Monitoring Fuzzy Capability Index $\widetilde{C}_{pk}$ by Using the EWMA Control Chart with Imprecise Data

A manufacturing process cannot be released to production until it has been proven to be stable. Also, we cannot begin to talk about process capability until we have demonstrated stability in our process. This means that the process variation is the result of random causes only and all assignable or special causes have been removed. In complicated manufacturing processes, such as drilling process, the natural instability of the process impedes the use of any control charts for the mean and standard deviation. However, a complicated manufacturing process can be capable in spite of this natural instability. In this paper we discuss the $\widetilde{C}_{pk}$ process capability index. We find the membership function of $\widetilde{C}_{pk}$ based on fuzzy data. Also, by using the definition of classical control charts and the method of V$\ddot{a}$nnman and Castagliola, we propose new control charts that are constructed by the $\alpha$-cut sets of $\widetilde{C}_{pk}$ for the natural instable manufacturing processes with fuzzy normal distributions. The results are concluded for $\alpha=0.6$, that is chosen arbitrarily.