Dual multivariate CUSUM mean charts

https://doi.org/10.1016/j.cie.2019.106028Get rights and content

Highlights

  • New dual and mixed dual multivariate CUSUM charts are proposed for the process mean.

  • The proposed charts are more sensitive than the existing charts.

  • Real datasets are used to explain the implementation of proposed charts.

Abstract

The multivariate CUSUM (MCUSUM) chart can be optimally designed to detect a specific shift in the process mean. In practice, the shift size is rarely known but it is known that it varies within a given interval. Thus, the MCUSUM chart may not perform well when detecting a range of the mean shift sizes. To overcome this issue, in this paper, we propose new dual MCUSUM (DMCUSUM) and mixed DMCUSUM (MDMCUSUM) charts for monitoring the mean of a multivariate normal process. The DMCUSUM (MDMCUSUM) chart integrates two similar (different) type MCUSUM charts into a single chart to provide an overall good detection for different sizes of shift in the process mean. The run length characteristics of these DMCUSUM charts are computed using the Monte Carlo simulation method. A detailed comparative study of the proposed and existing multivariate charts in terms of the average run length (ARL), extra quadratic loss and integral relative ARL is shown to be favorable to the former. Two real datasets are considered to explain the implementation of the proposed multivariate charts.

Introduction

Statistical Process Control (SPC) is a set of problem-solving tools that may be applied to any production/manufacturing process to monitor, control, improve the quality of the output, and increase process productivity. The aim of SPC is to establish a controlled manufacturing process by the use of statistical techniques to reduce process variation. The control chart is an important tool of SPC. It is extensively used in many manufacturing and service industries to track irregular changes in the process parameter(s). In many manufacturing industries, there are situations where joint monitoring of several quality characteristics is crucial. For example, the pharmaceutical industry employs batch processing with laboratory assays, including the identification of raw materials, homogeneity, moisture, particle size, hardness, dissolution testing, chemical composition, etc., carried out on finished products to evaluate the quality control. Hence, multivariate control charts are used to monitor the overall quality of a process (cf., Montgomery, 2009).

The multivariate charts are extensions of their univariate counterparts. The multivariate CUSUM (MCUSUM) and multivariate EWMA (MEWMA) charts are prime examples, to name a very few. Two MCUSUM charts were first suggested by Crosier (1988), called the Crosier MCUSUM (CMCUSUM) charts, for monitoring the mean of a multivariate normal process. Following Crosier, 1988, Pignatiello and Runger, 1990 have also suggested two MCUSUM charts for monitoring the process mean, called the Pignatiello and Runger MCUSUM (PRMCUSUM) charts. Hereafter, the CMCUSUM and PRMCUSUM charts are referred to as the C and P charts, respectively. Later, Lowry, Woodall, Champ, and Rigdon (1992) suggested an MEWMA chart for monitoring the process mean. Lee and Khoo (2006) suggested optimal statistical designs for the C chart based on the average run length (ARL) and median run length. Bersimis, Psarakis, and Panaretos (2007) have provided a comprehensive literature review of the multivariate charts for monitoring process parameters. Dai et al., 2011, Wang and Huang, 2016 proposed adaptive versions of the C and P charts for monitoring the process mean, respectively. Recently, Ajadi and Riaz (2017) have designed the C and P charts using the MEWMA statistic for monitoring the process mean. They have shown that their proposed charts are more sensitive than the existing counterparts when detecting very small shifts in the process mean. Adegoke, Smith, Anderson, Abbasi, and Pawley (2018) used shrinkage estimators of the covariance matrix to improve the run length performances of the MCUSUM dispersion charts. Adaptive multivariate double sampling and variable sampling interval Hotelling’s T2 charts have been suggested by Khatun, Khoo, Yeong, Teoh, and Chong (2018). In a recent work, Haq and Khoo (2019a) have considered an auxiliary-information-based estimator for the mean vector of a multivariate normal process, and used it to devise enhanced memory-type multivariate charts for the process mean. They have shown that the AIB MCUSUM and AIB MEWMA charts are more sensitive than their existing counterparts. In another work, Haq and Khoo (2019b) have suggested adaptive EWMA charts for monitoring the univariate and multivariate coefficients of variation when sampling from normal and multivariate normal processes, respectively. Adegoke, Abbasi, Smith, Anderson, and Pawley (2019) have proposed multivariate homogeneously weighted moving average chart for monitoring the mean vector of a multivariate normally distributed process. Using fixed and variable sample sizes, Chong, Khoo, Haq, and Castagliola (2019) constructed Hotelling’s T2 charts for monitoring the process mean vector in short production runs. For more related works on the multivariate charts, we refer to Woodall and Ncube, 1985, Healy, 1987, Qiu and Hawkins, 2001, Alkahtani and Schaffer, 2012, Lee et al., 2014, Abdella et al., 2017, Haq, 2018, Abbasi and Adegoke, 2018, and the references cited therein.

