Control theory for skewed distribution under operation side of the telecommunication industry and hard-bake process in the semiconductor manufacturing process

Abstract

Statistical process control basically involves inspecting a random sample of the output from a process and deciding whether the process is producing products with characteristics that fall within a predetermined range. It is used extensively in the field of reliability engineering. The reliability of the production process is thoroughly monitored for any internal variation using the SPC. The aim is always to settle such variations through a proper control monitoring. If the underlying distribution of the process is known to the researcher than the use of parametric control charts are useful but in many cases when there is doubt about the distribution of the process then it is preferred to use non parametric control charts. In this paper we propose the modified Exponentially weighted moving average (EWMA), Double Exponentially weighted moving average (DEWMA), Hybrid Exponentially weighted moving average (HEWMA), Extended Exponentially weighted moving average (EEWMA), Modified Exponentially weighted moving average (MEWMA) and mix- type control charts by mixing these control charts with Tukey control chart EWMA-TCC, DEWMA-TCC, HEWMA-TCC, EEWMA-TCC, MEWMA-TCC for the shape parameter of the Kumaraswamy Lehmann-2 Power function distribution (KL2PFD).

Introduction

Statistical quality control is an operative instrument to uphold the constancy of any manufacturing process such as the process to monitor the number of errors produced by any newly heaved software by a software manufacturing company or the reliability of any production process of the filling height of soft drink beverages. The reliability of such production processes is thoroughly monitored for any internal variation. The aim is to settle such variations through proper control monitoring. If the underlying distribution of the process is known to the researcher than the use of parametric control charts is useful but, in many cases, when there is doubt about the distribution of the process then it is advised to use non parametric control charts.

Many authors have proposed parametric control charts such as Shewhart control chart by Shewhart, ¹ EWMA control chart by Roberts, ² HEWMA control chart by Shamma and Shamma, ³ DEWMA control chart by Adeoti, ⁴ EEWMA control chart by Naveed et al. ⁵ Also the work on Nonparametric control chart are constructed to monitor a distribution free process such as Tukey Control chart (TCC) by Alemi. ⁶ The Tukey control chart (TCC) uses the principal of box plot and is efficient to monitor the process with less number of observations as compare to other control charts. Many authors further studied the TCC for example Borckardt et al. ⁷ and Torng and Lee ⁸ studied the performance of the chart for serially dependent variables. Many other authors also studied the TCC such as Lee ⁹ ; Lee et al. ¹⁰ ; Lee and Torng ¹¹ ; Tercero-Gomez et al. ¹² ; Torng. ¹³

The studies based on the mixture of TCC with some parametric control charts includes Khaliq et al., ¹⁴ Khaliq et al. ¹⁵ and Khaliq and Riaz ¹⁶ who studied the EWMA-TCC and CUSUM-TCC charts to monitor the mean of the process and showed their performance over simple control charts for symmetrical distributions. Similarly, Abu-Shawiesh, ¹⁷ Raza, ¹⁸ and Riaz, ¹⁹ Taboran ²⁰ proposed some mixed control charts with TCC and proved their efficiency for process mean. Phantu and Sukparungsee ²¹ also suggested mixing of doubly EWMA and TCC and showed its efficiency over EWMA–TCC. Khaliq ²² developed median based EWMA-TCC and concluded that it’s better to be used under rational subgrouping (Zaka et al. 2022). ²³

Although the skewed distribution occurs more often in different real-life scenarios but very less work has been discussed for such situations when the process distribution does not follow the normality but a skewed distribution. Interested readers may read Noorossana et al., ²⁴ Lin et al., ²⁵ Erto et al., ²⁶ Liang et al. ²⁷ and Khan et al. ²⁸ Also, the control monitoring for the shape parameter of Power function distribution, weighted power function distribution was done by Zaka et al., ²⁹ Zaka et al. ³⁰ The KL2PFD introduced by Zaka et al. ²⁹ is more flexible to use for the positively skewed data than Kumaraswamy distribution and Power function distribution. By considering this advantage of Kumaraswamy distribution, we will discuss the importance of the shape parameter distribution and its application in reliability engineering problems.

The shape parameter is a kind of numerical parameter, which directly affects the shape of a distribution rather than simply shifting or stretching it. So, the importance of the shape parameter cannot be ignored. Several authors have proposed the monitoring of the shape parameter for the skewed distributions such as Zaka et al.^29,30–32 and Zaka and Akhter. ³³

After reading the above literature, we see that the problem about the construction of proper control monitoring is unaddressed for the KL2PFD which has application in almost all fields of life. Also, the experts of statistical quality control provided either parametric control charts such as EWMA, DEWMA, HEWMA, EEWMA or non-parametric control charts are proposed but the mixture of these two types of control charts is rarely available in literature.

To overcome the gap discussed above and took motivation from the literature, we propose the modified EWMA, DEWMA, HEWMA, EEWMA, MEWMA and mix-type control charts by mixing these control charts with Tukey control chart (EWMA-TCC, DEWMA-TCC, HEWMA-TCC, EEWMA-TCC, MEWMA-TCC) for the shape parameter of the KL2PFD. In this paper we have not only modified the existing control charts for the shape parameter of KL2PFD but also proposed the mix-type control charts by mixing these control charts with Tukey control chart (EWMA-TCC, DEWMA-TCC, HEWMA-TCC, EEWMA-TCC, MEWMA-TCC) for the shape parameter of the KL2PFD.

The uniqueness of this paper comes from the fact that the construction and comparison of control charts for the shape parameter of KL2PFD has not been discussed so far in literature. In this paper, a brief discussion on the shape parameter of KL2PFD is presented in Section 2. The proposed parametric control charts for the shape parameter of KL2PFD (Section 2) is given in Section 3. The Mix-type control charts for the shape parameter of KL2PFD (Section 2) are proposed in Section 4.

Shape parameter of KL2PFD

Zaka et al. ²⁹ proposed the KL2PFD as a more consistent probability distribution which is easily used to model different real-life situations. The probability distribution function and cumulative distribution function is given by

$$\begin{array}{c}\multicolumn{1}{c}{f\left(x\right)=\alpha \phi \theta {\{1-{\{1-{\left(\frac{x}{\beta }\right)}^{\gamma }\}}^{\theta }\}}^{\alpha -1}{\{1-{\left(\frac{x}{\beta }\right)}^{\gamma }\}}^{\theta -1}}\\ \multicolumn{1}{c}{{\{1-{\{1-{\{1-{\left(\frac{x}{\beta }\right)}^{\gamma }\}}^{\theta }\}}^{\alpha }\}}^{\phi -1}\frac{\gamma x^{\gamma -1}}{{\beta }^{\gamma }},0}\\ \multicolumn{1}{c}{<x<\beta ,}\end{array}$$

and

$$F\left(x\right)=1-{\{1-{\{1-{\{1-{\left(\frac{x}{\beta }\right)}^{\gamma }\}}^{\theta }\}}^{\alpha }\}}^{\phi },0<x<\beta .$$

The percentile estimator (PE) for the shape parameter of KL2PFD is also discussed in Zaka ³⁴ is given by

$$\widehat{\gamma }=\frac{ln\frac{{(1-{(1-(H))}^{1/\phi })}^{1/\alpha })}{{(1-{(1-(H))}^{1/\phi })}^{1/\alpha })}}{ln\left(\frac{P_H}{P_L}\right)}.$$

Some proposed parametric control charts for the shape parameter of KL2PFD

Modified EWMA control chart for shape parameter of KL2PFD

The EWMA control chart for the shape parameter of power function distribution was first proposed by Jabeen and Zaka ³⁵ . We have proposed modified EWMA for the shape parameter of KL2PFD. The statistic for EWMA is given as

$${\mathrm{W}}_t=\mathrm{\lambda }{\hat{\mathrm{\gamma }}}_t+(1-\mathrm{\lambda })W_{t-1}.$$

and the control limits are given as

$$\mathrm{UC}{L}_{{\mathrm{W}}_t}=\gamma +L*\sqrt{V\left(W_t\right)},$$

$$\mathrm{LC}{L}_{{\mathrm{W}}_t}=\gamma -L*\sqrt{V\left(W_t\right)},$$

where “L” is chosen in such a way that the ARL₀ of EWMA control chart reaches a pre-specified level.

Modified DEWMA control chart for the shape parameter of KL2PFD

The usual DEWMA control chart was discovered by Shamma and Shamma ³ to control the average of any process. We modify Shamma and Shamma ³³ to study the performance of DEWMA control chart for the shape parameter of the KL2PFD

$$Z_t=(1-\lambda )Z_{t-1}+\lambda W_t,$$

where

$$W_t=(1-\lambda )W_{t-1}+\lambda {\hat{\gamma }}_t.$$

The control limits (Now, we can introduce the control limits in the following form)

$$\mathrm{LC}{L}_{Z_t}=\gamma -L\sqrt{\mathrm{var}\left(\hat{\gamma }\right)\left\{\frac{(2-2\lambda +{\lambda }^2)}{{(2-\lambda )}^3}\right\}},$$

$$C{L}_{Z_t}=\gamma ,$$

$$\mathrm{UC}{L}_{Z_t}=\gamma +L\sqrt{\mathrm{var}\left(\hat{\gamma }\right)\left\{\frac{(2-2\lambda +{\lambda }^2)}{{(2-\lambda )}^3}\right\}}.$$

Where L is chosen in such a way that the ARL₀ of DEWMA control chart reaches a pre-specified level.

