One-sided Shewhart control charts for monitoring the ratio of two normal variables in Short Production Runs

Quality control (QC) enables the manufacturer to produce high-quality products according to the needs of the customer. Its tool kit helps to improve product quality by minimizing product cost, increase the efficiency of the process by reducing product waste. Control charts are famous tools for monitoring the assignable causes, they also tell us when corrective action must be taken or timely notify us when corrective action must be taken to improve the process behavior (see Shewhart ²¹). With growing competition in customer markets, manufacturers extremely depend on the quality of their products and services for their survival. The short-run manufacturing production process is become quite common for achieving the satisfaction of the customers like job-shops, which are categorized by a high amount of flexibility and manufacture diversity. Therefore, the life cycle of the products is decreasing rapidly. Nowadays, the production lines in several manufacturings and engineering processes have limited. In short runs production, some sources are fixed or the time span of the product is too short, maybe one hour or day. For example, any warehouse which may be on the lease. This warehouse is operating in the short run because it has a limited place. The owner cannot extend their business or shifted it to another place. When the agreement expires, he will have in position to enlarge business or shift to a large place. The same is the case, in manufacturing industries, like robotics manufacturing industry incorporates the limited production runs of automatic parts within the flexible production cells and semiconductor industry assembly of electronic boards and in beverages industries where the high volume of production and filling of soft drinks in every $24$ to $48$ hour required the $20$ to $30$ inspection between consecutive session.

There is a myth among some manufacturing organizations, as they mostly feel that control charts are less applicable for the short-run process as the duration of their production cycle is too short. Recent studies signify control charts for the production process draw the attention of quality practitioners. Quality professionals are more concerned about the quality of the product. They are doing continuous improvement in product quality by reducing variability. The repute of any industry depends on the quality of goods or services that they offer to the customer. Quality is a major strategy that increases the productivity of any industry.

There are various SPC (statistical process control) short run control charts presented in literature to serve the purpose. Short run control charts are more effective and useful for the small lot manufacturing runs with limited production data. Ladany ¹² first introduced optimized-p chart for short production run. Later on, numerous authors introduced effective designs for such economic process, see for instance Ladany and Bedi ¹³, Del Castillo and Montgomery ⁸, Del Castillo and Montgomery ⁹, Tagaras ²², Nenes and Tagaras ¹⁵, Nenes and Tagaras ¹⁶, Celano et al. ⁷, Castagliola et al. ⁴, Castagliola et al. ³, Amdouni et al. ², Khatun et al. ¹⁰, Naseri et al. ¹⁴ and Khoo et al. ¹¹.

This study is planned to design the Shewhart control chart for monitoring the ratio of two variables for a short production run. There are several production or manufacturing processes, where the quality characteristics of interest formed the ratio of two normal variables. Production strategies, where numerous components need to be blended together to get a product composition can require monitoring the ratios of random variables when quality experts are frequently interested in the relative evaluation of the same property for two-components. In fact, guaranteeing the stable ratio between different components permits the product specifications to be encountered. Automotive, aerospace, electronics, pharmaceutical, materials production, food preparation, and packaging industries are typical applications of these manufacturing environments.

Celano and Castagliola ⁵ designed the ratio type Shewhart control chart by using the data from the food processing company. They took the ratio of two seeds (pumpkin and flex) to meet the nutrition facts. Tran et al. ²³ investigated the performance of the ratio chart using the run-rules scheme. Tran et al. ²³ designed EWMA type control charts for efficient monitoring of the ratio. Tran et al. ²⁴ assessed $RZ$ chart efficiency and performance in presence of measurement error. Tran and Knoth ²⁶ did an analysis to evaluate the steady-state ARL performance of the $RZ$ chart. Tran et al. ²⁵ designed the CUSUM control chart which was more efficient than its $RZ$ setup. Nguyen et al. ¹⁷ and Nguyen, Tran and Heuchenne ¹⁸ introduced EWMA and CUSUM design for ratio type control chart at several sampling intervals. Nguyen, Tran and Goh ²⁰ applied the variable sampling interval idea to $RZ$ chart to evaluate its efficacy and performance. Nguyen and Tran ¹⁹ presented the one-sided $RZ$ chart which was more efficient in case of measurement error present in the process.

