Change point estimation of high-yield processes experiencing monotonic disturbances

doi:10.1016/j.cie.2013.11.003

Computers & Industrial Engineering

Volume 67, January 2014, Pages 82-92

https://doi.org/10.1016/j.cie.2013.11.003 Get rights and content

Highlights

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A MLE is proposed to estimate monotonic change points of high-yield processes.
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The proposed estimator can be used without prior knowledge of the exact change type.
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The PAV algorithm is used to estimate the out-of-control parameter of the process.
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The proposed estimator works well under different change types.
•
The applicability of the proposed method is illustrated using a real data example.

Abstract

In this paper, we first propose a maximum likelihood estimator (MLE) of a change point in high-yield processes, where the only assumption is that the change belongs to a family of monotonic changes. Following a signal from the cumulative count of conforming (CCC) control chart, the performance of the proposed monotonic change-point estimator is next evaluated by comparing its performances to the ones designed for step-changes and linear-trend disturbances through extensive simulation experiments involving different single step-changes, linear-trend disturbances, and multiple-step changes. The results show that when the type of change is not known a priori, using the proposed change-point estimator is useful, because it provides accurate and precise estimates of the change points for almost all of the shift magnitudes and all of the change types considered in this paper. In addition, the applicability of the proposed method is illustrated using a real case.

Section snippets

Introduction and literature review

Control charts are one of the most important statistical process control (SPC) tools in industries used to monitor the state of processes and detect the shifts by distinguishing between common and special causes of process variation. Although control charts are very useful in determining out-of-control states of processes, they do not determine the time the process had really changed (change point) or provide specific information on the root causes of process variation. When a control chart

High-yield process monitoring and derivation of the MLE

Consider a monotonic-change disturbance model for the behavior of the process fraction non-conforming of a geometric process, in which a stream of independent Bernoulli data using a possible 100% inspection is available. We assume that the process is initially in-control where observations come from a Bernoulli process with known in-control fraction non-conforming parameter p = p₀ parts per million. To overcome the problem of employing either p or np chart in high-yield processes (Xie and Goh,

Comparison of the change point estimators

In this section, some Monte Carlo simulation experiments are used to evaluate the performance of the proposed change-point estimator designed for monotonic-change disturbances, ${\hat{τ}}_{ISO}$ , by comparing it to the change-point estimator of Noorossana et al. (2009), ${\hat{τ}}_{sc}$ , designed for step-changes and the change-point estimator of Niaki and Khedmati (2013), ${\hat{τ}}_{lt}$ , designed for linear-trend disturbances. Three types of non-decreasing changes including step-changes, linear-trend disturbances, and multiple

A real case

In this section, a real-world case provided by Chen et al. (2011) is used to illustrate the implementation of the proposed method. The data-set comes from an injection molding process that produces the micro-prism array of optical elements with an in-control non-conformity proportion at the level of p₀ = 5 × 10⁻⁶. Based on Type-I error (α) of 0.0027, the control limits of the control chart are obtained as: $UCL = \frac{\ln (α / 2)}{\ln (1 - p_{0})} = \frac{\ln (0.0027 / 2)}{\ln (1 - 0.000005)} = 1321526.8$ $LCL = 1 + \frac{\ln (1 - α / 2)}{\ln (1 - p_{0})} = 1 + \ln (1 - 0.0027 / 2$

Concluding remark

Although many researchers assume the type of the process change is known a priori, the type of the change is rarely known and consequently any deviation in its true form from the assumed change type is likely to affect the performances of the change-point estimator. Instead, process engineers may know the change belongs to a family of monotonic (non-decreasing or non-increasing) changes including single-step changes, linear-trend disturbances, nonlinear trends, multiple-step changes, and an

Acknowledgments

The authors are thankful for the constructive comments of anonymous reviewers. Taking care of the comments certainly improved the presentation.

References (37)

Y.K. Chen
Cumulative conformance count charts with variable sampling intervals for correlated samples
Computers & Industrial Engineering
(2013)
J.Y. Liu et al.
Cumulative count of conforming chart with variable sampling intervals
International Journal of Production Economics
(2006)
K. Atashgar et al.
Diagnosing the source(s) of a monotonic change in the process mean vector
International Journal of Advanced Manufacturing Technology
(2012)
M. Ayer et al.
An empirical distribution function for sampling with incomplete information
Annals of Mathematical Statistics
(1955)
M.J. Best et al.
Active set algorithms for monotonic regression: A unifying framework
Mathematical Programming
(1990)
T.W. Calvin
Quality control techniques for zero defects
IEEE Transactions on Components, Hybrids, and Manufacturing Technology
(1983)
L.Y. Chan et al.
Cumulative probability control charts for geometric and exponential process characteristics
International Journal of Production Research
(2002)
T.C. Chang et al.
Cumulative sum charts for high yield processes
Statistica Sinica
(2001)
Y.K. Chen et al.
Cumulative conformance count chart with variable sampling intervals and control limits
Applied Stochastic Models in Business and Industry
(2011)
H.M. Fahmy et al.
Drift time detection and adjustment procedures for processes subject to linear trend
International Journal of Production Research
(2006)

D.M. Hawkins et al.