Modified HEWMA control chart for shape parameter of KL2PFD

The HEWMA control chart to study the shape of any distribution was discussed by Zaka. ³⁰ The modified HEWMA for the shape parameter of KL2PFD is given below

$$\mathrm{HE}{\mathrm{W}}_t={\mathrm{\lambda }}_1{\mathrm{W}}_t+(1-{\mathrm{\lambda }}_1)\mathrm{HE}{\mathrm{W}}_{t-1},$$

where ${\mathrm{W}}_t={\mathrm{\lambda }}_2{\hat{\mathrm{\gamma }}}_t+(1-{\mathrm{\lambda }}_2)W_{t-1}.$

The control limits are given by

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{LC}{L}_{\mathrm{PE}{W}_t}}\\ \multicolumn{1}{c}{=\gamma }\\ \multicolumn{1}{c}{-L\frac{{\mathrm{\lambda }}_1{\mathrm{\lambda }}_2}{({\mathrm{\lambda }}_1-{\mathrm{\lambda }}_2)}\sqrt{\mathrm{var}\left(\hat{\gamma }\right)(\sum _{\mathrm{i}=1}^2\frac{{(1-{\mathrm{\lambda }}_{\mathrm{i}})}^2(1-{(1-{\mathrm{\lambda }}_{\mathrm{i}})}^{2\mathrm{t}})}{1-{(1-{\mathrm{\lambda }}_{\mathrm{i}})}^2}-\frac{2(1-{\mathrm{\lambda }}_1)(1-{\mathrm{\lambda }}_2)\{1-{(1-{\mathrm{\lambda }}_1)}^{\mathrm{t}}{(1-{\mathrm{\lambda }}_2)}^{\mathrm{t}}\}}{1-(1-{\mathrm{\lambda }}_1)(1-{\mathrm{\lambda }}_2)})}}\end{array}$$

$$C{L}_{\mathrm{PE}{W}_t}=\gamma$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{UC}{L}_{\mathrm{PE}{W}_t}=}\\ \multicolumn{1}{c}{\gamma +L\frac{{\mathrm{\lambda }}_1{\mathrm{\lambda }}_2}{({\mathrm{\lambda }}_1-{\mathrm{\lambda }}_2)}\sqrt{\mathrm{var}\left(\hat{\gamma }\right)(\sum _{\mathrm{i}=1}^2\frac{{(1-{\mathrm{\lambda }}_{\mathrm{i}})}^2(1-{(1-{\mathrm{\lambda }}_{\mathrm{i}})}^{2\mathrm{t}})}{1-{(1-{\mathrm{\lambda }}_{\mathrm{i}})}^2}-\frac{2(1-{\mathrm{\lambda }}_1)(1-{\mathrm{\lambda }}_2)\{1-{(1-{\mathrm{\lambda }}_1)}^{\mathrm{t}}{(1-{\mathrm{\lambda }}_2)}^{\mathrm{t}}\}}{1-(1-{\mathrm{\lambda }}_1)(1-{\mathrm{\lambda }}_2)})},}\end{array}$$

where the value of L is chosen in a manner that ARL₀ of HEWMA control chart reaches a pre-specified level.

Modified MEWMA control chart for the shape parameter of KL2PFD

The MEWMA was proposed by Khan. ²⁸ The modified MEWMA is given by

$$M_t=(1-\lambda )M_{t-1}+\lambda X_t+K(X_t-X_{t-1}).$$

We have proposed the modification to Khan et al. ²⁸ using the shape parameter of the underlying probability distribution of process

$$M_t=(1-\lambda )M_{t-1}+\lambda {\hat{\gamma }}_t+K({\hat{\gamma }}_t-{\hat{\gamma }}_{t-1}),$$

$$M_t=(\lambda +K){\hat{\gamma }}_t-K{\hat{\gamma }}_{t-1}+(1-\lambda )[(\lambda +K){\hat{\gamma }}_{t-1}-K{\hat{\gamma }}_{t-2}+(1-\lambda )M_{t-2}],$$

$$M_t=(\lambda +K)[{\hat{\gamma }}_t+(1-\lambda ){\hat{\gamma }}_{t-1}]-K[{\hat{\gamma }}_{t-1}+(1-\lambda ){\hat{\gamma }}_{t-2}]+(1-\lambda )M_{t-2,}$$

$$\begin{array}{c}\multicolumn{1}{c}{M_t=(\lambda +K)[{\hat{\gamma }}_t+(1-\lambda ){\hat{\gamma }}_{t-1}]-K[{\hat{\gamma }}_{t-1}+(1-\lambda ){\hat{\gamma }}_{t-2}]}\\ \multicolumn{1}{c}{+(1-\lambda {)}^2[(\lambda +K){\hat{\gamma }}_{t-2}-K{\hat{\gamma }}_{t-3}+(1-\lambda )M_{t-3}]}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{M_t=(\lambda +K)[{\hat{\gamma }}_t+(1-\lambda ){\hat{\gamma }}_{t-1}+{(1-\lambda )}^2{\hat{\gamma }}_{t-2}]-K[{\hat{\gamma }}_{t-1}+(1-\lambda ){\hat{\gamma }}_{t-2}+{(1-\lambda )}^2{\hat{\gamma }}_{t-3}]}\\ \multicolumn{1}{c}{+(1-\lambda {)}^3{M}_{t-3}}\end{array}.$$

On generalizing, we get

$$M_t=(\lambda +K)\sum _{j=1}^t{(1-\lambda )}^{t-j}{\hat{\gamma }}_j-K\sum _{j=0}^{t-1}{(1-\lambda )}^{t-j-1}{\hat{\gamma }}_j+{(1-\lambda )}^t{M}_0,$$

$$M_t=(\lambda +K){\sum }_{j=1}^t{(1-\lambda )}^{t-j}{\hat{\gamma }}_j-K{\sum }_{j=0}^{t-1}{(1-\lambda )}^{t-j-1}{\hat{\gamma }}_j-K{(1-\lambda )}^{t-1}{\hat{\gamma }}_0+{(1-\lambda )}^t{M}_0,$$

$$\begin{array}{c}\multicolumn{1}{c}{M_t=(\lambda +K)\sum _{j=1}^t{(1-\lambda )}^{t-j}{\hat{\gamma }}_j-K\sum _{j=0}^{t-1}{(1-\lambda )}^{t-j-1}{\hat{\gamma }}_j+(1-\lambda -K){(1-\lambda )}^{t-1}{\hat{\gamma }}_0}\end{array},$$

$$\begin{array}{c}\multicolumn{1}{c}{E\left(M_t\right)=[(\lambda +K)\sum _{j=1}^{t-1}{(1-\lambda )}^j-K\sum _{j=0}^{t-2}{(1-\lambda )}^j{\hat{\gamma }}_j+(1-\lambda -K){(1-\lambda )}^{t-1}]\gamma .}\end{array}$$

The variance is given as $\mathrm{var}\left({\hat{\gamma }}_j\right)=v$

$$\mathrm{Var}\left(M_t\right)=[{(\lambda +K)}^2+{\lambda }^2{(1-\lambda -K)}^2\sum _{j=1}^{t-1}{(1-\lambda )}^{2(t-j-1)}]v,$$

$$\mathrm{Var}\left(M_t\right)=[{(\lambda +K)}^2+{\lambda }^2{(1-\lambda -K)}^2\sum _{j=0}^{t-2}{(1-\lambda )}^{2j}]v,$$

$$\mathrm{Var}\left(M_t\right)=[{(\lambda +K)}^2+{\lambda }^2{(1-\lambda -K)}^2\frac{{(1-\lambda )}^{2(t-1)}}{\lambda (2-\lambda )}]v,$$

$$\mathrm{Var}\left(M_t\right)=\left[\frac{(\lambda +2\lambda K+2K^2)-\lambda {(1-\lambda -K)}^2{(1-\lambda )}^{2(t-1)}}{(2-\lambda )}\right]v.$$

The control limits are given as

$$\mathrm{LC}{L}_{M_t}=\gamma -L\sqrt{\mathrm{var}\left(\hat{\gamma }\right)\left\{\frac{(\lambda +2\lambda k+k^2)}{(2-\lambda )}\right\},}$$

$$C{L}_{M_t}=\gamma$$

$$\mathrm{UC}{L}_{M_t}=\gamma +L\sqrt{\mathrm{var}\left(\hat{\gamma }\right)\left\{\frac{(\lambda +2\lambda k+k^2)}{(2-\lambda )}\right\}.}$$

Where the value of L is calculated in such a way that the ARL₀ of HEWMA control chart reaches a pre-specified level.

Modified EEWMA control chart for shape parameter of KL2PFD

The traditional EEWMA control chart was proposed by Naveed et al. ⁵ The EEWMA was first modified by Jabeen and Zaka ³⁵ using the shape parameter of some skewed distribution. We have modified it for the shape parameter of KL2PFD as:

$${\mathrm{E}}_t={\lambda }_1{\hat{\mathrm{\gamma }}}_{(\mathrm{t})}-{\lambda }_2{\hat{\mathrm{\gamma }}}_{(\mathrm{t}-1)}+(1-{\lambda }_1+{\lambda }_2){\mathrm{E}}_{t-1}$$

The control limits are given as

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{UC}{L}_{{\mathrm{E}}_t}}\\ \multicolumn{1}{c}{=\gamma +L\sqrt{\mathrm{Var}(\gamma )\{({{\lambda }_1}^2+{{\lambda }_2}^2)\left(\frac{1-{(1-{\lambda }_1+{\lambda }_2)}^{2t}}{1-{(1-{\lambda }_1+{\lambda }_2)}^2}\right)-2a{\lambda }_1{\lambda }_2\left(\frac{1-{(1-{\lambda }_1+{\lambda }_2)}^{2t-2}}{1-{(1-{\lambda }_1+{\lambda }_2)}^2}\right)\}}}\end{array}$$

$$C{L}_{{\mathrm{E}}_t}=\gamma$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{LC}{L}_{{\mathrm{E}}_t}}\\ \multicolumn{1}{c}{=\gamma -L\sqrt{\mathrm{Var}(\gamma )\{({{\lambda }_1}^2+{{\lambda }_2}^2)\left(\frac{1-{(1-{\lambda }_1+{\lambda }_2)}^{2t}}{1-{(1-{\lambda }_1+{\lambda }_2)}^2}\right)-2a{\lambda }_1{\lambda }_2\left(\frac{1-{(1-{\lambda }_1+{\lambda }_2)}^{2t-2}}{1-{(1-{\lambda }_1+{\lambda }_2)}^2}\right)\}}}\end{array}$$

