Modified side-sensitive synthetic double sampling monitoring scheme for simultaneously monitoring the process mean and variability

Highlights

• Joint monitoring of the process mean and standard deviation.

• The proposed MSS SDS $\overline{X}$ scheme presents very interesting properties.

• Efficient in detecting both mean and standard deviation shifts simultaneously.

• Cost effective.

• Outperforms the competing schemes in many situations.

Abstract

This paper develops a new synthetic double sampling (SDS) monitoring scheme based on the modified side-sensitive (MSS) design for jointly monitoring the process mean and variability. We first give the operation of the proposed scheme and secondly, the closed-form expression of the probability of declaring a specific sampling stage as “conforming” or “nonconforming”. Thirdly, we evaluate the performance of the newly proposed scheme in terms of the zero- and steady-state out-of-control average run-length, standard deviation of the run-length, average number of observations to signal, average extra quadratic loss, average ratio of the average run-length and performance comparison index. Finally, we compare the performance of the new monitoring scheme with some existing monitoring schemes. It is observed that the proposed scheme has attractive run-length properties and outperforms the existing joint $\overline{X}$ and $S$ SDS monitoring scheme as well as all other competing schemes in many situations.

Introduction

A synthetic chart (or scheme) is a combination of a traditional chart (such as a Shewhart, CUSUM or EWMA chart) and a conforming run-length (CRL) chart (cf. Wu and Spedding, 2000, Guo et al., 2015). Bourke (1991) defined a CRL as the number of inspected units between two consecutive nonconforming units. Wu and Spedding (2000) were the first authors to propose a synthetic Shewhart $\overline{X}$ scheme for monitoring the process mean. Their scheme gives an out-of-control (OOC) signal at the $i^{\mathit{th}}$ sampling point if the sample mean plots above the upper control limit ($\mathit{UCL}$) or below the lower control limit ($\mathit{LCL}$) of the basic $\overline{X}$ scheme and the $\mathit{CRL}$ value is smaller or equal to some positive integer $W>$ 0. The threshold $W$ represents the lower limit of the CRL sub-chart. Wu and Spedding (2000) found that the basic synthetic $\overline{X}$ scheme not only outperforms the basic $\overline{X}$ scheme regardless of the magnitude of the shift (or change) in the process mean, it also outperforms the $\overline{X}$-EWMA in detecting moderate and large shifts in the process mean. More recently, Haq, Brown, and Moltchanova (2015) proposed synthetic control schemes for monitoring the process mean and dispersion based on the ranked set sampling and its modifications. Lee and Khoo (2017) designed a synthetic |S| chart for monitoring the process dispersion. Guo et al. (2015) proposed an optimal design of a synthetic scheme for monitoring the process dispersion with an unknown in-control variance. Attracted by the strengths of the synthetic scheme, researchers extended the use of the synthetic chart to different types of control schemes such as the EWMA and CUSUM (see, for example, Haq, Brown, and Moltchanova (2016)); including the two stage monitoring schemes such as the univariate and multivariate double sampling (DS) schemes (cf. Khoo et al., 2013, Teoh et al., 2014, You et al., 2015, You, 2017), to name a few.

The original DS scheme is a two-stage scheme based on unconnected samples (cf. Croasdale, 1974). To improve Croasdale (1974)’s DS scheme, Daudin et al., 1990, Daudin, 1992) developed DS control schemes based on two connected samples. Daudin (1992) showed that the DS scheme is superior to the basic Shewhart-type scheme. Khoo, Lee, Wu, Chen, and Castagliola (2011) proposed a synthetic DS (hereafter SDS) scheme for monitoring the process mean. They showed that the SDS is more sensitive to process shifts than the synthetic Shewhart scheme. Khoo et al. (2013) proposed a SDS scheme using the Hotelling $T^2$ statistic of a multivariate process. The SDS np monitoring scheme for attributes data was designed by Chong, Khoo, and Castagliola (2014). More recently, You (2017) studied the performance of the SDS $\overline{X}$ in terms of the average run-length ($\mathit{ARL}$) and the expected $\mathit{ARL}$ (denoted by $\mathit{EARL}$). Lee and Khoo (2016) developed a SDS scheme for monitoring the process standard deviation of a univariate process. Costa and Machado (2015) used a Markov chain approach to investigate the steady-state performance of non-side-sensitive (NSS) and the revised side-sensitive (RSS) SDS $\overline{X}$ schemes. They found that the RSS SDS scheme performs better than the NSS scheme. For more details on the variety of synthetic Shewhart schemes, readers are referred to Shongwe and Graham, 2016, Shongwe and Graham, 2018.

