Control charts for random and fixed components of variation in the case of split-unit structure of data

Abstract

Control charts are widely used in production processes to detect the occurrence of assignable causes so that corrective actions may be taken to achieve process stability. In a standard control charting method, a subgroup of units is regarded as a random sample from a homogeneous process, and the within-subgroup variation is supposed to represent all variation attributable to chance causes. However, in many practical situations, the within-subgroup variation is traceable to various causes which cannot be eliminated or reduced to an extent that their presence could be considered negligible. For such a case, the notion of standard control charting method is not adequate, and the within-subgroup variation needs to be decomposed into relevant components of the common-cause system for more meaningful and effective control charting. The objective of this thesis is to develop control charts for monitoring various fixed and random components of variation in a split-unit structure of data with an emphasis on a LPCVD process in semiconductor manufacturing. Two models are developed. One is the mixed-effects model and the other the split-unit model. For the two models, control charting methods are developed for the overall process average, for random components, and for the fixed differences due to the wafer location and measurement position separately. Performance of the proposed sum-of-squares control chart is compared with that of the orthogonal-contrast charts by Roes and Does in terms of the Type-II error probability of detecting changes in the fixed differences due to the wafer locations and measurement positions. Computational results indicate that the proposed sum-of-squares approach yields a smaller Type-II error probability than the orthogonal-contrast approach for a given overall false-alarm rate. In addition, Roes and Does approach tends to increase the number of control charts to maintain since a control chart is constructed for each contrast. The proposed st...