Modified DMEWMA control chart for shape parameter of KL2PFD

The DEWMA as a mixture of two control charts is first proposed by Alevizakosa et al. ³⁶ We have modified Alevizakosa et al. ³⁶ for the shape parameter of any probability distribution as given by

$$M_t=(1-{\lambda }_1)M_{t-1}+{\lambda }_1{\hat{\gamma }}_t+K_1({\hat{\gamma }}_t-{\hat{\gamma }}_{t-1})$$

$$D{M}_t=(1-{\lambda }_2)D{M}_{t-1}+{\lambda }_2{\hat{\gamma }}_t+K_2({\hat{\gamma }}_t-{\hat{\gamma }}_{t-1})$$

Where $0<{\lambda }_1,{\lambda }_2\le 1$ arethe smoothing parameters, $K_1\mathrm{and}{K}_2$ are additional parameters and ${\hat{\gamma }}_0=M_0=D{M}_0=\gamma$ are starting values. Now by assuming ${\lambda }_1={\lambda }_2=\lambda ,K_1=K_2=K$ .

$$D{M}_t=(\lambda +K)M_t-\lambda (\lambda +K-1)\sum _{j=1}^{t-j}(1-\lambda {)}^{t-j-1}{M}_j-(\lambda +K-1)(1-\lambda {)}^{t-1}\gamma$$

$$\begin{array}{c}\multicolumn{1}{c}{D{M}_t={(\lambda +K)}^2{\hat{\gamma }}_j-2\lambda (\lambda +K)(\lambda +K-1)\sum _{j=1}^{t-j-1}(1-\lambda {)}^{t-j-1}{\hat{\gamma }}_j+{\lambda }^2{(\lambda +K-1)}^2\sum _{j=1}^{t-j}(t-j-1)(1-\lambda {)}^{t-j-2}{\hat{\gamma }}_j}\\ \multicolumn{1}{c}{+(\lambda +K-1)(1-\lambda {)}^{t-2}[K(\lambda t-1)-(1-\lambda )(1+\lambda t)]\gamma .}\end{array}$$

In the same way following Alevizakosa et al., ³⁶ we find variance as

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{var}(D{M}_t)}{v}={(\lambda +K)}^4+4{\lambda }^2{(\lambda +K)}^2{(\lambda +K-1)}^2}\\ \multicolumn{1}{c}{+4{\lambda }^2{(\lambda +K)}^2{(\lambda +K-1)}^2(1-\lambda {)}^2\sum _{l=1}^{t-2}(1-\lambda {)}^{l-1}}\\ \multicolumn{1}{c}{-4{\lambda }^3(\lambda +K)(1-\lambda ){(\lambda +K-1)}^3\sum _{l=1}^{t-2}l{(1-\lambda )}^{l-1}}\\ \multicolumn{1}{c}{+{\lambda }^4{(\lambda +K-1)}^4\sum _{l=1}^{t-2}{l}^2{(1-\lambda )}^{l-1}}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{var}(D{M}_t)}{v}}\\ \multicolumn{1}{c}{={(\lambda +K)}^4+4{\lambda }^2{(\lambda +K)}^2{(\lambda +K-1)}^2}\\ \multicolumn{1}{c}{+4{\lambda }^2{(\lambda +K)}^2{(\lambda +K-1)}^2(1-\lambda {)}^2\frac{1-{(1-\lambda )}^{t-2}}{\lambda }}\\ \multicolumn{1}{c}{-4{\lambda }^3(\lambda +K)(1-\lambda ){(\lambda +K-1)}^3\left(\frac{1-{(1-\lambda )}^{t-1}}{{\lambda }^2}\right)-\left(\frac{(t-1){(1-\lambda )}^{t-2}}{\lambda }\right)}\\ \multicolumn{1}{c}{+{\lambda }^4{(\lambda +K-1)}^4\frac{1+(1-\lambda )-{(t-1)}^2{(1-\lambda )}^{t-2}+(2t^2-6t+3){(1-\lambda )}^{t-1}-{(t-2)}^2{(1-\lambda )}^t}{{\lambda }^3}}\end{array}$$

The control limits are:

$$\mathrm{LC}{L}_{D{M}_t}=\gamma -L*\sqrt{\mathrm{var}\left(D{M}_t\right)}$$

$$C{L}_{D{M}_t}=\gamma$$

$$\mathrm{UC}{L}_{D{M}_t}=\gamma +L*\sqrt{\mathrm{var}\left(D{M}_t\right)}$$

Construction of some mix-type control charts for the shape parameter of KL2PFD

In this section, we discuss the construction of some mix-type control chart to control the shape of any underlying process.

EWMA-Tukey Control Chart for the shape parameter

The control limits for the mixture of two control charts EWMA for the shape parameter of Power function distribution proposed by Zaka et al. ³⁰ and Tukey control chart proposed to monitor non normal distribution is given as

$$\mathrm{LC}{L}_{E-T}=Q_1-K(\mathrm{IQR})\sqrt{\frac{\lambda }{2-\lambda }}$$

$$C{L}_{E-T}=Q_2$$

$$\mathrm{UC}{L}_{E-T}=Q_3+K(\mathrm{IQR})\sqrt{\frac{\lambda }{2-\lambda }}.$$

Where $Q_1\mathrm{and}{Q}_3$ are the lower and upper quartiles of any data set, $K=\frac{\lambda }{2}$ and $\mathrm{IQR}=Q_3-Q_1$ is Inter quartile range.

DEWMA-Tukey Control Chart for the shape parameter

The control limits using DEWMA-Tukey Control Chart for the shape parameter is given as

$$\mathrm{LC}{L}_{D-T}=Q_1-L\left(\mathrm{IQR}\right)\sqrt{\frac{(2-2\lambda +{\lambda }^2)}{{(2-\lambda )}^3}}$$

$$C{L}_{D-T}=Q_2$$

$$\mathrm{UC}{L}_{D-T}=Q_3+L(\mathrm{IQR})\sqrt{\frac{(2-2\lambda +{\lambda }^2)}{{(2-\lambda )}^3}}.$$

Where $Q_1,Q_3$ are lower and upper quartiles of any data set, $K=\frac{\lambda }{2}$ and $\mathrm{IQR}=Q_3-Q_1$ is inter quartile range.

Construction of HEWMA-TCC Control chart for the shape parameter of KL2PFD

The control limits for HEWMA-TCC for the shape parameter of KL2PFD is given below

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{LC}{L}_{H-T}}\\ \multicolumn{1}{c}{=Q_1}\\ \multicolumn{1}{c}{-K(\mathrm{IQR})\frac{{\mathrm{\lambda }}_1{\mathrm{\lambda }}_2}{({\mathrm{\lambda }}_1-{\mathrm{\lambda }}_2)}\sqrt{(\sum _{\mathrm{i}=1}^2\frac{{(1-{\mathrm{\lambda }}_{\mathrm{i}})}^2(1-{(1-{\mathrm{\lambda }}_{\mathrm{i}})}^{2\mathrm{t}})}{1-{(1-{\mathrm{\lambda }}_{\mathrm{i}})}^2}-\frac{2(1-{\mathrm{\lambda }}_1)(1-{\mathrm{\lambda }}_2)\{1-{(1-{\mathrm{\lambda }}_1)}^{\mathrm{t}}{(1-{\mathrm{\lambda }}_2)}^{\mathrm{t}}\}}{1-(1-{\mathrm{\lambda }}_1)(1-{\mathrm{\lambda }}_2)})}}\end{array}$$

$$C{L}_{H-T}=Q_2$$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{UC}{L}_{H-T}=}\\ \multicolumn{1}{c}{Q_3+K(\mathrm{IQR})\frac{{\mathrm{\lambda }}_1{\mathrm{\lambda }}_2}{({\mathrm{\lambda }}_1-{\mathrm{\lambda }}_2)}\sqrt{(\sum _{\mathrm{i}=1}^2\frac{{(1-{\mathrm{\lambda }}_{\mathrm{i}})}^2(1-{(1-{\mathrm{\lambda }}_{\mathrm{i}})}^{2\mathrm{t}})}{1-{(1-{\mathrm{\lambda }}_{\mathrm{i}})}^2}-\frac{2(1-{\mathrm{\lambda }}_1)(1-{\mathrm{\lambda }}_2)\{1-{(1-{\mathrm{\lambda }}_1)}^{\mathrm{t}}{(1-{\mathrm{\lambda }}_2)}^{\mathrm{t}}\}}{1-(1-{\mathrm{\lambda }}_1)(1-{\mathrm{\lambda }}_2)})}.}\end{array}$$

Where $Q_1,Q_3$ are lower and upper quartiles of any data set, $K=\frac{\lambda }{2}$ and $\mathrm{IQR}=Q_3-Q_1$ is Inter quartile range.

MEWMA-Tukey Control Chart for the shape parameter

We modified the MEWMA proposed by Khan et al. ²⁸ for the shape parameter of any distribution and Tukey control chart (TCC) proposed to monitor non normal distribution is given as

$$\mathrm{LC}{L}_{M-T}=Q_1-K(\mathrm{IQR})\sqrt{\frac{(\lambda +2\lambda k+k^2)}{(2-\lambda )}}$$

$$C{L}_{M-T}=Q_2$$

$$\mathrm{UC}{L}_{M-T}=Q_3+K(\mathrm{IQR})\sqrt{\frac{(\lambda +2\lambda k+k^2)}{(2-\lambda )}}.$$

Where $Q_1,Q_3$ are lower and upper quartiles of any data set, $K=\frac{\lambda }{2}$ and $\mathrm{IQR}=Q_3-Q_1$ is Inter quartile range.