The articles cited above are planned to observe the ratio over a production horizon considered as infinite. However, there are numerous situations, where the production run no longer is infinite. As far as we know, no research has been done regarding the observing of the ratio (of two normal variables) infinite horizon framework. The aim of this study is to fill this gap by introducing two one-sided $RZ$ charts aiming to observe the decrease or increase in the finite runs. In the remainder of the study, they are denoted by “${Sh}^{+}_{RZ}$ Chart” for the upper sided chart and by “${Sh}^{-}_{RZ}$ Chart” for the lower sided one.

The rest of the article organized as follows, Section 2 denotes the main properties of ratio distribution. Section 3 deals with the design structure of one-sided $RZ$ (${Sh}^{+}_{RZ}$ and ${Sh}^{-}_{RZ}$) Shewhart charts for a limited production run. The truncated run length (TRL), as the performance measure of the proposed short run charts, is briefly described in Section 4. To assess the behavior of the new charts, a numerical analysis is done in Section 5. Real-life illustration of the proposed technique presented in Section 6. Finally, conclusion and recommendation for future research are given in Section 7.

Suppose that $X$ and $Y$ be the two normal random variables such that $\mathbf{W}=(X,Y)^{T}\sim N(\boldsymbol{\mu}_{\mathbf{W}},\boldsymbol{\Sigma}_{\mathbf{W}}),$ i.e. $\mathbf{W}$ is a bivariate normal random vector with mean vector and variance-covariance matrix as follows:

$\displaystyle{\boldsymbol{\mu}}_{\mathbf{W}}=\begin{pmatrix}\mu_{X}\\ \mu_{Y}\end{pmatrix}\quad\text{and}\quad{\boldsymbol{\Sigma}}_{\mathbf{W}}=\begin{pmatrix}\sigma^{2}_{X}&\rho\sigma_{X}\sigma_{Y}\\ \rho\sigma_{X}\sigma_{Y}&{\sigma}^{2}_{Y}\end{pmatrix}$

(1)

where $\mu_{X}$ and $\mu_{Y}$ are the means of two variables and $\rho$ is the correlation coefficient between them. Coefficients of variation $\left(\gamma_{X},\gamma_{Y}\right)$ and standard-deviation ratio ($\omega$) of $X$ and $Y$ are denoted by $\gamma_{X}=\frac{\sigma_{X}}{\mu_{Y}}$, $\gamma_{Y}=\frac{\sigma_{Y}}{\mu_{Y}}$ and $\omega=\frac{\sigma_{X}}{\sigma_{Y}}$, respectively. Let $Z$ be the ratio of $X$ to $Y\left(Z={X}/{Y}\right)$. Celano and Castagliola ⁵ derived an adequate approximation for the c.d.f (cumulative distribution function) of $Z$ as a function of $\gamma_{X},\gamma_{Y},\omega,$ and $\rho$ as:

$\displaystyle F_{Z}(z|\gamma_{X},\gamma_{Y},\omega,\rho)\simeq\Phi\left(\frac{A}{B}\right),$

(2)

where

$\displaystyle A=\frac{z}{\gamma_{Y}}-\frac{\omega}{\gamma_{X}},\qquad\text{and}\qquad B=\sqrt{\omega^{2}-2\rho\omega z+{z}^{2}},$

and $\Phi$ is the c.d.f of standard normal distribution (Sdn). After some tedious derivations, approximated p.d.f (probability density function) of the ratio $Z$ is

$\displaystyle f_{Z}(z|\gamma_{X},\gamma_{Y},\omega,\rho)\simeq\left(\frac{1}{B\gamma_{Y}}-\frac{\left(z-\rho\omega\right)A}{B^{3}}\right)\times\phi\left(\frac{A}{B}\right),$

(3)

where $\phi(\cdot)$ is the p.d.f. of Sdn. Solving the equation $F_{\mathrm{Z}}(z\left|\gamma_{X},\gamma_{Y},\omega,\rho\right)=p$ allows obtaining an approximate expression for the i.d.f (inverse distribution function) $F^{-1}_{\mathrm{Z}}(p|\gamma_{X},\gamma_{Y},\omega,\rho)$. We have

