VSSI $\bar X$ control charts for processes with multiple assignable causes

This research investigates the statistical efficiency of variable sample size and sampling interval (VSSI) X macr control charts under multiple assignable causes. Algorithms for calculating the average run length (ARL) and average time to signal (ATS) of a VSSI X macr control chart are proposed by employing the Markov chain method. States of a process are defined in vector forms according to the location of a control statistic and the occurrence states of the assignable causes. Initial probabilities and transition probabilities are carefully derived from the definitions of the states. Statistical properties of the proposed control chart are also investigated. Illustrative examples show that the VSSI X macr control chart is superior to the VSS or VSI X macr control chart as well as to the Shewhart X macr control chart in a statistical sense, even under multiple assignable causes.