DMEWMA-Tukey Control Chart for the shape parameter

We modified the DMEWMA by Alevizakosa et al. ³⁶ for the shape parameter of any distribution and Tukey control chart (TCC) proposed to monitor non normal distribution is given as

$$\mathrm{LC}{L}_{\mathrm{DM}-T}=Q_1-K(\mathrm{IQR})\sqrt{\mathrm{var}\left(D{M}_t\right)}$$

$$C{L}_{\mathrm{DM}-T}=Q_2$$

$$\mathrm{UC}{L}_{\mathrm{DM}-T}=Q_3+K(\mathrm{IQR})\sqrt{\mathrm{var}\left(D{M}_t\right)}.$$

Where $Q_1,Q_3$ are lower and upper quartiles of any data set, $K=\frac{\lambda }{2}$ and $\mathrm{IQR}=Q_3-Q_1$ is inter quartile range.

Construction of EEWMA-TCC Control Charts

We have mixed the EEWMA control chart proposed by Zaka et al. ³⁰ and Tukey control chart to produce a flexible control chart that can be used for the early detection of any minor change in the process. The control limits are given as

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{LC}{L}_{\mathrm{EE}-T}}\\ \multicolumn{1}{c}{=Q_1}\\ \multicolumn{1}{c}{-K(\mathrm{IQR})\sqrt{\{({{\lambda }_1}^2+{{\lambda }_2}^2)\left(\frac{1-{(1-{\lambda }_1+{\lambda }_2)}^{2t}}{1-{(1-{\lambda }_1+{\lambda }_2)}^2}\right)-2a{\lambda }_1{\lambda }_2\left(\frac{1-{(1-{\lambda }_1+{\lambda }_2)}^{2t-2}}{1-{(1-{\lambda }_1+{\lambda }_2)}^2}\right)\}}}\end{array}$$

$$C{L}_{\mathrm{EE}-T}=Q_2$$

$$\mathrm{UC}{L}_{\mathrm{EE}-T}=Q_3+K(\mathrm{IQR})\sqrt{\{({{\lambda }_1}^2+{{\lambda }_2}^2)\left(\frac{1-{(1-{\lambda }_1+{\lambda }_2)}^{2t}}{1-{(1-{\lambda }_1+{\lambda }_2)}^2}\right)-2a{\lambda }_1{\lambda }_2\left(\frac{1-{(1-{\lambda }_1+{\lambda }_2)}^{2t-2}}{1-{(1-{\lambda }_1+{\lambda }_2)}^2}\right)\}}.$$

Where $Q_1,Q_3$ are lower and upper quartiles of any data set, $K=\frac{\lambda }{2}$ and $\mathrm{IQR}=Q_3-Q_1$ is Inter quartile range.

DMEWMA-Tukey Control Chart for the shape parameter

We modified the DMEWMA by Alevizakosa et al. ³⁶ for the shape parameter of any distribution and Tukey control chart (TCC) proposed to monitor non normal distribution is given as

$$\mathrm{LC}{L}_{\mathrm{DM}-T}=Q_1-K(\mathrm{IQR})\sqrt{\mathrm{var}\left(D{M}_t\right)}$$

$$C{L}_{\mathrm{DM}-T}=Q_2$$

$$\mathrm{UC}{L}_{\mathrm{DM}-T}=Q_3+K(\mathrm{IQR})\sqrt{\mathrm{var}\left(D{M}_t\right)}.$$

Where $Q_1,Q_3$ are lower and upper quartiles of any data set, $K=\frac{\lambda }{2}$ and $\mathrm{IQR}=Q_3-Q_1$ is Inter quartile range.

Performance evaluation of proposed parametric and mix-type control chart and algorithm

Average run length

The ability of the control charts is evaluated through Average run length (ARL) criterion. It is well-defined as the average number of samples used to plot before any signal of an out-of-control sample. ARL is marked by ARL0 for the expected number of samples until a control chart signals, given that the process is in control and ARL1 for the expected number of samples until a control chart signals, given the process is out of control.

By definition

$$\mathrm{ARL}=\frac{\sum \mathrm{RL}}{m}$$

Where $\mathrm{RL}=\mathrm{Run}\mathrm{Length}=\mathrm{the}\mathrm{first}\mathrm{observation}\mathrm{recorded}\mathrm{as}\mathrm{out}\mathrm{of}\mathrm{control}$

Algorithm

For Parametric control charts we use the following algorithm to generate Tables 1 to 6. The steps are explained as follows

1. Generate a random sample of size n = 150 from KL2PFD.

2. Calculate ${\widehat{\gamma }}_{}$ using (3), (5) and (7). Also compute $E\left({\widehat{\gamma }}_{}\right)$ and $V\left(\hat{\gamma }\right)$ by repeating step 2 10,000 times.

3. Decide the value of $\lambda .$ Compute control limits to construct EWMA, DEWMA, HEWMA, MEWMA, EEWMA and DMEWMA control chart for the Process shape parameter ( $\hat{\gamma })$ .

4. Compute ARL value and fix it as 350 for each control chart given that process is in-control state.

5. Now assume if the process parameter $\gamma$ is shifted by 0.05 from its true value and compute ARL₁. This step is repeated for different shift values 0.10, 0.15, 0.25 & 0.50, and compute ARL₁ in each case of shift values.

Table 1.

The ARL_S for the shape parameter of modified EWMA Control chart.

		0	0.01	0.02	0.03	0.1	0.2	0.3	0.8	1.0	1.5
$\lambda =0.25$ L = 3.168	ARL	300.378	224.042	172.511	131.569	32.566	10.077	5.476	1.83	1.491	1.058
	SDRL	284.851	224.970	165.5102	125.2333	28.0587	6.5176	2.86175	0.5995	0.5236	0.2338
	P10	36.00	26.00	20	17	7.00	4	3	1	1	1
	P25	94.75	62.00	52	40	13.00	6	3	1	1	1
	P50	215.50	156.00	118	93	24.00	8	5	2	1	1
	P75	409.25	302.75	241	177	43.25	13	7	2	2	1
	P90	697.10	526.50	402	295	70.00	18	9	3	2	1
$\lambda =0.60$ L = 3.263	ARL	300.626	254.115	215.53	182.579	55.547	17.313	7.425	1.451	1.169	1.006
	SDRL	291.224	249.288	221.4715	179.17	54.25	15.934	5.823	0.61	0.3802	0.0773
	P10	30.00	23.0	21.9	19.00	7.0	3	2	1	1	1
	P25	86.00	77.0	61.0	55.00	16.0	6	3	1	1	1
	P50	209.00	180.0	152.0	129.50	41.0	12	6	1	1	1
	P75	408.25	355.0	285.0	249.25	77.0	23	10	2	1	1
	P90	692.60	591.1	496.2	402.10	124.1	38	15	2	2	1
$\lambda =0.75$ L = 3.2839	ARL	300.012	255.411	223.248	195.796	69.945	22.719	9.302	1.44	1.138	1.002
	SDRL	296.460	250.956	224.4594	198.93	68.17	22.090	7.924	0.69	0.3564	0.0447
	P10	30.90	23.0	21.00	19.0	9.00	3	2	1	1	1
	P25	86.00	72.5	61.00	54.0	20.00	7	3	1	1	1
	P50	213.00	189.0	157.00	133.0	51.00	16	7	1	1	1
	P75	386.25	352.5	310.25	268.5	97.25	31	13	2	1	1
	P90	705.10	592.7	522.40	438.0	159.10	50	20	2	2	1

Table 2.

The ARL_S for the shape parameter of modified DEWMA Control chart.

		0	0.01	0.02	0.03	0.1	0.2	0.3	0.8	1.0	1.5
$\lambda =0.25$ L = 2.905	ARL	299.628	206.178	146.962	108.194	25.916	9.614	6.258	2.974	1.001	1
	SDRL	300.442	207.537	141.0467	97.792	19.786	4.387	2.056	0.511	0.5102	0.1743
	P10	36.90	26.00	21.00	16	8.9	5	4	2	1	1
	P25	82.75	59.75	46.00	34	12.0	7	5	3	1	1
	P50	212.50	138.00	100.50	81	19.0	9	6	3	1	1
	P75	415.00	280.25	198.25	147	34.0	12	7	3	1	1
	P90	725.30	478.10	343.10	242	50.1	15	9	4	1	1
$\lambda =0.60$ L=3.1599	ARL	298.172	238.816	195.795	156.724	42.027	12.20	5.701	1.684	1.382	1.031
	SDRL	292.884	232.287	185.4137	150.57	38.741	10.06	3.811	0.573	0.4902	0.1734
	P10	35.00	25.90	22.0	19.0	7	3	2	1	1	1
	P25	92.75	72.75	59.0	50.0	14	5	3	1	1	1
	P50	209.00	161.00	139.5	110.0	30	9	5	2	1	1
	P75	402.50	341.25	279.0	220.0	57	15	7	2	2	1
	P90	683.90	554.10	434.1	346.1	94	25	11	2	2	1
$\lambda =0.75$ L = 3.239	ARL	298.794	252.627	215.495	183.99	53.847	16.48	7.142	1.467	1.19	1.006
	SDRL	287.101	250.077	215.8264	181.39	50.694	15.25	5.470	0.599	0.3975	1
	P10	30.0	23.00	21.90	19.00	8.0	3	2	1	1	1
	P25	86.0	74.00	64.50	56.75	17.0	6	3	1	1	1
	P50	210.5	177.00	151.50	130.00	40.0	12	6	1	1	1
	P75	421.0	354.25	287.25	254.25	74.0	22	10	2	1	1
	P90	663.5	583.60	486.50	402.10	119.1	36	14	2	2	1

Table 3.

The ARL_S for the shape parameter of modified HEWMA Control chart.