$\displaystyle F^{-1}_{Z}(p|\gamma_{X},\gamma_{Y},\omega,\rho)\simeq\begin{cases}\frac{-C_{2}-\sqrt{C^{2}_{2}-4C_{1}C_{3}}}{2C_{1}},\quad\text{if}\quad p\in(0,0.5],\\ \frac{-C_{2}+\sqrt{C^{2}_{2}-4C_{1}C_{3}}}{2C_{1}},\quad\text{if}\quad p\in[0.5,1),\end{cases}$

(4)

where $C_{1}$, $C_{2}$, and $C_{3}$ are functions of $\omega,\rho,\gamma_{X},\gamma_{Y},p$ and they are:

	$\displaystyle C_{1}$	$\displaystyle=\frac{1}{\gamma^{2}_{Y}}-\left(\Phi^{-1}(p)\right)^{2},$
	$\displaystyle C_{2}$	$\displaystyle=2\omega\left(\rho\left(\Phi^{-1}(p)\right)^{2}-\frac{1}{\gamma_{X}\gamma_{Y}}\right)$
	$\displaystyle C_{3}$	$\displaystyle=\omega^{2}\left(\frac{1}{\gamma^{2}_{X}}-{\left({\Phi}^{-1}\left(\mathrm{p}\right)\right)}^{2}\right),$

and where $\Phi^{-1}(\cdot)$ is the i.d.f. of Sdn.

The production run is planned to produce small size lot having $N$ parts after a fixed rolling length ${H}$. Let $I$ be the number of planned inspections of rolling horizon $H$ and assume that no inspection takes place at the end of the run. By these settings, the sampling frequency (the time interval between two consecutive inspections) will be $\mathcal{S}_{h}={{\frac{{H}}{\left(I+1\right)}}}$ hours. The observed values of random variables $X$ and $Y$ are used to calculate the ratio for two one sided Shewhart charts for short run. In order to monitor the ratio $Z$, samples of size $n$ will be taken at every sampling interval from the process and the quality characteristic $\mathbf{W}$ will be measured for each item. Let $[\mathbf{W}_{i,1},\mathbf{W}_{i,2},\dots,\mathbf{W}_{i,n}]$ be the collected sample in which the couples $\mathbf{W}_{i,j}=(X_{i,j},Y_{i,j})^{T}$ for $j=1,2,...n$ follow the bivariate normal model $N(\boldsymbol{\mu}_{\mathbf{W},i},\boldsymbol{\Sigma}_{\mathbf{W},i})$ where:

$\displaystyle{\boldsymbol{\mu}}_{\mathbf{W},i}=\begin{pmatrix}\mu_{X,i}\\ \mu_{Y,i}\end{pmatrix}\quad\text{and}\quad{\boldsymbol{\Sigma}}_{\mathbf{W},i}=\begin{pmatrix}\sigma^{2}_{X,i}&\rho\sigma_{X,i}\sigma_{Y,i}\\ \rho\sigma_{X,i}\sigma_{Y,i}&{\sigma}^{2}_{Y,i}\end{pmatrix},\quad i=1,2,3,...$

(5)

To design the charts, let $\gamma_{X}$ and $\gamma_{Y}$ be the known and constant coefficients of variation and let $z_{0}=\frac{\mu_{X,i}}{\mu_{Y,i}}$ and $\rho_{0}$ be the known in-control values of the ratio and the coefficient of correlation that will ensure the stability of process. We further assume that linear relationships are held between the sample standard deviation and the sample mean for both variables $X$ and $Y$, i.e., $\sigma_{X,i}=\gamma_{X}\mu_{X,i}$ and $\sigma_{Y,i}=\gamma_{Y}\mu_{Y,i}$ for every $i\geq 1$. This implies that the standard deviation of each sample can change proportionally to its mean such that their ratio remains constant. There are several quality characteristics in practice (such as weights, tensile strengths and linear dimensions) that can have a dispersion proportional to the population mean. Finally, the sample units are considered free to vary from sample to sample and thus it is possible to have $\boldsymbol{\mu}_{\mathbf{W},i}\neq\boldsymbol{\mu}_{\mathbf{W},k}$ and $\boldsymbol{\Sigma}_{\mathbf{W},i}\neq\boldsymbol{\Sigma}_{\mathbf{W},k}$, for $i\neq k.$ The monitoring statistic of the proposed chart is the ratio of sample means that should be calculated at the inspections $i=1,2,,3,...$ as:

$\displaystyle\hat{Z}_{i}=\frac{\hat{\mu}_{X,i}}{\hat{\mu}_{Y,i}}=\frac{\bar{X}_{i}}{\bar{Y}_{i}}=\frac{\sum_{j=1}^{n}X_{i,j}}{\sum_{j=1}^{n}Y_{i,j}}$

(6)

To obtain the c.d.f and i.d.f of the statistic $\hat{Z}_{i}$, one needs to some distributional proportions of the sample means $\bar{X}_{i}$ and $\bar{Y}_{i}$. It is easy to see that $\bar{X}_{i}\sim N(\mu_{X,i},\frac{\sigma_{X,i}}{\sqrt{n}})$ and $\bar{Y}_{i}\sim N(\mu_{Y,i},\frac{\sigma_{Y,i}}{\sqrt{n}})$ having the constant coefficients of variation $\gamma_{\bar{X}}=\frac{\gamma_{X}}{\sqrt{n}}$ and $\gamma_{\bar{Y}}=\frac{\gamma_{Y}}{\sqrt{n}}$. By the definition and the mentioned assumptions, the standard deviation ratio $\omega_{i}$ in each inspection can be calculated as:

$\displaystyle\omega_{i}=\frac{\sigma_{X,i}}{\sigma_{Y,i}}=\frac{\mu_{X,i}}{\mu_{Y,i}}\frac{\gamma_{X}}{\gamma_{Y}}=z_{0}\times\frac{\gamma_{X}}{\gamma_{Y}}=\omega_{0},$

(7)

where $\omega_{0}$ is the in-control standard deviations ratio. We are now in a position to obtain the c.d.f and i.d.f of $\hat{Z}_{i}$ based on the c.d.f and i.d.f of $Z$ in (2) and (4) as (Nguyen, Tran and Goh ²⁰):

	$\displaystyle F_{\hat{Z}_{i}}(z|n,\gamma_{X},\gamma_{Y},z_{0},\rho_{0})$	$\displaystyle=F_{Z}\left(z|\frac{\gamma_{X}}{\sqrt{n}},\frac{\gamma_{Y}}{\sqrt{n}},\frac{z_{0}\gamma_{X}}{\gamma_{Y}},\rho_{0}\right),$		(8)
	$\displaystyle F^{-1}_{\hat{Z}_{i}}(p|n,\gamma_{X},\gamma_{Y},z_{0},\rho_{0})$	$\displaystyle=F^{-1}_{Z}\left(p|\frac{\gamma_{X}}{\sqrt{n}},\frac{\gamma_{Y}}{\sqrt{n}},\frac{z_{0}\gamma_{X}}{\gamma_{Y}},\rho_{0}\right).$		(9)

The control limits of two separate one sided ratio charts are as follows

1. 1.

   One sided Shewhart-$R{Z}^{-}$ chart (${Sh}^{-}_{RZ}$ Chart) The downward ratio uses to monitor the decrease in ratio of two variables. The lower and the upper control limits $LCL^{-}$ and $UCL^{-}$ of said chart are:

   	$\displaystyle LCL^{-}$	$\displaystyle=F^{-1}_{\hat{Z}_{i}}(\alpha_{0}|n,\gamma_{X},\gamma_{Y},z_{0},\rho_{0})$		(10)
   	$\displaystyle UCL^{-}$	$\displaystyle=+\infty.$		(11)

   where $\alpha_{0}$ is the probability of type I error.

2. 2.

