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A CUSUM scheme with variable sample sizes for monitoring process shifts

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Abstract

The cumulative sum scheme (CUSUM) and the adaptive control chart are two approaches to improve chart performance in detecting process shifts. A weighted loss function CUSUM scheme (WLC) is able to monitor both the mean shift and the increasing variance shift by manipulating a single chart. This paper investigates the WLC scheme with a variable sample sizes (VSS) feature. A design procedure is firstly proposed for the VSS WLC scheme. Then, the performance of the chart is compared with that of four other competitive control charts. The results show that the VSS WLC scheme is more powerful than the other charts from an overall viewpoint. More importantly, the VSS WLC scheme is simpler to design and operate. A case study in the manufacturing industry is used to illustrate the chart application. The proposed VSS WLC scheme suits the scenario where the strategy of varying sample sizes is feasible and preferable to pursue a high capability of detecting process variations.

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Abbreviations

CUSUM:

Cumulative sum chart

WL :

Weighted loss function chart or statistic used by this chart

WLC :

Weighted loss function CUSUM scheme

CCC :

Joint three one-sided CUSUM chart (I, D and V charts)

I :

CUSUM chart for detecting increasing mean shifts

D :

CUSUM chart for detecting decreasing mean shifts

V :

CUSUM chart for detecting increasing variance shifts

VSS:

Variable sample sizes

VSI:

Variable sampling intervals

VSSI:

Variable sample sizes and variable sampling intervals

A t :

Statistic used by a WLC scheme

λ :

Weighting factor in a WLC scheme

w A , H A :

Warning and control limits of a WLC scheme

k A :

Reference parameter of a WLC scheme

\(LWL_{{\overline{X} }} ,\;UWL_{{\overline{X} }} \) :

Lower and upper warning limits of an \(\overline{X} \) chart

\(LCL_{{\overline{X} }} ,\;UCL_{{\overline{X} }} \) :

Lower and upper control limits of an \(\overline{X} \) chart

UWL S1, UCL S1 :

Warning and control limits of an S chart when using relax samples

UWL S2, UCL S2 :

Warning and control limits of an S chart when using alert samples

w I , H I , k I :

Warning limit, control limit and reference parameter of an I chart

w D , H D , k D :

Warning limit, control limit and reference parameter of a D chart

w V , H V , k V :

Warning limit, control limit and reference parameter of a V chart

x :

Quality parameter

μ, σ 2 :

Mean and variance of x

\(\overline{x} ,\;s\) :

Sample mean and standard deviation

μ 0, δ 0 :

In-control mean and standard deviation

μ d , δ d :

Out-of-control mean and standard deviation

n 1, n 2 :

Relax and alert sample sizes

h :

Sampling interval

B 2 :

Stabilised probability for a process being in a warning zone

ATS :

Average time to signal

ATS 0 :

In-control average time to signal

ATS min :

Used to store the minimum ATS during a chart design

ARATS :

Average ratios of ATSs

ARATS s :

ARATS in a small shift region

ARATS m :

ARATS in a moderate shift region

ARATS l :

ARATS in a large shift region

ARATS o :

ARATS in an overall shift region

\(\overline{{ARATS}} \) :

Average of ARATS values for a chart over all runs

τ :

Allowed minimum in-control ATS

R :

Allowed maximum in-control average inspection ratio

n max :

Allowed maximum sample size

\(\delta _{{\mu d}} ,\;\delta _{{\sigma d}} \) :

Selected mean shift and variance shift for chart design

\(\widehat{\delta }_{\mu } ,\;\widehat{\delta }_{\sigma } \) :

Estimated mean shift and variance shift

m 1, m 2 :

Numbers of relax and alert samples taken after a change point

\(\overline{x} _{{ij}} ,\;s^{2}_{{ij}} \) :

Sample mean and variance of the jth sample of size n i after a change point

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Authors and Affiliations

  1. School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore, 639798, Singapore

    Sheng Zhang & Zhang Wu

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  1. Sheng Zhang
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  2. Zhang Wu
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Correspondence to Sheng Zhang.

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Zhang, S., Wu, Z. A CUSUM scheme with variable sample sizes for monitoring process shifts. Int J Adv Manuf Technol 33, 977–987 (2007). https://doi.org/10.1007/s00170-006-0544-0

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