		0	0.01	0.02	0.03	0.1	0.2	0.3	0.8	1.0	1.5
${\lambda }_1=0.25,$ ${\lambda }_2=0.30$ L = 2.936	ARL	299.628	208.865	153.663	110.639	26.526	9.109	4.312	1.234	1.066	1
	SDRL	306.1087	214.7259	149.8949	102.1742	20.8133	5.4661	2.60	0.464	0.2484	0
	P10	28.90	22.0	17.0	14.00	6	2	1	1	1	1
	P25	75.00	54.0	44.0	33.00	10	4	2	1	1	1
	P50	209.50	133.0	103.0	80.50	18	7	4	1	1	1
	P75	425.25	284.5	205.5	147.25	33	11	6	1	1	1
	P90	720.10	480.2	374.1	248.10	51	15	8	2	1	1
${\lambda }_1=0.25,$ ${\lambda }_2=0.60$ L = 3.036	ARL	299.32	216.888	159.01	117.818	27.655	8.427	4.448	1.257	1.078	1
	SDRL	305.1447	225.3824	154.5968	114.1382	24.6935	5.8773	2.7977	0.4787	0.2683	0
	P10	28.0	24.0	18	15.0	5	2	1	1	1	1
	P25	80.0	61.0	48	34.0	10	4	2	1	1	1
	P50	209.0	145.5	108	86.0	20	7	4	1	1	1
	P75	424.0	287.5	216	165.0	38	11	6	1	1	1
	P90	707.4	514.1	387	269.1	58		8	2	1	1
${\lambda }_1=0.25,$ ${\lambda }_2=0.75$ L = 3.09	ARL	299.32	220.786	166.511	121.825	29.188	8.775	4.556	1.269	1.084	1
	SDRL	303.4593	230.1437	171.2577	117.9973	26.4560	6.3228	2.855	0.490	0.277	0
	P10	30.00	23.00	18.00	15.00	5.0	2	1	1	1	1
	P25	82.00	61.00	48.75	35.00	11.0	4	3	1	1	1
	P50	207.50	151.50	113.50	89.50	21.0	7	4	1	1	1
	P75	418.25	295.25	229.25	168.25	40.0	12	6	1	1	1

Table 4.

The ARL_S for the shape parameter of modified MEWMA Control chart.

		0	0.01	0.02	0.03	0.1	0.2	0.3	0.8	1.0	1.5
$\lambda =0.25$ L = 3.057	ARL	296.842	215.179	159.154	120.051	29.476	9.165	4.874	1.674	1.357	1.025
	SDRL	301.656	214.444	150.2006	115.22	25.99	6.1370	2.643	0.606	0.487	0.1562
	P10	30.0	23.0	19.0	16.0	6	3	2.0	1	1	1
	P25	85.0	63.0	50.0	37.0	11	5	3.0	1	1	1
	P50	209.0	149.0	110.5	87.5	21	8	4.0	2	1	1
	P75	418.0	290.5	219.0	163.0	41	12	6.0	2	2	1
	P90	697.1	498.3	362.2	270.0	63	17	8.1	2	2	1
$\lambda =0.60$ L = 3.15294	ARL	296.842	244.228	197.799	164.42	43.875	12.647	5.355	1.222	1.064	1
	SDRL	288.857	235.962	188.5229	154.98	42.12	11.698	4.228	0.452	0.244	0
	P10	28.0	24.90	20.00	17.0	6	2.0	1	1	1	1
	P25	88.0	73.00	55.75	49.0	13	4.0	2	1	1	1
	P50	210.0	170.00	139.00	119.0	30	9.0	4	1	1	1
	P75	398.5	340.75	283.00	238.0	61	17.0	7	1	1	1
	P90	707.0	560.00	455.00	371.1	101	27.1	11	2	1	1
$\lambda =0.75$ L = 3.156	ARL	296.841	242.494	200.796	166.24	48.52	14.135	5.809	1.149	1.037	1
	SDRL	292.554	237.818	194.25	155.83	47.57	13.964	5.244	0.393	0.188	0
	P10	29.0	23.00	19.00	16.0	5.00	2	1	1	1	1
	P25	90.0	74.75	56.75	48.0	13.75	4	2	1	1	1
	P50	217.0	166.50	139.50	119.5	33.00	10	4	1	1	1
	P75	393.0	339.25	286.25	240.5	70.00	19	8	1	1	1
	P90	705.2	554.30	460.00	378.2	112.1	32	13	2	1	1

Table 5.

The ARL_S for the shape parameter of modified EEWMA Control chart.

Estimation Method	Shift
		0	0.01	0.02	0.03	0.1	0.2	0.3	0.8	1	1.5
${\lambda }_1=0.30,$ ${\lambda }_2=0.25$ L = 3.47	ARL	296.75	220.13	167.753	128.579	40.948	18.299	11.356	2.267	1.563	1.036
	SDRL	280.54	203.85	149.83	106.414	24.633	8.7700	4.8911	1.2575	0.8004	0.191
	P10	53.00	43.00	35.00	30.0	15	8.0	5	1	1	1
	P25	99.00	79.00	64.75	54.0	24	13.0	8	1	1	1
	P50	215.50	156.00	121.50	94.0	36	17.0	11	2	1	1
	P75	392.25	287.75	220.00	171.0	53	23.0	14	3	2	1
	P90	650.80	477.50	365.10	270.1	70	29.1	18	4	3	1
${\lambda }_1=0.60,$ ${\lambda }_2=0.25$ L = 3.3	ARL	296.59	244.44	207.756	171.29	53.931	16.845	7.69	1.51	1.18	1.006
	SDRL	293.41	242.22	206.918	166.997	49.455	14.273	5.7441	0.7241	0.414	0.077
	P10	30.00	25.0	22.00	20.9	8.0	3.0	2	1	1	1
	P25	85.75	71.0	61.75	50.0	18.0	7.0	4	1	1	1
	P50	208.00	174.5	145.00	118.0	40.0	13.0	6	1	1	1
	P75	386.25	326.5	283.50	237.0	76.0	23.0	10	1	1	1
	P90	705.10	578.0	482.10	387.0	114.1	34.1	15	2	2	1
${\lambda }_1=0.75,$ ${\lambda }_2=0.25$ L = 3.297	ARL	296.841	253.289	219.404	178.529	69.391	22.365	9.522	1.492	1.162	1.004
	SDRL	290.4474	249.9306	212.5275	177.3818	65.65314	21.02503	7.786536	0.7513325	0.4048703	0.06315052
	P10	30.00	25.00	23.0	20.00	9.0	4.0	2	1	1	1
	P25	83.75	74.75	62.0	54.00	20.0	8.0	4	1	1	1
	P50	210.00	186.00	160.5	124.00	50.5	15.0	7	1	1	1
	P75	387.00	349.50	309.0	246.25	97.0	31.0	13	2	1	1
	P90	698.10	597.20	508.2	401.20	159.2	49.1	20	2	2	1

Table 6.

The ARL_S for the shape parameter of modified DEWMA Control chart.

		0	0.01	0.02	0.03	0.1	0.2	0.3	0.8	1.0	1.5
$\lambda =0.25$ L = 3.69	ARL	295.702	202.626	146.984	107.293	25.381	9.844	6.501	3.201	2.82	2.124
	SDRL	297.523	200.436	139.6844	97.113	19.48	4.5433	2.096	0.60	0.5290	0.3503
	P10	33.00	26.9	21.90	17	8.9	5	4	3	2	2
	P25	80.75	59.0	46.00	36	12.0	7	5	3	3	2
	P50	207.50	137.5	103.00	79	19.0	9	6	3	3	2
	P75	405.75	279.0	198.25	144	32.0	12	8	4	3	2
	P90	720.50	460.5	343.00	242	49.1	16	9	4	3	3
$\lambda =0.60$ L = 3.377	ARL	295.537	230.322	175.241	139.64	36.41	10.383	4.919	1.51	1.207	1.006
	SDRL	286.308	223.290	169.6488	137.01	34.06	8.4693	3.100	0.62	0.4222	0.0772
	P10	30.00	24.0	18	15.00	6.0	3	2	1	1	1
	P25	84.75	70.0	51	40.75	12.0	4	3	1	1	1
	P50	210.00	161.0	122	97.50	26.0	8	4	1	1	1
	P75	420.50	312.0	254	194.00	50.0	13	6	2	1	1
	P90	663.40	523.3	398	321.70	80.1	22	9	2	2	1
$\lambda =0.75$ L = 3.043	ARL	296.984	229.961	188.572	153.65	41.53	11.479	4.915	1.21	1.058	1
	SDRL	280.730	228.898	180.425	149.49	39.68	10.647	3.762	0.48	0.2381	0
	P10	28.00	21.00	19.00	14.00	5	2	1	1	1	1
	P25	89.00	68.00	55.00	45.75	13	4	2	1	1	1
	P50	216.50	161.00	131.50	108.00	29	8	4	1	1	1
	P75	423.25	318.25	277.25	219.00	58	15	7	1	1	1
	P90	671.00	524.20	425.30	344.00	94	26	10	2	1	1

For non-parametric control charts, we used the following algorithm to generate Tables 7 to 12. The steps are explained are as follows

1. Repeat the Step 1–2 and find the quartiles $Q_1,Q_2\mathrm{and}{Q}_3$ .

2. Find the control limits of the EWMA-TCC, DEWMA-TCC, HEWMA-TCC, MEWMA-TCC, EEWMA-TCC and DMEWMA-TCC based on shape parameter of KL2 PFD.

3. Repeat Step 4 and then take the shift in process parameter and Repeat Step 5.

Table 7.

The ARL_S for the shape parameter of KL2 PFD using EWMA-TCC.

