqrt{n(n-1)}}\sqrt{n\left(\frac{1}{3}\pi +2\sqrt{3}-4\right)+\left(6-4\sqrt{3}+\frac{1}{3}\pi \right)} \end{aligned}$$
(5.5)

Replacing an estimate of \(\sigma \) (i.e. \(\widehat{\sigma } = \overline{D})\) in Eq. (5.3), we obtain \(\widehat{\sigma }_D = z_3 \overline{D}\)

Hence the widely used 3-sigma control limits for the proposed \(D\) chart are defined as

$$\begin{aligned} LCL&= max(0, \overline{D} - 3 z_3\overline{D}), \quad CL = \overline{D}\quad \text{ and} \quad UCL = \overline{D} + 3 z_3\overline{D} \end{aligned}$$
(5.6)
$$\begin{aligned} LCL&= Z_3 \overline{D}, \quad CL = \overline{D}\quad \text{ and} \quad UCL = Z_4 \overline{D} \end{aligned}$$
(5.7)

where \(Z_3 = max(0, 1-3z_3)\) and \(Z_4 = 1+3 z_3\). The coefficients \(z_3, Z_3\) and \(Z_4\) entirely depends on sample size \(n\) and are given in Table 5.1 for some representative values of sample size \(n\). After setting up control limits, sample statistic \(D\) is plotted against time or sample number. If all the plotted points lie inside the control limits we can say that the process variability is in statistical control otherwise if one or more points lie outside the control limits, the process variability is said to be out-of-control.

The use of 3-sigma limits is based on the symmetric assumption of the plotted statistic, we will see that the distribution of \(D\) is not symmetric atleast for small to moderate values of \(n\). Hence there is a need to develop the probability limit structure for the proposed \(D\) chart. Probability limits for \(D\) chart can be computed by using the quantile points of the distribution of \(Z\). Let \(\alpha \) be the specified probability of making Type-I error, denoting \(\alpha \)-quantile of the distribution of \(Z\) by \(Z_\alpha \), the probability limits based on \(D\) are given as:

$$\begin{aligned} LCL&= Z_{\small {(\alpha /2})}\overline{D}\quad \quad \text{ with} \quad \text{ Pr}(Z \le Z_{(\alpha /2)})\quad = \alpha /2 \nonumber \\ UCL&= Z_{\small {(1-\alpha /2})}\overline{D}\quad \text{ with} \quad \text{ Pr}(Z \ge Z_{(1-\alpha /2)}) = 1-\alpha /2 \end{aligned}$$
(5.8)

These quantile points have been computed through extensive Monte Carlo simulation routines. The distribution of \(Z\) is obtained by generating 10,000 samples of size \(n = 2, 3, \cdots , 15, 20, 25, 35, 50, 75\) and \(100\) from standard normal distribution. For a specified Type-I error probability \(\alpha \), \((\alpha /2)^{th}\) and \(((1-\alpha )/2)^{th}\) quantile points have been computed from the distribution of \(Z\) for every combination of \(\alpha \) and \(n\). The same procedure is repeated \(1000\) times and the mean values of the quantile points together with their standard errors are reported in Table 5.1. The 3-sigma and probability limit structure of \(R\) and \(S\) charts with their respective control chart constants and quantile points can be seen in [13].

5.3 Comparison of \(D\), \(R\) and \(S\) Charts for Normal Processes

In this section we provide comparison of the \(D\), \(R\) and \(S\) Charts for normally distributed quality characteristic using probability to signal as the performance measure. For a fair comparison, different competing procedures needs to be adjusted to have the same false alarm probability and then comparison is made with respect to out-of-control detection probabilities. In our case, the process is said to be out-of-control whenever process standard deviation \(\sigma \) shifts from an in-control value, say \(\sigma _0\) to another value say \(\sigma _1\), where \(\sigma _1\) is defined as \(\sigma _1=\sigma _0 + \delta \sigma _0\). For a fixed false alarm rate, control chart structure which gives highest probability to signal for out-of-control situations will indicate best performance as compared to other charts.

By setting up probability limits for \(\alpha = 0.002\), probability to signal have been computed for both in-control and out-of-control situations for \(D\), \(R\) and \(S\) charts using their respective control chart coefficients and quantile points. To save space and to aid in visual clarity, power curves have been constructed instead of presenting results in tabular form. The power curves of the three charts for normally distributed quality characteristics for \(n = 5, 10\) and \(15\) are shown in Fig. 5.1.