   One sided Shewhart-$R{Z}^{+}$ chart ($Sh^{+}_{RZ}$ Chart) The upward ratio uses to monitor the increase in ratio of two variables. The lower and the upper control limits $LCL^{+}$ and $UCL^{+}$ of said chart are:

   	$\displaystyle LCL^{+}$	$\displaystyle=0$		(12)
   	$\displaystyle UCL^{+}$	$\displaystyle=F^{-1}_{\hat{Z}_{i}}(1-\alpha_{0}|n,\gamma_{X},\gamma_{Y},z_{0},\rho_{0}).$		(13)

The ratio of two normal variables $\hat{Z}_{i}$ will be plotted against the limits. We will count all points which fall outside of the limits of both charts. The process is consider to be out-of-control (occ) if $\hat{Z}_{i}$ fulfills either of these conditions, $\hat{Z}_{i}<LCL^{-}\,\text{or}\,\hat{Z}_{i}>UCL^{+}$ and assignable causes will be removed. As the production run is too small, the traditional run length will not be applicable to determine the run length properties of the control charts. To serve the purpose truncated run-length TRL and its average (TARL) will be used to determine the performance of the charts. The details and computations of TRL and TARL are provided in what follows.

In certain manufacturing processes, it could not be practical to gather a sufficient amount of data at the beginning of the manufacturing process for trial limits in phase-I. For example, in the aerospace industry, the production rate of huge components can be finite. It takes too much time to gather enough samples for the applying control charts. On the other hand, it is often needed that the quality control procedure starts as early as possible since the rate of each product is high. In such circumstances, traditional quality charts are not as effective. It may need more samples to create control limits until they are precise adequate to screen the whole process. Traditional run-length does not give the reliable monitoring of assignable cause for such a process in which a finite (or limited) number I of samples is available in the production horizon having finite length $H$. Therefore, TRL as a modified version of traditional run length is used to determine the behavior of the process short production runs. It is define as “the number of samples until a shift is identified or until completion of process either happen first”. TRL is a discreet random variable with support $TRL\in\{1,2,...,I,I+1\}$. The event $TRL=l$ for $l\in\{1,2,...,I\}$ means that an ooc signal is declared by the chart before the termination of the production run and $TRL=I+1$ expresses that the run is completed without detecting any shift in $I$ inspections. The p.m.f (probability mass function) $f_{TRL}(l)$ and the c.d.f $F_{TRL}(l)$ of TRL can be derived as follows:

$\displaystyle f_{TRL}(l)=\begin{cases}p(1-p)^{l-1},\quad\text{if}\quad l=1,2,...,I,\\ (1-p)^{I},\quad\quad\,\,\,\text{if}\quad l=I+1,\end{cases}$

(14)

and

$\displaystyle F_{TRL}(l)=\begin{cases}1-(1-p)^{l},\quad\text{if}\quad l=1,2,...,I,\\ 1,\qquad\qquad\quad\,\,\,\,\text{if}\quad l=I+1,\end{cases}$

(15)

where $p$ is the probability of an occ signal. Let TARL be the expected value of TRL that can be used in place of the traditional ARL to evaluate the performance of the control charts in finite horizon production. Moreover, let $\textit{TARL}_{0}$ and $\textit{TARL}_{1}$ be the in-control and out-of-control values of TARL, respectively. Substituting $p$ by $\alpha$ ($\alpha$ is the probability of type I error) or $1-\beta$ ($\beta$ is the probability of type II error) in (14) and taking expectation over the TRL support, $\textit{TARL}_{0}$ and $\textit{TARL}_{1}$ will be derived as (Nenes and Tagaras ¹⁶):

	$\displaystyle TARL_{0}$	$\displaystyle=\sum^{I}_{l=1}l(1-\alpha)^{l-1}\alpha+(I+1)(1-\alpha)^{I}=\frac{1-(1-\alpha)^{I+1}}{\alpha},$		(16)
	$\displaystyle TARL_{1}$	$\displaystyle=\sum^{I}_{l=1}l\beta^{l-1}(1-\beta)+(I+1)\beta^{I}=\frac{1-\beta^{I+1}}{1-\beta}.$		(17)

The error probabilities $\alpha$ and $\beta$ can be calculated for the proposed charts by:

$\displaystyle\alpha=\begin{cases}F_{\hat{Z}_{i}}(LCL^{-}|n,\gamma_{X},\gamma_{Y},z_{0},\rho_{0}),\quad\quad\,\,\,\,\text{for $\mathcal{S}h^{-}_{RZ}$ Chart}\\ 1-F_{\hat{Z}_{i}}(UCL^{+}|n,\gamma_{X},\gamma_{Y},z_{0},\rho_{0}),\quad\text{for $\mathcal{S}h^{+}_{RZ}$ Chart}\end{cases}$

(18)

and

$\displaystyle\beta=\begin{cases}1-F_{\hat{Z}_{i}}(LCL^{-}|n,\gamma_{X},\gamma_{Y},z_{1},\rho_{1}),\quad\text{for $\mathcal{S}h^{-}_{RZ}$ Chart}\\ F_{\hat{Z}_{i}}(UCL^{+}|n,\gamma_{X},\gamma_{Y},z_{1},\rho_{1}),\quad\quad\,\,\,\,\text{for $\mathcal{S}h^{+}_{RZ}$ Chart}\end{cases}$

(19)

where $z_{1}=\tau z_{0}\,(\tau>0)$ and $\rho_{1}$ are the out-of-control values of the ratio and the coefficient of correlation, respectively. While the values of shift size $\tau\in(0,1)$ correspond to decrease of nominal ratio $z_{0}$, the values of $\tau>1$ correspond to an increase of nominal ratio $z_{0}$.

In this section, we evaluate the statistical performance of the two one-sided ${Sh}^{-}_{RZ}$ and ${Sh}^{+}_{RZ}$ charts for the short production run. The performance of the chart is evaluated by the startup of the short run when a particular shift $(\tau)$ happens. To serve the purpose, the TARL metric is used as the performance measure instead of using traditional ARL. Initially, let ${TARL}_{0}=I$ when the process expected to be in control. Statistical performance has been calculated by considering the following parameters values.

• •

  $z_{0}$=1 (in-control ratio of two variables)

• •

  $\gamma_{X}\in\{0.01,0.2\}\,\text{and}\,\gamma_{Y}\in\{0.01,0.2\}$

• •

  $\rho_{0}\in\{0.0,\pm 0.4,\pm 0.8\}$

• •

  $n=\{1,5,7,10,15\}$

• •

  $I=\{10,30,50\}$

• •

  $\tau=\left\{\begin{array}[]{c}0.9,0.95,0.98,0.99;\ \ \ \ \ \ for\ {Sh}^{-}_{RZ}\ chart\\ 1.01,1.02,1.05,1.1\ \ \ \ \ for\ {Sh}^{+}_{RZ}\ chart\end{array}\right.$

Tables 1-3 illustrate the probability-type control limits $LCL^{-}$ and $UCL^{+}$ values designing such that $TARL_{0}=I$ (Nenes and Tagaras ¹⁵) by varying $\gamma_{X},\gamma_{Y},\rho_{0}$ and $n$ at $\tau=\tau_{0}$ (when process is in control) and $z_{0}=1$. Tables 4-9 exhibit the $TARL_{1}$ values for two charts (${Sh}^{-}_{RZ}$ and ${Sh}^{+}_{RZ}$) for $I=10,30$ and $50$, when $\rho_{1}=\rho_{0}$ using $n\in\left\{1,5,7,15,30\right\},\gamma_{X}=\gamma_{Y},\gamma_{X}\neq\gamma_{Y}$ and $\tau=\left\{0.9,0.95,0.98,0.99,1.01,1.02,1.05,1.1\right\}$. Tables 10-15 demonstrate the $TARL_{1}$ values for two charts (${Sh}^{-}_{RZ}$ and ${Sh}^{+}_{RZ}$) for $I=10,30$ and $50$, when $\rho_{1}\neq\rho_{0}$ using $n\in\{1,5,7,15,30\},\tau=\{0.9,0.95,0.98,0.99,1.01,1.02,1.05,1.1\},\gamma_{X}=\gamma_{Y}$, and $\gamma_{X}\neq\gamma_{Y}$.