		0	0.01	0.02	0.03	0.1	0.2	0.3	0.8	1.0	1.5
$\lambda =0.25$ L = 9.396	ARL	299.319	223.429	171.958	131.568	32.491	10.391	5.482	1.828	1.491	1.058
	SDRL	284.359	224.593	164.4993	125.17	28.07	6.9198	2.857	0.5923	0.5236	0.23386
	P10	36.00	26.00	20	17.00	7.00	4	3	1	1	1
	P25	94.00	63.00	52	40.75	12.00	6	3	1	1	1
	P50	213.50	155.00	118	93.00	24.00	9	5	2	1	1
	P75	409.25	301.25	241	176.25	43.25	13	7	2	2	1
	P90	696.10	515.60	402	295.00	70.00	19	9	2.1	2	1
$\lambda =0.60$ L = 6.321	ARL	300.016	252.621	214.277	181.456	55.18	17.286	52.39	1.45	1.169	1.006
	SDRL	289.582	246.974	218.0091	178.04	53.51	15.892	49.10	0.6195	0.3802	0.0772
	P10	30.00	23.00	22.00	19.00	7.0	3.0	8	1	1	1
	P25	86.00	77.00	62.00	54.75	16.0	6.0	17	1	1	1
	P50	208.50	179.00	151.00	129.00	41.0	12.0	38	1	1	1
	P75	408.25	354.25	282.25	249.00	77.0	23.0	75	2	1	1
	P90	692.60	589.00	491.40	393.20	124.1	37.1	114	2	2	1
$\lambda =0.75$ L = 5.6735	ARL	300.625	255.41	223.247	196.33	70.05	22.742	9.301	1.44	1.138	1.002
	SDRL	296.628	250.956	224.4601	199.13	68.15	22.074	7.925	0.6934	0.3564	0.04469
	P10	30.90	23.0	21.00	19.9	9.0	3	2	1	1	1
	P25	86.00	72.5	61.00	54.0	20.0	7	3	1	1	1
	P50	213.50	189.0	157.00	133.0	51.0	16	7	1	1	1
	P75	387.25	352.5	310.25	270.0	98.0	31	13	2	1	1
	P90	705.10	592.7	522.40	438.2	159.1	50	20	2	2	1

Table 8.

The ARL_S for the shape parameter of KL2 PFD using DEWMA-TCC.

		0	0.01	0.02	0.03	0.1	0.2	0.3	0.8	1.0	1.5
$\lambda =0.25$ L = 12.197	ARL	299.857	205.74	146.814	108.191	25.801	9.67	6.262	2.973	2.576	2.024
	SDRL	302.557	207.268	140.3786	97.773	19.71	4.5103	8.193	0.51	0.5084	0.1716
	P10	36.90	27.0	20.0	16	8	5.0	8	2	2	2
	P25	82.75	60.0	46.0	34	12	7.0	11	3	2	2
	P50	212.50	138.0	100.5	81	19	8.5	14	3	3	2
	P75	414.25	279.0	199.0	147	34	12.0	19	3	3	2
	P90	725.30	478.1	342.1	242	50	15.0	27	4	3	2
$\lambda =0.60$ L = 7.541	ARL	299.627	238.651	194.938	156.189	41.93	12.191	5.704	1.683	1.382	1.031
	SDRL	292.982	231.539	184.448	150.28	38.66	10.074	3.815	0.57	0.4902	0.1734
	P10	35.0	26.0	22.0	19.0	7	3	2	1	1	1
	P25	93.0	73.0	59.0	50.0	14	5	3	1	1	1
	P50	212.5	161.0	139.0	109.5	29	9	5	2	1	1
	P75	404.0	340.0	279.0	220.0	57	15	7	2	2	1
	P90	692.3	554.1	430.1	345.1	94	25	11	2	2	1
$\lambda =0.75$ L = 6.415	ARL	299.933	252.907	216.331	183.995	53.881	16.508	7.15	1.468	1.19	1.006
	SDRL	286.597	250.122	216.1802	181.39	50.83	15.243	5.486	0.599	0.397	0.0772
	P10	31.9	23.0	22.00	19.00	8.0	3	2	1	1	1
	P25	88.0	74.0	65.00	56.75	17.0	6	3	1	1	1
	P50	215.0	177.0	152.50	130.00	40.0	12	6	1	1	1
	P75	421.0	355.0	290.25	254.25	74.0	22	10	2	1	1
	P90	663.5	583.6	491.00	402.10	119.1	36	14	2	2	1

Table 9.

The ARL_S for the shape parameter of KL2 PFD using EWMA-TCC.

		0	0.01	0.02	0.03	0.1	0.2	0.3	0.8	1.0	1.5
${\lambda }_1=0.25,$ ${\lambda }_2=0.30$ L = 2.936	ARL	299.857	207.74	148.814	110.191	27.801	10.67	7.262	3.973	3.576	2.024
	SDRL	302.5576	207.2686	140.3786	97.77309	19.7143	4.5103	8.193	0.5	0.5084	0.171
	P10	36.90	27.0	20.0	16	8	5.0	8	2	2	2
	P25	82.75	60.0	46.0	34	12	7.0	11	3	2	2
	P50	212.50	138.0	100.5	81	19	8.5	14	3	3	2
	P75	414.25	279.0	199.0	147	34	12.0	19	3	3	2
	P90	725.30	478.1	342.1	242	50	15.0	27	4	3	2
${\lambda }_1=0.25,$ ${\lambda }_2=0.60$ L = 3.036	ARL	299.627	240.651	196.938	158.189	43.93	14.191	6.704	1.7	1.482	1.031
	SDRL	292.9823	231.5395	184.448	150.2832	38.6602	10.0743	3.812	0.5	0.4902	0.173
	P10	35.0	26.0	22.0	19.0	7	3	2	1	1	1
	P25	93.0	73.0	59.0	50.0	14	5	3	1	1	1
	P50	212.5	161.0	139.0	109.5	29	9	5	2	1	1
	P75	404.0	340.0	279.0	220.0	57	15	7	2	2	1
	P90	692.3	554.1	430.1	345.1	94	25	11	2	2	1
${\lambda }_1=0.25,$ ${\lambda }_2=0.75$ L = 3.09	ARL	299.933	254.907	218.331	185.995	55.881	17.508	8.15	2.4	1.19	1.006
	SDRL	286.597	250.1227	216.1802	181.3977	50.8362	15.2435	5.486	0.5	0.3975	0.077
	P10	31.9	23.0	22.00	19.00	8.0	3	2	1	1	1
	P25	88.0	74.0	65.00	56.75	17.0	6	3	1	1	1
	P50	215.0	177.0	152.50	130.00	40.0	12	6	1	1	1
	P75	421.0	355.0	290.25	254.25	74.0	22	10	2	1	1
	P90	663.5	583.6	491.00	402.10	119.1	36	14	2	2	1

Table 10.

The ARL_S for the shape parameter of KL2 PFD using MEWMA-TCC.

		0	0.01	0.02	0.03	0.1	0.2	0.3	0.8	1.0	1.5
$\lambda =0.25$ L = 9.838	ARL	296.169	215.178	159.01	119.05	29.38	9.148	4.873	1.66	1.353	1.024
	SDRL	302.628	214.1657	150.1037	113.6613	25.98056	6.135776	2.643456	0.6083964	0.4864439	0.1531256
	P10	30.00	23.00	19.9	16.0	6	3	2	1	1	1
	P25	85.00	64.75	50.0	37.0	11	5	3	1	1	1
	P50	208.50	148.00	109.0	87.0	21	8	4	2	1	1
	P75	414.25	290.00	219.0	163.0	40	12	6	2	2	1
	P90	689.80	498.30	362.2	264.1	63	17	8.1	2	2	1
$\lambda =0.60$ L = 7.08927	ARL	296.841	244.227	197.798	164.367	43.874	12.622	5.351	1.221	1.064	1
	SDRL	288.719	235.965	188.5236	155.03	42.13	11.710	4.223	0.44	0.2448	0
	P10	28.00	24.90	20.00	17.0	6	2.0	1	1	1	1
	P25	88.00	73.00	55.75	49.0	13	4.0	2	1	1	1
	P50	210.00	170.00	139.00	119.0	30	9.0	4	1	1	1
	P75	403.25	340.75	283.00	238.0	61	17.0	7	1	1	1
	P90	707.00	560.00	455.00	371.1	101	27.1	11	2	1	1
$\lambda =0.75$ L = 6.5512	ARL	296.982	242.799	200.795	166.276	48.507	14.111	5.807	1.149	1.037	1
	SDRL	291.323	237.546	194.0202	155.85	47.43	13.969	5.245	0.39	0.1888	0
	P10	29.9	23.00	19.00	16.00	5	2	1	1	1	1
	P25	91.0	75.00	57.75	47.75	14	4	2	1	1	1
	P50	217.5	167.50	139.50	119.50	33	10	4	1	1	1
	P75	393.0	339.25	286.25	240.50	70	19	8	1	1	1
	P90	699.8	552.20	460.00	378.20	112	32	13	2	1	1

Table 11.

The ARL_S for the shape parameter of KL2PFD using DMEWMA-TCC.

		0	0.01	0.02	0.03	0.1	0.2	0.3	0.8	1.0	1.5
$\lambda =0.25$ L = 15.918	ARL	294.346	202.034	146.321	105.46	25.31	9.821	6.489	3.196	2.814	2.125
	SDRL	296.438	199.363	139.2053	95.610	19.43	4.5417	2.080	0.60	0.526	0.3514
	P10	33.00	26.90	21.0	17.00	8.0	5	4	3	2	2
	P25	80.75	58.75	46.0	35.75	12.0	7	5	3	3	2
	P50	205.50	137.00	101.0	78.00	19.0	9	6	3	3	2
	P75	405.00	278.00	198.0	141.00	32.0	12	8	4	3	2
	P90	708.60	460.50	339.2	232.30	49.1	16	9	4	3	3
$\lambda =0.60$ L = 9.168	ARL	294.345	226.616	172.167	138.254	35.93	10.334	4.874	1.511	1.207	1.006
	SDRL	286.672	220.432	167.5795	135.38	33.75	8.4204	3.084	0.62	0.4222	0.07726
	P10	30.0	23.9	18.00	15.00	6	3	2	1	1	1
	P25	84.0	70.0	51.00	39.75	12	4	3	1	1	1
	P50	209.0	158.0	120.00	96.00	25	8	4	1	1	1
	P75	419.0	309.0	249.25	189.00	49	13	6	2	1	1
	P90	667.5	516.1	393.10	328.00	78	21	9	2	2	1
$\lambda =0.75$ L = 7.325	ARL	294.718	231.936	187.663	150.41	41.36	11.441	4.914	1.21	1.057	1
	SDRL	277.709	221.709	177.3723	147.47	39.72	10.605	3.764	0.48	0.236	0
	P10	28.00	22.9	18.90	14.0	5	2	1	1	1	1
	P25	87.50	68.0	55.00	44.0	13	4	2	1	1	1
	P50	213.00	160.5	131.50	105.5	29	8	4	1	1	1
	P75	423.25	332.0	277.25	214.0	58	15	7	1	1	1
	P90	668.30	554.0	424.10	340.0	94	25	10	2	1	1

Table 12.

