ARL-unbiased geometric and CCCG control charts

In order to control increases and decreases in a parameter in a timely fashion, a chart should be set in such way that the average run length (ARL) curve attains a maximum in the in-control situation.

This article proposes not only a geometric (or cumulative count of conforming, CCC) chart but also a CCC chart under group inspection (CCC_G) for which all out-of-control ARL values are smaller than the in-control ARL and thus provides an improvement on the designs described in L. Zhang et al. (2004 Zhang, L., Govindaraju, K., Bebbington, M. and Lai, C. D. (2004). On the Statistical Design of Geometric Control Charts, Quality Technology &; Quantitative Management 2: 233–243.Taylor &; Francis Online] , Google Scholar]) and C. W. Zhang et al. (2012 Zhang, C. W., Xie, M., and Jin, T. (2012). An Improved Self-starting Cumulative Count of Conforming Chart for Monitoring High-Quality Processes under Group Inspection, International Journal of Production Research 50: 7026–7043.Taylor &; Francis Online], Web of Science ®] , Google Scholar]). Moreover, by exploring the notions of uniformly most powerful unbiased tests with randomization probabilities, we are able not only to eliminate the bias of the ARL function of the geometric charts but also to bring their in-control ARL exactly to a prespecified value.

Instructive examples are provided to illustrate the e?ciency of the proposed charts.