Two perspectives for designing a phase II control chart with estimated parameters: The case of the Shewhart X¯ Chart

Jardim, Felipe Schoemer; Chakraborti, S.; Epprecht, Eugenio K.

doi:10.1080/00224065.2019.1571345

Cited by 49 publications

(55 citation statements)

References 23 publications

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“…Albers and Kallenberg defined the “exceedance probability” (denoted by EP ) as the probability that a performance measure of interest exceeds some specified limit. Some authors (eg, Jardim et al) have studied the particular EP given by

P ((), {CARL}_{0} \geq {ARL}_{0}^{Lim})

and have defined acceptable performances in terms of this probability; in other words, they required that

P ((), {CARL}_{0} \geq {ARL}_{0}^{Lim}) = 1 - p,

where p is a specified low probability such as .10, for example, and where, most often, the smaller tolerated CARL 0 ,

{ARL}_{0}^{Lim}

has been defined in terms of a percent reduction relative to the nominal ARL 0 (

{ARL}_{0}^{Nom}

), ie,

{ARL}_{0}^{Lim} = ((), 1 - ε_{0}) {ARL}_{0}^{Nom},

so that

{ARL}_{0}^{Lim}

is 100 ε 0 % smaller than the nominal

{ARL}_{0}^{Nom}

.…”

Section: Background and Underlying Assumptionsmentioning

confidence: 99%

“…This choice is justified by the recommendation of Mahmoud et al 22 of using either S p , S p /c 4 , or c 4 S p (where c 4 has argument m(n − 1)+1) instead of the estimator S=c 4 (where c 4 has argument n) that is the usual practice. We stick thus to S p for simplicity, as did recent authors on this topic of the effect of parameter estimation on control charts performance (eg, Castagliola et al, 23 Epprecht et al 11 , Faraz et al, 13,14 Jardim et al, 17 Loureiro et al, 18 and Guo and Wang 19 ). The estimator S p is given by the following:…”

Section: Background and Underlying Assumptionsmentioning

confidence: 99%

“…See, for example, Gandy and Kvaløy, Faraz et al, Saleh et al, Goedhart et al, Jardim et al, and Guo and Wang . An overview and comparison of different proposals of adjustments to the limits of the

\overset{X}{true}

chart is given in Jardim et al…”

Section: Introductionmentioning

confidence: 99%

“…Considering the X chart, Saleh et al 10 have recommended the numbers of phase I samples that limit the standard deviation of the conditional ARL 0 to a specified value; at the same time, Epprecht et al 11 recommended the numbers of phase I samples that limit to a specified value the probability that the conditional false-alarm rate of an S chart exceeded some maximum tolerated value specified by its user. Later, a considerable number of papers appeared that analyze the effect of parameter estimation on the conditional IC performance of several types of control charts (eg, Saleh et al, 12 Faraz et al, 13,14 Goedhart et al, 15,16 Jardim et al, 17 Loureiro et al, 18 and Guo and Wang 19 ). A common finding of those works is that the number of samples required is orders of magnitude larger than the classic recommendation of 20 to 30 samples (see, for example, Montogomery 1 ), and often unpractical.…”

mentioning

confidence: 99%

“…As a response to this problem, some authors proposed adjustments to the control limits of the charts that would enable achieving acceptable IC performance with practicable numbers of phase I samples. See, for example, Gandy and Kvaløy, 9 Faraz et al, 13,14 Saleh et al, 12 Goedhart et al, 15,16 Jardim et al, 17 and Guo and Wang. 19 An overview and comparison of different proposals of adjustments to the limits of the X chart is given in Jardim et al 20 The recommended adjustments were thought to guarantee an acceptable IC performance.…”

mentioning

confidence: 99%

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Design of an X̅ control chart with guaranteed joint in‐control and out‐of‐control performances when the mean and variance are estimated

Mosquera

Aparisi

Epprecht

2019

Quality & Reliability Eng

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During phase I analysis, the mean and variability of the process are estimated, in order to obtain the control limits for the chart. The effect of the estimation errors in the performance of the chart (phase II) has been widely studied in the literature. However, the papers have been mainly orientated to the in‐control performance of the chart. In this paper, the in‐control and out‐of‐control performances of the trueX¯ control chart are considered at the same time. Therefore, the number of samples to be taken in the phase I analysis is calculated, taking into account the possible deterioration of both metrics. In addition, an adjustment in the control limit is proposed, to reduce this number of samples.