The ARL_S for the shape parameter of KL2 PFD using EEWMA-TCC.

Estimation Method	Shift
		0	0.01	0.02	0.03	0.1	0.2	0.3	0.8	1	1.5
${\lambda }_1=0.30,$ ${\lambda }_2=0.25$ L = 10.911	ARL	295.38	223.47	170.18	135.809	46.972	25.063	17.326	6.034	4.219	1.929
	SDRL	271.2582	202.5086	145.5811	104.6355	23.2664	7.8357	4.6563	1.8039	1.5663	0.8802
	P10	61.0	53.00	46.0	40.90	23	16	12	4	2	1
	P25	105.0	84.75	71.0	63.00	30	20	14	5	3	1
	P50	218.0	157.00	123.5	102.00	43	24	17	6	4	2
	P75	389.5	290.00	224.0	178.25	57	29	20	7	5	3
	P90	634.8	472.20	359.4	274.10	76	35	23	8	6	3
${\lambda }_1=0.60,$ ${\lambda }_2=0.25$ L = 6.53	ARL	295.53	244.22	207.75	171.44	54.415	17.358	7.922	1.624	1.241	1.009
	SDRL	292.57	242.23	205.89	166.87	49.393	14.977	5.7778	0.7702	0.46166	0.0944
	P10	30.00	25.0	22.0	21.90	8.0	4	2	1	1	1
	P25	85.00	71.0	62.0	50.75	19.0	7	4	1	1	1
	P50	208.00	173.5	145.0	118.00	41.0	13	6	1	1	1
	P75	385.25	326.5	285.0	237.00	77.0	23	11	2	1	1
	P90	699.60	578.0	474.3	387.00	115.1	37	15	3	2	1
${\lambda }_1=0.75,$ ${\lambda }_2=0.25$ L = 5.732	ARL	295.46	251.74	219.40	194.00	69.61	22.448	9.791	1.561	1.206	1.006
	SDRL	288.16	248.98	212.17	191.74	65.469	20.870	7.9860	0.79177	0.44017	0.0772
	P10	30.00	25.0	23.0	21.00	9.00	4	2	1	1	1
	P25	83.75	74.0	63.0	54.00	21.00	8	4	1	1	1
	P50	209.50	182.0	160.5	139.50	51.00	16	8	1	1	1
	P75	387.00	349.0	309.0	269.25	97.25	31	13	2	1	1
	P90	698.00	592.5	503.5	428.00	159.20	49	20	3	2	1

Performance evaluation and interpretations

In this section, the assessment from the previously mentioned control charts is made using the criterion of ARL. The evaluations have been done using the Algorithm mentioned in Section 5 using the shifts 0.01, 0.02, 0.03, 0.1, 0.2, 0.3, 0.8, 1.0 and 1.5.

Parametric control charts

We have constructed Tables 1 to 6 to evaluate the performance of some parametric control charts such as EWMA, MEWMA, HEWMA, DEWMA, EEWMA and DMEWMA.

From Table 1, we observe the ARL value as 224.042 at a shift value of 0.01 in the shape parameter. Also for different choices of $\lambda$ , we see a rapid increase in the value of ARL and EWMA tends to Shewhart control chart as $\lambda \to 1$ .

Table 2 shows the ARL values for DEWMA control chart for the shape parameter of KL2PFD. We see that DEWMA control chart for the shape parameter outperforms EWMA and provide early detection. For different choices of, we see an increase in ARL values such as at $\lambda =0.25$ the ARL is 206.178 and it increases when we take $\lambda =0.75$ as 252.

Now if we see Table 3, we see that HEWMA gives ARL as 208.865 which is >DEWMA but <EWMA. It shows HEWMA can detect early as compared to EWMA DEWMA is better to use than HEWMA. Also, for different choices of ${\lambda }_1\mathrm{and}{\lambda }_2$ , we see an increase in ARL values such as at ${\lambda }_1=0.25,{\lambda }_2=0.60$ , the ARL is 216 and it increases when we take ${\lambda }_2=0.75$ as 220.786.

Now if we see Table 4, which is based on MEWMA, we see that MEWMA provides the ARL value of 215 at a shift of 0.01 which is <EWMA but >HEWMA and DEWMA. We also see the same behavior of increasing ARL for MEWMA by the increase in $\lambda .$

Table 5 discusses the performance of EEWMA control chart; the ARL at shift of 0.01 is 220.136 which are higher than HEWMA, MEWMA and DEWMA but <EWMA. Also, we see that when ${\lambda }_1\to 1,{\lambda }_2\to 1$ the ARL increase rapidly so best choice of $\lambda$ is small. The ARL values of DMEWMA are presented in Table 6. We observe the ARL value as 202.626 for a shift of 0.01 in the shape parameter which is lowest among all the proposed control charts. So, the use of DMEWMA control chart is recommended for the monitoring of any process shape parameter having KL2 PFD as underlying distribution.

Mix-Type Control Charts

The Non- parametric control charts are also proposed in section 4 that is, EWMA-TCC, MEWMA-TCC, HEWMA-TCC, DEWMA-TCC, EEWMA-TCC and DMEWMA –TCC. Tables 7 to 12 discusses the ARL values of these proposed control charts for the shape parameter of KL2PFD. From Table 7, we see that ARL value for the shape parameter at shift of 0.01 is 223.429. For the different choices of $\lambda =0.25$ and $0.75$ , the ARL values are 207.2686 and 252.907 which shows the increase of ARL value with the increase in $\lambda .$

From Table 8 that shows the performance of DEWMA-TCC, we see that at the shift of 0.01 from the true shape parameter is 205.74 which shows its efficiency over EWMA-TCC. From Table 9, it is clear that HEWMA-TCC is providing the ARL of 207.74 which is a little higher than DEWMA-TC.

Tables 10 and 11 provide the performance chart of MEWMA–TCC and EEWMA-TCC. We see that for a shift of 0.01 in the true value of the shape parameter of KL2PFD, we get the ARL as 215.178 for MEWMA-TCC and 223.473. We see that both MEWMA-TCC and EEWMA-TCC gives large ARL values when  $\lambda$ increases. From Table 12, for DMEWMA-TCC, we see that at a shift value of 0.01, we get the ARL value as 202.03. At $\lambda =0.25$ , the ARL is 202.03 and $\lambda =0.75$ , the ARL is 231.936. We observe that DMEWMA-TCC shows minimum ARL value as compared to the other proposed nonparametric control chart.

Parametric control charts versus Mix-Type Control Charts

From all the above detailed discussion about the performance of all proposed control charts, we see that the performance of the parametric and mix- type control charts vary. The use of mix-type control chart is suggested for EWMA. DEWMA, HEWMA and EEWMA control charts because mix-type control chart provides early detection in this case. Overall DMEWMA proprietarily detects for the monitoring of the shape of any process having KL2PFD.

Simulation study

From Section 6, we have observed that DMEWMA control chart for and DMEWMA Tukey control chart helps in early detection of any shift for Parametric control charts and Mix-Type control chart. We have further tried to evaluate the performance of each control chart by generating 50 observations from KL2PFD out of which 25 observations are generated using the shift of 0.40. The results are presented in Tables 13 and 14.

Table 13.

The ARLS for the shape parameter of KL2 PFD using DMEWMA-TCC.

		0	0.01	0.02	0.03	0.1	0.2	0.3	0.8	1.0	1.5
$\lambda =0.25$ L=15.918	ARL	294.346	202.034	146.321	105.465	25.315	9.821	6.489	3.196	2.814	2.125
	SDRL	296.438	199.363	139.2053	95.610	19.43	4.5417	2.080	0.60	0.5269	0.35142
	P10	33.00	26.90	21.0	17.00	8.0	5	4	3	2	2
	P25	80.75	58.75	46.0	35.75	12.0	7	5	3	3	2
	P50	205.50	137.00	101.0	78.00	19.0	9	6	3	3	2
	P75	405.00	278.00	198.0	141.00	32.0	12	8	4	3	2
	P90	708.60	460.50	339.2	232.30	49.1	16	9	4	3	3
$\lambda =0.60$ L=9.168	ARL	294.345	226.616	172.167	138.254	35.937	10.334	4.874	1.511	1.207	1.006
	SDRL	286.672	220.432	167.5795	135.38	33.75	8.4204	3.084	0.62	0.4222	0.0772
	P10	30.0	23.9	18.00	15.00	6	3	2	1	1	1
	P25	84.0	70.0	51.00	39.75	12	4	3	1	1	1
	P50	209.0	158.0	120.00	96.00	25	8	4	1	1	1
	P75	419.0	309.0	249.25	189.00	49	13	6	2	1	1
	P90	667.5	516.1	393.10	328.00	78	21	9	2	2	1
$\lambda =0.75$ L=7.325	ARL	294.718	231.936	187.663	150.41	41.36	11.441	4.914	1.21	1.057	1
	SDRL	277.709	221.709	177.3723	147.47	39.72	10.605	3.764	0.48	0.2362	0
	P10	28.00	22.9	18.90	14.0	5	2	1	1	1	1
	P25	87.50	68.0	55.00	44.0	13	4	2	1	1	1
	P50	213.00	160.5	131.50	105.5	29	8	4	1	1	1
	P75	423.25	332.0	277.25	214.0	58	15	7	1	1	1
	P90	668.30	554.0	424.10	340.0	94	25	10	2	1	1

Table 14.

