Design and Analysis of Cpm and Cpmk Indices for Uncertainty Environment by Using Pythagorean Fuzzy Sets

Process capability analysis (PCA) is a statistical analysis tool to examine variability of the process that causes faults for outputs and reduces customer satisfaction level. So, it is a completely effective method to improve the process’ quality. One of the effective methods is process capability indices (PCIs) that are used to analyze the capability of any process by using specification limits (SLs) and process’ variation. Especially in real case problems, there are many factors that causing uncertainty for the process. Although traditional PCIs are effective tools to analyze variation of process, they caused some misleading results and incorrect interpretations when the process has uncertainties. To overcome the problem, the PCIs have been re-designed under uncertainty to increase their effectiveness by using fuzzy sets (FSs). In recently, some fuzzy set extensions (FSEs) have been derived to deal with uncertainty and they can model uncertainties of process more effectively. In this paper, Pythagorean Fuzzy Sets (PFSs), one of the most common FSEs, are used to analyze process capability bu improving some PCIs based on PFSs. For this aim, generally used PCIs called ${\tilde C_{pm}}$ and ${\tilde C_{pmk}}$ are re-designed by using PFSs as the first time in the literature. The mathematical structures of these two indices are re-formulated and PCIs based on PFSs (PFPCIs) have been derived. Additionally, an application related with dimensions of a gear for a piston is also applied to analyze usage of proposed PFPCIs. The obtained results confirmed that the indices ${\tilde C_{pm}}$ and ${\tilde C_{pmk}}$ based on PFSs are more capable for modelling uncertainty and give more information and have more sensitiveness than the traditional PCIs.