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References 379
is only limited to the origin point whereas the two-chart V-mask’s FIR is added into
the consecutive CUSUM.
24.2.4 ARL comparisons
Using the Markov chain approach, Lucas and Crosier 11 analyzed the average run
length properties of the tabular FIR CUSUM scheme. Similarly, by comparing ARL
properties, Atienza et al. 12 further established the equivalence of the tabular FIR
CUSUM and the V-mask FIR CUSUM. The ARL values for the tabular FIR CUSUM are
11
given by Lucas and Crosier, while those for the V-mask are obtained by Monte Carlo
12
simulation. The results of the comparison by Atienza et al. are replicated here in Ta-
ble 24.2. It is clear that, in addition to the analysis concerning the tabular and V-mask
FIR CUSUMs, the ARL comparison in Table 24.2 further supports the equivalence of
the two FIR CUSUM schemes.

24.3 CONCLUSIONS

The FIR feature is found to be useful in implementing the CUSUM chart in its tabular
form. In this chapter, we have discussed a procedure for developing an equivalent
scheme to the V-mask CUSUM. The V-mask FIR CUSUM requires only a simple
modification of the first point in calculating the CUSUM but requires plotting two
n
separate charts. An alternative CUSUM, CUSUM , is thus proposed to overcome the
need for this. That is, instead of using the point at the origin (0,0), two points are
plotted, namely (0,−FIR) and (0,+FIR). This makes the V-mask FIR CUSUM looks
like a fork at the start-up stage. This procedure of implementing the FIR feature in
V-mask form is also applicable to all other forms of CUSUM masks.

REFERENCES

1. Champ, C.W. and Woodall, W.H. (1987) Exact results for Shewhart control charts with
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2. Page, E.S. (1954) Continuous inspection schemes. Biometrika, 41, 100--115.
3. Gan, F.F. (1991) An optimal design of CUSUM quality control charts. Journal of Quality
Technology, 23, 278--286.
4. Lucas, J.M. (1976) The design and use of V-mask control schemes. Journal of Quality Tech-
nology, 8, 1--12.
5. Hawkins, D.M. and Owell, D.H. (1998) Cumulative Sum Charts and Charting for Quality
Improvement. New York: Springer-Verlag.
6. Kemp, K.W. (1961) The average run length of the cumulative sum chart when a V-mask is
used. Journal of the Royal Statistical Society B, 23, 149--153.
7. Bissel, A.F. (1969) CUSUM techniques for quality control (with discussion). Applied Statistics,
18, 1--30.
8. Montgomery, D.C. (2005) Introduction to Statistical Quality Control, 5th edition. Hoboken,
NJ: John Wiley & Sons Inc.
9. Barnard, G.A. (1959) Control charts and stochastic processes. Journal of the Royal Statistical
Society B, 21, 239--257.