Elsevier

Computers & Industrial Engineering

Volume 135, September 2019, Pages 426-439
Computers & Industrial Engineering

A comparative study of some EWMA schemes for simultaneous monitoring of mean and variance of a Gaussian process

https://doi.org/10.1016/j.cie.2019.06.021Get rights and content

Highlights

  • EWMA schemes based on the ‘max’ and the ‘distance’ combining functions are introduced.

  • Two different ways of introducing the functions into the EWMA structure are examined.

  • Results show that the distance-type schemes outperform the max-type schemes.

  • The proposed schemes appear to be more useful than some existing schemes.

  • Two industrial datasets are used to show the implementation strategies of the schemes.

Abstract

In this paper, we introduce four different combinations of EWMA schemes, each based on a single plotting statistic for simultaneous monitoring of the mean and variance of a Gaussian process. We compare the four schemes and address the problem of adopting the best combining mechanism. We consider that the actual process parameters are unknown and estimated from a reference sample. We take into account the effects of estimation of unknown parameters in designing the proposed schemes. We consider the maximum likelihood estimators based pivot statistics for monitoring both the parameters and combine them into a single statistic through the ‘max’ and the ‘distance’ type combining functions. Also, we examine two different adaptive approaches to introduce pivot statistics into the EWMA-structure. Results show that the distance-type schemes outperform the max-type schemes. Generally, the proposed schemes are useful in detecting small-to-moderate shifts in either or both of the process parameters. Computational studies reveal that the proposed schemes can identify a process shift more quickly compared to some of the existing schemes. We illustrate the implementation strategies of the schemes using two industrial datasets.

Introduction

Statistical process monitoring (SPM) comprises techniques to study and assess the quality of manufacturing and other industrial processes. During the days of the early development of process monitoring schemes, practitioners used to apply the SPM schemes mainly in monitoring the item quality in production lines. In recent years, applications process monitoring techniques have reached various other business domains, for example, services and health sectors, among others (Mukherjee, 2017, Qiu, 2014). In most cases, manufacturing processes assume that the underlying process distribution is normal. The normal distribution has two parameters; namely, the location and scale parameters. A large number of process monitoring schemes are intended for monitoring the location parameter only (Adegoke et al., 2017, Sanusi et al., 2017, Sanusi et al., 2018, Zwetsloot et al., 2016). Significant research has also been done on monitoring the scale parameter of a process (Abbasi et al., 2012, Sanusi et al., 2017, Zaman et al., 2016). Several researchers have recommended monitoring both location and scale parameters simultaneously to ensure the stability of the process. It is worth noting that some process distributions, for example, time to event processes, do not follow the normality assumption. One should not use the SPM schemes designed for Gaussian processes in such situations. It is well-known that the performance of SPM schemes based on the normality assumption is severely affected when this assumption is not valid. See, for example, Qiu and Li (2011). In the absence of appropriate knowledge about the underlying process distribution, one should use the distribution-free schemes. See, for example, Graham et al., 2012, Mukherjee, 2017, Chong et al., 2017, Song et al., 2019. In this paper, we, however, consider a parametric set-up and assume that the normality assumption is valid.

Simultaneous monitoring of location and scale parameters are of paramount importance primarily when particular causes of variation affect a process, resulting in concurrent shifts in the location and the scale parameters. For example, a wrongly fixed stencil in a circuit manufacturing may lead to a change in the mean and variance of the thickness of the solder paste printed on circuit boards (Gan et al., 2004, Li et al., 2016). Joint monitoring of process parameters also has widespread applications in the fields like climate dynamics (Zhang, Xu, & Yang, 2009), supply chain (Shu & Barton, 2012), maintenance (Morales, 2013), and service management (Mukherjee & Marozzi, 2017). The location parameter, for example, the mean of a Gaussian process and the scale parameter, for example, the standard deviation of the same process, are sometimes jointly monitored with two different schemes following a two-chart scheme. As opposed to the two-charting mechanism, a single charting scheme is more effective, worthy and attractive. For more details, see, for example, McCracken, Chakraborti, and Mukherjee (2013), and Li et al. (2016). For a good overview of joint monitoring schemes, interested readers may see Cheng and Thaga, 2006, Mccracken and Chakraborti, 2013. The simplicity in the implementation of a single charting scheme makes it relevant and applicable in many sectors such as healthcare, manufacturing, among others.

In this article, we propose some exponentially weighted moving average (EWMA) based single charting schemes for joint monitoring of parameters of Gaussian processes. The EWMA scheme has an excellent performance in monitoring different manufacturing processes. Practitioners use the EWMA schemes in monitoring numerous stochastic processes, such as, the production process of a cylindrical bores of an engine block (Chen, Cheng, & Xie, 2001); the footwear manufacturing process (Jiang, Wang, & Tsung, 2012) in a production line; the sintering process that manufactures mechanical parts (Yeong, Khoo, Tham, Teoh, & Rahim, 2017); the non-isothermal continuous tank chemical reactor model (Sanusi, Riaz, Adegoke, & Xie, 2017); the condition monitoring of an electro-pump (Lampreia, Requeijo, Dias, Vairinhos, & Barbosa, 2018); among others. It is useful for detecting small and persistent variations in a process parameter. Some classical works on the improvement of this scheme may be found in Zou et al., 2007, Liu et al., 2007. To this end, two types of monitoring strategies are possible; Phase-I and Phase-II. The Phase-I analysis aims at studying a process average, variability, and overall stability. We use the reference sample selected via appropriate Phase-I analysis as a gold standard, and often compare the incoming sequence of test samples in respect of such gold standard in Phase-II. Interested readers may see Jones-Farmer et al., 2014, Li et al., 2019 and the references therein for more understanding of phase-I analysis.