A shortcoming of C and P charts is that they can only be used when the shift sizes are known in advance against which the protection is required. However, in practice, the shift sizes are not known but it is expected that they lie within a certain interval. Thus, these multivariate charts may not perform well when detecting a range of shift sizes. In order to overcome this issue, following Zhao, Tsung, and Wang (2005), we integrate two same-type or different-type MCUSUM charts into a single chart, named the dual MCUSUM (DMCUSUM) and mixed DMCUSUM (MDMCUSUM) charts, for monitoring the infrequent changes in the mean of multivariate normal process. The DMCUSUM or MDMCUSUM chart uses one MCUSUM chart for detecting small-to-moderate shifts whilst the other MCUSUM chart for detecting moderate-to-large shifts. The DMCUSUM chart combines either two C charts, called the dual C (DC) chart, or two P charts, called the dual P (DP) chart, into a single chart. Similarly, the MDMCUSUM chart combines C and P charts, called the dual Crosier and Pignatiello and Runger MCUSUM (CP) chart, or P and C charts, called the dual Pignatiello and Runger and Crosier MCUSUM (PC) chart, into a single chart. Using extensive Monte Carlo simulations, the ARL, extra quadratic loss (EQL) and integrate relative ARL (IRARL) profiles of the proposed multivariate charts are computed. It is shown that the DMCUSUM and MDMCUSUM charts are more sensitive than the existing MCUSUM charts when detecting a range of mean shift sizes.

The rest of the paper is organized as follows: In Section 2, some existing MCUSUM charts are briefly reviewed. In Section 3, the proposed MCUSUM charts are given. The run length computation and comparisons are given in Section 4. The proposed MCUSUM charts are applied on real datasets in Section 5. Finally, Section 6 summarizes the main findings and concludes the paper.

Section snippets

The existing MCUSUM charts

In this section, a brief overview of some existing MCUSUM charts is given when the interest lies in detecting small-to-moderate shifts in the process mean and the quality characteristics to be monitored jointly follow a multivariate normal distribution.

Let X=[X1,X2,,Xp]1×p be a p×1 vector that represents p2 quality characteristics, which are to be monitored simultaneously. Assume that the process {Xt} for t1 follows a multivariate normal distribution with the mean vector μ and the

The proposed MCUSUM charts

An important shortcoming of the C and P charts is that they can only be used when the mean shift sizes against which the protection is required are known in advance. In practice, however, the shift sizes may not be known. Thus, these MCUSUM charts may not perform well when detecting a range of the mean shift sizes. It is thus vital to have such an MCUSUM chart that can effectively guard against a range of mean shift sizes. Getting motivation from the work of Zhao et al. (2005) on the univariate

Run length evaluation and comparison

In this section, we compute the ARL profiles of the proposed and existing control charts using extensive Monte Carlo simulations. A detailed comparative study is also conducted in order to gain insight into the detection abilities of the considered multivariate charts.

Generally, the performance of a control chart is evaluated in terms of its run length characteristics, which include the ARL, EQL, IRARL, etc. The ARL is a frequently used performance measure when comparing the run length

Real data application

It is a common practice to apply the proposed control charts on simulated and/or real datasets. In what follows, two real datasets are considered here to explain the implementation of existing and proposed multivariate charts.

The first real dataset is taken from Santos-Fernández (2013) that was originated from a process capability study for turning aluminium pins. The multivariate data on six diameter and length measurements were recorded from seventy pins, where the first three variables are

Conclusions

The optimal MCUSUM chart can only detect a specific shift in the process mean. In practice, the shift sizes vary within a given interval. Thus, the MCUSUM chart may perform worse than optimal when detecting a range of the mean shift sizes. To overcome this issue, in this paper, we have proposed DMCUSUM and MDMCUSUM charts for monitoring the mean of a multivariate normal process. The DMCUSUM (MDMCUSUM) chart integrates two similar (different) MCUSUM charts into a single chart to provide an

Acknowledgements

The authors are thankful to three anonymous reviewers for providing useful comments that led to an improved version of the article.

References (26)

  • N.A. Adegoke et al.

    Shrinkage estimates of covariance matrices to improve the performance of multivariate cumulative sum control charts

    Computers & Industrial Engineering

    (2018)
  • A. Haq et al.

    New adaptive EWMA control charts for monitoring univariate and multivariate coefficient of variation

    Computers & Industrial Engineering

    (2019)
  • S.A. Abbasi et al.

    Multivariate coefficient of variation control charts in phase I of SPC

    The International Journal of Advanced Manufacturing Technology

    (2018)
  • G.M. Abdella et al.

    Variable selection-based multivariate cumulative sum control chart

    Quality and Reliability Engineering International

    (2017)
  • N.A. Adegoke et al.

    A multivariate homogeneously weighted moving average control chart

    IEEE Access

    (2019)
  • J.O. Ajadi et al.

    Mixed multivariate EWMA-CUSUM control charts for an improved process monitoring

    Communications in Statistics – Theory and Methods

    (2017)
  • S. Alkahtani et al.

    A double multivariate exponentially weighted moving average (dEWMA) control chart for a process location monitoring

    Communications in Statistics – Simulation and Computation

    (2012)
  • S. Bersimis et al.

    Multivariate statistical process control charts: An overview

    Quality and Reliability Engineering International

    (2007)
  • N.L. Chong et al.

    Hotelling’s T2 control charts with fixed and variable sample sizes for monitoring short production runs

    Quality and Reliability Engineering International

    (2019)
  • R.B. Crosier

    Multivariate generalizations of cumulative sum quality-control schemes

    Technometrics

    (1988)
  • Y. Dai et al.

    A new adaptive CUSUM control chart for detecting the multivariate process mean

    Quality and Reliability Engineering International

    (2011)
  • A. Haq

    Weighted adaptive multivariate CUSUM control charts

    Quality and Reliability Engineering International

    (2018)
  • A. Haq et al.

    Memory-type multivariate control charts with auxiliary information for process mean

    Quality and Reliability Engineering International

    (2019)
  • Cited by (0)

    View full text