Fig. 5.1
figure 1

Power curves of \(D, R\) and \(S\) charts for \(n = 5, 10\) and \(15\) under Normal distribution when \(\alpha = 0.002\)

Full size image

From power curves in Fig. 5.1 we can observe that for zero sigma shift in process standard deviation, the probability of signaling is very close to \(0.002\) for all the charts and for every sample size, representing the case for an in-control process. When the process is out-of-control, \(D\) chart is equally efficient to \(S\) chart for detecting shifts in process variability and have significantly higher probability to signal as compared to \(R\) chart, as the power curves of \(D\) chart coincides with that of \(S\) chart and always higher than the power curves of \(R\) chart for every choice of \(n\). Hence we can say that under the ideal assumption of normality \(D\) chart is more efficient than \(R\) chart and is a close competitor to \(S\) chart.

Table 5.1 Control chart constants and quantile points of the distribution of \(Z\) (standard errors)
Full size table

5.4 Comparison of \(D\), \(R\) and \(S\) Charts for Non-Normal Processes

Normal distribution have wide applications in statistics and almost all SPC charts are based on this assumption. But in practice data from many real world processes follow non-normal distributions. To mention a few of such cases: [4] and [8] pointed out that quality characteristics such as capacitance, insulation resistance, surface finish, roundness, mold dimensions follow non-normal distributions. [9] indicates that impurity levels in semiconductor process chemicals follow Gamma distribution. Many other characteristics such as straightness, flatness, cycle time are not distributed normally. Hence there is a need to study the performance of these variability charts for different parent non normal distributions. To represent the case of non-normal processes, the performance of \(D, R\) and \(S\) charts is investigated by assuming that the quality characteristic follows heavy tailed symmetric Student’s t and skewed Gamma and Weibull distributions. The density function of these non-normal distributions are given below:

Student’s t (\(t_k):\quad f(x|k) = \frac{\Gamma {[(k+1)/2]}}{\sqrt{k\pi }\Gamma {(k/2)}}\left(1+\frac{x^2}{k}\right)^{-(k+1)/2}, \quad -\infty < x< \infty ,k > 0 \)

Gamma(\(\alpha , \beta ): \quad f(x|\alpha ,\beta ) = \frac{\beta ^\alpha }{\Gamma {(\alpha )}}x^{\alpha -1}e^{-\beta x}, \quad \quad \quad \quad \quad \quad \; x > 0, \alpha > 0, \beta > 0 \)

Weibull(\(\alpha , \beta ): \quad f(x|\alpha ,\beta ) = \frac{\alpha }{\beta }x^{\alpha -1}e^{-x^\alpha /\beta }, \qquad \quad \qquad \qquad x \ge 0, \alpha > 0, \beta > 0 \)

Probability to signal of \(D, R\) and \(S\) charts have been computed for these non-normal distributions using similar simulation routines as were used earlier for the case of normal distribution. In our simulation study we used Student’s t distribution with \(k = 5\), Gamma distribution with \(\alpha = 2\) and \(\beta = 1\), and finally Weibull distribution with \(\alpha = 1.5\) and \(\beta = 1\). The power curves of the three charts when quality characteristic is assumed to follow Student’s t, Gamma and Weibull distributions are presented in Figs. 5.2–5.4 respectively. From Figs. 5.2–5.4 we can clearly see that the power curves of \(D\) chart are always higher than the power curves of both \(R\) and \(S\) charts for all non-normal cases and for every choice of sample size \(n\). This indicates that \(D\) chart has higher probability to signal shifts in process variability as compared to both \(R\) and \(S\) charts when the assumption of normality is violated. We can also observe that the difference in the detection ability of these charts increases with an increase in \(n\). Relatively \(R\) chart is extremely affected while \(D\) chart is least affected by non-normality. Hence for non-normal processes, we can easily say that \(D\) chart is always superior than both \(R\) and \(S\) charts.

Fig. 5.2
figure 2

Power curves of \(D, R\) and \(S\) charts for \(n = 5, 10\) and \(15\) under Student’s \(t\) distribution when \(\alpha = 0.002\)

Full size image
Fig. 5.3
figure 3

Power curves of \(D, R\) and \(S\) charts for \(n = 5, 10\) and \(15\) under Gamma distribution when \(\alpha = 0.002\)

Full size image
Fig. 5.4
figure 4

Power curves of \(D, R\) and \(S\) charts for \(n = 5, 10\) and \(15\) under Weibull distribution when \(\alpha = 0.002\)

Full size image

5.5 Conclusions

This study proposes an efficient control chart, namely the \(D\) chart, to monitor changes in process dispersion. The performance of the \(D\) chart is compared to the widely used \(R\) and \(S\) charts. It has been shown that for normally distributed quality characteristic, \(D\) chart is equally efficient to the \(S\) chart in terms of detecting shifts in process variability and has significantly better detection ability as compared to the \(R\) chart. For non-normal processes, \(D\) chart clearly showed superiority over both \(R\) and \(S\) charts. Quality control practitioners can now easily choose \(D\) chart as a superior alternative to both \(R\) and \(S\) charts due to its efficient detection ability.