In this article, we focus only on Phase-II analysis and subsequently, we assume that an appropriate reference sample of size m is available. A large number of Phase-II analyses assume that the true process parameters are known (Case-K) or they are precisely estimated from Phase-I sample. See, for example, Razmy, 2005, Mukherjee et al., 2015. In real life scenarios, more often process parameters are unknown and are estimated from a reference sample. Thereafter, the estimate of a parameter itself is considered as the true value of the same parameter just as the gold standard (Mukherjee, 2017). Nevertheless, there is always an effect of estimation of parameters on the run-length distribution. For example, Saleh, Mahmoud, and Abdel-Salam (2013) showed that the performance of an adaptive EWMA control scheme becomes worse when parameters are estimated. Also, plugging estimated parameters in a Phase-II, Case-K analysis often increases false alarm rate and reduces the in-control (IC) average run length (ARL) performance of the SPM schemes. To minimize the effect of estimation bias and to maintain the desired IC performance, we often require a reference sample of large size (Aly, Saleh, Mahmoud, & Woodall, 2015). It is sometimes difficult in practice to obtain such a large number of Phase-I observations. This problem has long been recognized. For example, Shewhart (1939) wrote that “In the majority of practical instances, the most difficult job of all is to choose the sample that is to be used as the basis for establishing the tolerance range (control limits)”. To circumvent this problem McCracken et al. (2013) introduced Shewhart-type control scheme for joint monitoring of mean and variance of a Gaussian process assuming that the process parameters are unknown and taking the effect of estimation into account. We classify such modified schemes as the Phase-II, Case-U type schemes. Li et al. (2016) considered CUSUM version of the Phase-II, Case-U schemes designed by McCracken et al. (2013).

In this study, we propose four Phase-II EWMA-type control schemes for the simultaneous monitoring of the mean and variance of a Gaussian process under Case-U. The schemes are based on specific single plotting statistics. The process parameters are assumed to be unknown and are estimated from a suitable reference sample. Further, we examine the effects of parameter estimation on the performance of Case-K control schemes. Subsequently, we suggest some remedial measures. We also discuss and compare two different adaption mechanisms, one based on the maximum of two (max-type) and the other based on the Euclidean distance between two (distance-type), (Haq, 2017, Javaid et al., 2018, Sanusi and Mukherjee, 2019), to combine the individual monitoring statistics for mean and variance. We also examine and compare two different approaches of applying the adaptive techniques during the implementation of the EWMA schemes. In the first approach, we initially introduce the two suitably transformed individual EWMA schemes, one for the mean and the other for the variance, and subsequently, combine them using either the max-type or distance-type measures. In the second approach, we first combined the two appropriately transformed individual statistics, one for monitoring the mean and the other for the variance, using either the max-type or the distance-type adaptive tools and subsequently design the EWMA schemes based on the single combined statistic.

The rest of this article is arranged as follows: Section 2 gives the details of the proposed schemes, studies the effect of parameter estimation on Case-K control schemes, and subsequently gives some remedial measures. In Section 3, we present the design and implementation steps of the four EWMA schemes in four subsections and afterwards discuss the problem of determination of control limits. Section 4 contains our findings of numerical studies via Monte-Carlo. In this context, we compare the proposed schemes with their existing counterparts, study the effect of the combining mechanisms and also investigate the impact of sample size. We illustrate the proposed schemes in Section 5 using two real datasets from manufacturing industries. Finally, Section 6 offers a summary and conclusion with some possible directions for future research.

Section snippets

Development of the monitoring statistics

Let X1,X2,,Xj,,Xm be a reference sample of size m related to a certain quality characteristic from an IC process. Similarly, let the set of random observations Y1,Y2,,Yk,,Yn of size n be the test sample of the quality characteristic. As in Chen et al., 2001, Khoo et al., 2010, McCracken et al., 2013, and Li et al. (2016), suppose that X's and Y's follow a normal distribution with means μX and μY, respectively, and variances σX2 and σY2, respectively, where -<μX,μY< and σX2,σY2>0. Further,

Design and implementation of the proposed charting schemes

This section outlines the step-by-step charting procedures of the proposed modified EWMA schemes for monitoring unknown mean and variance of a normally distributed process. Subsequently, we explain the computational methods of the UCL values of our proposed plans.

Performance analysis and comparisons

In this Section, we comprehensively analyse and compare the performance of the proposed schemes. We examine the performance of the different combining approaches used in constructing the proposed schemes. We compare different adaptive strategies for constructing EWMA schemes based on the two combining mechanisms. We also compare the performance of the newly designed EWMA schemes with their existing Shewhart and CUSUM counterparts. Moreover, we examine the effect of sample size on the proposed

Illustrative examples

This section presents illustrative examples by using real datasets from a forged automobile engine piston rings and from a medium-sized timber industry that manufactures plastic plywood.

Summary and conclusion

We introduce four variations of the EWMA-type parametric schemes for the joint monitoring of mean and variance of a normally distributed process and compare their relative performances. We assume a more general situation that the true process parameters are unknown. We propose to estimate unknown parameters from a suitable reference sample from Phase-I and develop modified SPM schemes to circumvent the effects of estimation of parameters on the performance of these schemes.

Our proposed SPM

Acknowledgement

Authors are grateful to the two anonymous reviewers for their careful reading of the earlier version of the manuscript and also for their valuable comments and suggestions that lead to significant improvement.

This study was supported by a grant from University Grants Committee of Hong Kong (CityU 11213116) and National Natural Science Foundation of China under a key Project (No. 71532008).

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