Let us acknowledge that synthetic charts have received a lot of criticism in the literature (especially by Knoth (2016)). Knoth (2016) advised against the use of synthetic charts, however, Knoth (2016) only considered one type of synthetic scheme and, it has been shown in Shongwe and Graham (2016), that there are actually four types of synthetic charts and that the other three types outperform the type considered by Knoth (2016). It is highly recommended that the use of synthetic schemes be investigated further, i.e. a thorough investigation of the other three types of synthetic schemes should be done and compared to Knoth (2016)’s findings. Thus, it is of our opinion that synthetic schemes should not yet be discarded, as recommended by Knoth (2016), and the abovementioned reasons are motivations to continue developing synthetic monitoring schemes even after Knoth (2016)’s warning not to do so. It is of great interest to investigate the performance of a synthetic scheme based on each of the four possible designs i.e. the NSS, standard side-sensitive (SSS), RSS and modified side-sensitive (MSS). Shongwe and Graham (2016) reported that SSS and RSS designs have similar performances (with the RSS marginally better) but the design structure of SSS schemes is way more complicated and time consuming than that of RSS schemes. However, the focus should be on the MSS design since it outperforms the other three designs (cf Shongwe & Graham, 2016).

In statistical process monitoring (SPM), control schemes are mostly designed for monitoring either the mean (cf. Wu et al., 2010, Costa and Machado, 2015, You, 2017, Lee and Khoo, 2018, Saha et al., 2018) or the standard deviation (or variance) (cf. He and Grigoryan, 2002, He and Grigoryan, 2006, Guo et al., 2015, Lee and Khoo, 2017) of the process. Monitoring the process mean alone would mean ignoring the changes in the process variance (or standard deviation) despite the latter being closely linked to the mean. Moreover, it is well known that the variance (or standard deviation) can be greatly affected when the mean value gives a poor measure of central tendency. Therefore, it would be prudent to monitor both the process mean and variance at the same time for an effective monitoring scheme design. Motivated by this, many authors recommended the investigation of both mean and variance simultaneously. Works for joint monitoring of the process mean and variance (or standard deviation) were discussed by Costa and Rahim, 2006, Costa and De Magalhães, 2007, Hawkins and Deng, 2009, Teh et al., 2012, Tasias and Nenes, 2012, Haq et al., 2015, Ou et al., 2015, to count a few. For an overview on the earlier literature covering the joint monitoring of the process mean and variance, readers are referred to the review paper by McCracken and Chakraborti (2013).

Up to this day, very few researchers have shown interest in designing DS and SDS schemes for a joint monitoring of the process mean and variance (see for example, Lee (2013)), and none has shown interest in designing side-sensitive SDS scheme for a joint monitoring of both process mean and variance (or standard deviation) in both the zero- and steady-state modes. Since assignable causes can result in a change in either the process mean or variance or in both mean and variance (see for example, DeVor, Chang, & Sutherland, 2007), the design of joint monitoring schemes is very useful in order to avoid many false alarms. Therefore, the aim of this paper is to propose a MSS SDS monitoring scheme, for a joint statistical monitoring of the process mean and standard deviation. Many authors reported that synthetic schemes perform poorly under the steady-state scenario (see for example Costa and Machado, 2015, Knoth, 2016). Consequently, most of the research works on these types of schemes are based on the zero-state mode only. However, in this paper, we investigate both the zero-state and steady-state properties of the proposed control scheme.

The remainder of this paper is organised as follows: Section 2 introduces the operation of the new monitoring scheme. In Section 3, closed-form expressions of the zero-state and steady-state characteristics of the run-length are derived and measures of the overall performance are given. Section 4 discusses the performance of the proposed MSS SDS monitoring scheme for a joint monitoring of the process mean and standard deviation. Moreover, in Section 4, the proposed scheme is compared with other monitoring schemes. A real-life example is given in Section 5 to illustrate the design and implementation of the proposed scheme. Section 6 gives a summary and some recommendations.