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P ((), {CARL}_{0} \geq {ARL}_{0}^{Lim})

and have defined acceptable performances in terms of this probability; in other words, they required that

P ((), {CARL}_{0} \geq {ARL}_{0}^{Lim}) = 1 - p,

where p is a specified low probability such as .10, for example, and where, most often, the smaller tolerated CARL 0 ,

{ARL}_{0}^{Lim}

has been defined in terms of a percent reduction relative to the nominal ARL 0 (

{ARL}_{0}^{Nom}

), ie,

{ARL}_{0}^{Lim} = ((), 1 - ε_{0}) {ARL}_{0}^{Nom},

so that

{ARL}_{0}^{Lim}

is 100 ε 0 % smaller than the nominal

{ARL}_{0}^{Nom}

.…”

Section: Background and Underlying Assumptionsmentioning

confidence: 99%

Section: Background and Underlying Assumptionsmentioning

confidence: 99%

\overset{X}{true}

chart is given in Jardim et al…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Design of an X̅ control chart with guaranteed joint in‐control and out‐of‐control performances when the mean and variance are estimated

Mosquera

Aparisi

Epprecht

2019

Quality & Reliability Eng

Self Cite

View full text Add to dashboard Cite

show abstract

Control Charts, Synthetic

Chakraborti

Rakitzis

2021

Wiley StatsRef: Statistics Reference Online

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It is well known that the Shewhart control charts are not very sensitive for detecting small and moderate shifts in the process parameters. A synthetic chart, which is a combination of a Shewhart control chart and a conforming run‐length (CRL) chart, has been proposed and studied as a viable alternative for that purpose. The advantage of a synthetic chart is its ease of design, operation, and interpretation. In this contribution, we discuss the most common types of synthetic charts, along with some illustrations of their practical implementation. Both parametric and nonparametric synthetic charts are considered.

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Finite Horizon Process Monitoring

Chakraborti

Celano

Castagliola

et al. 2021

Wiley StatsRef: Statistics Reference Online

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In some industrial environments flexibility is a key for the manufacturer, that is, the ability to switch frequently from the production of one product code to another and produce batches of different items during consecutive short production time intervals, called finite production horizons; such a setting is called a finite horizon process (FHP). The frequent process changeovers may require new setup activities and machine reconfigurations every time before the beginning of a new production run. An erroneous setup or a process failure can negatively impact the quality of the manufactured products. In order to perform statistical quality control for these types of processes, practitioners have to deal with the challenging issues of designing, scheduling, and performing only a limited number of inspections. This is typically not the case in traditional quality control and monitoring applications. In this article, we highlight various aspects of statistical process monitoring (SPM) implemented in an FHP setting that need to be considered by quality practitioners. We describe available statistical process monitoring tools, such as the control charts, properly designed to monitor quality characteristics in the FHP setting. Finally, we discuss some of the still open challenges in the area of FHP monitoring.

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Two perspectives for designing a phase II control chart with estimated parameters: The case of the Shewhart X¯ Chart

Cited by 49 publications

References 23 publications

Design of an X̅ control chart with guaranteed joint in‐control and out‐of‐control performances when the mean and variance are estimated

Design of an X̅ control chart with guaranteed joint in‐control and out‐of‐control performances when the mean and variance are estimated

Control Charts, Synthetic

Finite Horizon Process Monitoring

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