Cumulative sum charts and charting for quality improvements

(1988)

Y. He et al.

A new statistical high-quality process monitoring method: The counted number between omega-event control charts

Quality and Reliability Engineering International

(2012)

F.C. Kaminsky et al.

Statistical control charts based on a geometric distribution

Journal of Quality Technology

(1992)

V. Kuralmani et al.

A conditional decision procedure for high-yield processes

IIE Transactions

(2002)

E. Mavroudis et al.

EWMA control charts for monitoring high yield processes

Communications in Statistics—Theory and Methods

(2013)

S.T.A. Niaki et al.

Change point estimation of high-yield processes with a linear trend disturbance

International Journal of Advanced Manufacturing Technology

(2013)

R. Noorossana et al.

Identifying the period of a step change in high-yield processes

Quality and Reliability Engineering International

(2009)

R. Noorossana et al.

Estimating the change point of a normal process mean with a monotonic change

Quality and Reliability Engineering International

(2009)

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However, as this paper focuses on the application change point analysis in statistical process control and more specifically in simple linear profile monitoring, we only report research on change point identification in statistical process control. Most of these investigatins use maximum likelihood to tackle the problem of change point identification (Samuel et al. [38,39]), Hawkins et al. [40], Pignatiello and Samuel [41], Pignatiello and Simpson [42], Pery and Pignatiello [43], Niaki and Khedmati [44,45], Dogu [46], Movaffagh and Amiri [47]). Beside maximum likelihood method, non-parametric methods (Zhou et al. [48], Ross et al. [49], Ross and Adams [50], Huang et al. [51]) and approaches based on new techniques, such as Artificial Neural Networks, Fuzzy Clustering, etc., are also proposed to estimate the change point in statistical process control problems (Wu [52], Fazel Zarandi et al. [53], Ghazanfari et al. [54], Alaeddini et al. [55], Harnish et al. [56], Fazel Zarandi and Alaeddini [57], Atashgar and Noorossana [58], Ghiasabadi et al. [59], Amiri et al. [60], Kazemi et al. [61], Lu et al. [62]).
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Knowing the real time of changes, called change-point, in a process is essential for quickly identifying and removing special causes. Many change-point methods in statistical process control assume the distribution and the in-control parameters of the process known, however, they are rarely known accurately. Small errors accompanied with estimated parameters may lead to unfavorable change-point estimates. In this paper, a new method, called fuzzy shift change-point algorithm, which does not require the knowledge of the distribution nor the parameter of the process, is proposed to detect change-points for shifts in process mean. The fuzzy c-partition concept is embedded into change-point formulation in which any possible collection of change-points is considered as a partitioning of data with a fuzzy membership. These memberships are then transferred into the pseudo memberships of observations belonging to each individual cluster, so the fuzzy c-means clustering can be used to obtain the estimates for shifts. Subsequently, the fuzzy c-means algorithm is used again to obtain new iterates of change-point collection memberships by minimizing an objective function concerning the deviations between observations and the corresponding cluster means. The proposed algorithm is nonparametric and applicable to normal and non-normal processes in both phase I and II. The performance of the proposed fuzzy shift change-point algorithm is discussed in comparison with powerful statistical methods through extensive simulation studies. The results demonstrate the superiority and usefulness of our proposed method.
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View full text

Change point estimation of high-yield processes experiencing monotonic disturbances☆

Highlights

Abstract

Section snippets

Introduction and literature review

High-yield process monitoring and derivation of the MLE

Comparison of the change point estimators

A real case

Concluding remark

Acknowledgments

Computers & Industrial Engineering

International Journal of Production Economics

Diagnosing the source(s) of a monotonic change in the process mean vector

International Journal of Advanced Manufacturing Technology

An empirical distribution function for sampling with incomplete information

Annals of Mathematical Statistics

Active set algorithms for monotonic regression: A unifying framework

Mathematical Programming

Quality control techniques for zero defects

IEEE Transactions on Components, Hybrids, and Manufacturing Technology

Cumulative probability control charts for geometric and exponential process characteristics

International Journal of Production Research

Cumulative sum charts for high yield processes

Statistica Sinica

Cumulative conformance count chart with variable sampling intervals and control limits

Applied Stochastic Models in Business and Industry

Drift time detection and adjustment procedures for processes subject to linear trend

International Journal of Production Research

Cumulative sum charts and charting for quality improvements

A new statistical high-quality process monitoring method: The counted number between omega-event control charts

Quality and Reliability Engineering International

Statistical control charts based on a geometric distribution

Journal of Quality Technology

A conditional decision procedure for high-yield processes

IIE Transactions

EWMA control charts for monitoring high yield processes

Communications in Statistics—Theory and Methods

Change point estimation of high-yield processes with a linear trend disturbance

International Journal of Advanced Manufacturing Technology

Identifying the period of a step change in high-yield processes

Quality and Reliability Engineering International

Estimating the change point of a normal process mean with a monotonic change

Quality and Reliability Engineering International