Simulated data.

DMEWMA ( $D{M}_t$ )			DMEWMA-TCC ( $\mathrm{DM}-T$ )
${\lambda }_1=0.90$		${\lambda }_2=0.75$	${\lambda }_1=0.90$	${\lambda }_2=0.75$
L = 12			L = 7.66
$\mathrm{DMEWMP}{\mathrm{E}}_t$	LCL	UCL	$\mathrm{DEWM}\mathrm{P}{\mathrm{E}}_t$	LCL	UCL
2.017248	1.868460	2.131540	1.997681	1.868460	2.131540
2.032753	1.848728	2.151272	2.000240	1.848728	2.151272
2.082240	1.835943	2.164057	2.037633	1.835943	2.164057
2.062794	1.827293	2.172707	2.011376	1.827293	2.172707
2.022564	1.821304	2.178696	1.967336	1.821304	2.178696
2.040718	1.817099	2.182901	1.978608	1.817099	2.182901
2.075273	1.814120	2.185880	2.005952	1.814120	2.185880
2.101648	1.811997	2.188003	2.026590	1.811997	2.188003
2.100406	1.810478	2.189522	2.021486	1.810478	2.189522
2.074516	1.809388	2.190612	1.993269	1.809388	2.190612
2.072141	1.808604	2.191396	1.988105	1.808604	2.191396
2.052613	1.808040	2.191960	1.967061	1.808040	2.191960
2.026051	1.807634	2.192366	1.939233	1.807634	2.192366
1.996827	1.807340	2.192660	1.911444	1.807340	2.192660
2.075796	1.807129	2.192871	1.983682	1.807129	2.192871
2.099065	1.806976	2.193024	2.005896	1.806976	2.193024
2.099456	1.806866	2.193134	2.005738	1.806866	2.193134
2.056843	1.806786	2.193214	1.964617	1.806786	2.193214
2.085972	1.806729	2.193271	1.991746	1.806729	2.193271
2.087100	1.806687	2.193313	1.992585	1.806687	2.193313
2.156410	1.806657	2.193343	2.057674	1.806657	2.193343
2.155110	1.806635	2.193365	2.055984	1.806635	2.193365
2.173715	1.806620	2.193380	2.072615	1.806620	2.193380
2.200592	1.806609	2.193391	2.098813	1.806609	2.193391
2.261086	1.806600	2.193400	2.155442	1.806600	2.193400
2.146604	1.806594	2.193406	2.046232	1.806594	2.193406
2.185490	1.806590	2.193410	2.083346	1.806590	2.193410
2.148622	1.806587	2.193413	2.047346	1.806587	2.193413
2.183985	1.806585	2.193415	2.081138	1.806585	2.193415
2.181744	1.806583	2.193417	2.078903	1.806583	2.193417
2.159838	1.806582	2.193418	2.057063	1.806582	2.193418
2.215260	1.806581	2.193419	2.110747	1.806581	2.193419
2.222658	1.806581	2.193419	2.116725	1.806581	2.193419
2.195272	1.806580	2.193420	2.091045	1.806580	2.193420
2.193018	1.806580	2.193420	2.089801	1.806580	2.193420
2.184159	1.806580	2.193420	2.080296	1.806580	2.193420
2.204701	1.806579	2.193421	2.100051	1.806579	2.193421
2.195714	1.806579	2.193421	2.091239	1.806579	2.193421
2.157754	1.806579	2.193421	2.056106	1.806579	2.193421
2.172195	1.806579	2.193421	2.069127	1.806579	2.193421
2.121605	1.806579	2.193421	2.021060	1.806579	2.193421
2.121196	1.806579	2.193421	2.020867	1.806579	2.193421
2.110891	1.806579	2.193421	2.009967	1.806579	2.193421
2.083255	1.806579	2.193421	1.985412	1.806579	2.193421
2.043096	1.806579	2.193421	1.946221	1.806579	2.193421
2.102057	1.806579	2.193421	2.001374	1.806579	2.193421
2.093582	1.806579	2.193421	1.993960	1.806579	2.193421
2.160051	1.806579	2.193421	2.056331	1.806579	2.193421
2.114314	1.806579	2.193421	2.013163	1.806579	2.193421
2.129418	1.806579	2.193421	2.028419	1.806579	2.193421
2.124392	1.806579	2.193421	2.023339	1.806579	2.193421
2.094214	1.806579	2.193421	1.995809	1.806579	2.193421
2.074404	1.806579	2.193421	1.976514	1.806579	2.193421
2.076705	1.806579	2.193421	1.978191	1.806579	2.193421
2.075617	1.806579	2.193421	1.976331	1.806579	2.193421
2.102646	1.806579	2.193421	2.001816	1.806579	2.193421
2.076179	1.806579	2.193421	1.976117	1.806579	2.193421
2.102455	1.806579	2.193421	2.001032	1.806579	2.193421
2.136972	1.806579	2.193421	2.033813	1.806579	2.193421
2.147813	1.806579	2.193421	2.043793	1.806579	2.193421

From Table 14, Figures 1 and 2, we see that DMEWMA control chart helps in the early detection of any shifted process ${\lambda }_1=0.90,{\lambda }_2=0.75$ .

Figure 1.

Graph of simulated data of the proposed DMEWMA-TCC control chart under PE.

Figure 2.

Graph of simulated data of the proposed DMEWMA control chart under PE.

Real life application

Filling height of soft drink beverage bottles

We have used the data of filling height of soft drink beverage bottles given by Montgomery ³⁷ . The data set has 15 samples each of size $n=10$ .

We plotted for both DMEWMA-TCC and DMEWMA control charts under PE, as shown in Figures 3 and 4. We have constructed DMEWMA and DMEWMA-TCC control charts under PE in Figures 3 and 4. We see that DMEWMA detects the process shift early compared to DMEWMA-TCC, which shows that DMEWMA is better to be used in filling height data of soft drinks when the distribution of the underlying process is KL2PFD.

Figure 3.

Graph of real data of the DMEWMA-TCC control chart under PE.

Figure 4.

Graph of real data of the DMEWMA control chart under PE.

Hard-bake process in the semiconductor manufacturing process

The second data set is also taken from Montgomery. ³⁷ The data consists of a hard-bake process in conjunction with photolithography in the semiconductor manufacturing process. Our aim of using this data is to provide a better process monitoring for the flow width of the resist. Thirty samples each of size 5 wafers are taken with a time interval of 1 h and flow width measurements are in microns. We used DMEWMA control chart to monitor the shape of the process following KL2PFD. From Figures 5 and 6, we see that at ${\lambda }_1=0.90,{\lambda }_2=0.75$ , the DMEWMA control chart is early detected for any minor change in the shape of the process.

Figure 5.

Graph of real data of the DMEWMA-TCC control chart under PE.

Figure 6.

Graph of real data of the DMEWMA control chart under PE.

Operation side of the telecommunication industry

The Real-life data is taken from the operation side of the telecommunication industry, where errors in the software frequently occur regarding the billing amount, dispatch issue, contact and login details. The per-day frequency of errors in the software is reported below: 1124, 1013, 1187, 1153, 1141, 1051, 1178, 1145, 1124, 1132, 1141, 1136, 1241, 1301, 1214, 1421, 1258, 1109, 1321, 1121, 1114, 1021, 1131, 1142, 1165, and 1184. The data followed the KL2PFD and plotted for DMEWMA control charts. We have constructed DMEWMA control charts on real-life dat. For example, in Figures 7 and 8, we see that DMEWMA under PE predicts the process shift at early levels compared to DMEWMA-TCC, which indicates that DMEWMA under PE provides a better explanation of the distributions when the underlying process is based on KL2PFD.

Figure 7.

Graph of real data of the DMEWMA-TCC control chart under PE.

Figure 8.

Graph of real data of the DMEWMA control chart under PE.

Final remarks

Some parametric and mix-type control charts have been developed for monitoring processes where the information about the quality characteristic is available. We assume that the quality characteristics follow KL2PFD. The shape parameter of the process is estimated through percentile estimator and further used to construct the parametric and mix type control charts.

We have fixed the ARL at 300 and then constructed the tables of ARL, SDRL using EWMA, DEWMA, HEWMA, EEWMA, MEWMA and DMEWMA for the shape parameter of KL2PFD.

We have made a comparison of these control charts and we observe that the DMEWMA control chart proves to be more efficient to detect any little change in the shape of process.

To investigate the performance of mix type control charts such as EWMA-Tukey, DEWMA-Tukey, HEWMA-Tukey, EEWMA-Tukey, MEWMA-Tukey and DMEWMA-Tukey. We construct control monitoring charts for the shape parameter of KL2PFD and finds DMEWMA- Tukey control chart more efficient in the early detection of any little change in the shape of process having KL2PFD.

By comparing the performance of all parametric and mix-type control charts, we find out that DMEWMA effectively detects any shift as compared to other control charts. After this, DMEWMA-Tukey shows better results to perceive any early shift in the process following KL2PFD.

We also showed the performance of proposed control charts using real-life data and simulation studies. We used three different real life data sets such as the data of filling height of soft drink beverage bottles given by Montgomery. The data consist of a hard-bake process in conjunction with photolithography in the semiconductor manufacturing process given by Montgomery and the data of operation side of the telecommunication industry, where errors in the software frequently occurred regarding the billing amount, dispatch issue, contact and login details. For all these three data sets it’s clear from Figures 3 to 8 that the proposed control chart DMEWMA for the shape parameter performs outclass by providing minimum ARL value (early shift in the process).

The main objective of the paper is to provide the reader with the wide application of KL2PFD in the field of Statistical quality control. The shape parameter is mostly ignored when we come to the application of probability distributions. The current paper is an innovation in the field of probability distribution and statistical quality control which not only provide new control charts named as parametric and Mix-type control charts but also proves their validity by taking the different real life data sets and the shape parameter of KL2PFD is controlled using the proposed monitoring.