Design and analysis of Cpm and Cpmk indices for uncertainty environment by using two dimensional fuzzy sets

Process capability analysis (PCA) is an important stage to check variability of process by using process capability indices (PCIs) that are very effective statistics to summarize process' performance. Traditional PCIs can produce some incorrect results and declare misinterpretation about process' quality if the process includes uncertainties. Additionally, definitions of process' parameters with exact values is not possible when there are uncertainty caused by measurement errors, sensitivities of measuring instruments or quality engineers' hesitancies. Although the fuzzy set theory (FST) has been successfully used in PCA, it is the first time to use of Pythagorean fuzzy sets (PFSs) to model uncertainties of process more than traditional fuzzy sets in PCA. Since the PFSs has two-dimensional configurations by defining membership and nonmembership values, they also have a huge ability to model uncertainty that arises from the human's thinking and hesitancies, and has brought flexibility, sensitivity and reality for PCA. In this paper, specification limits (SLs), mean (mu(p)), standard deviation (s) and target value (T) main parameters of PCIs have been analyzed by using PFSs and Pythagorean fuzzy process capability indices (PFPCIs) for two well-known PCIs such as ((C) over tilde (pm)) and ((C) over tilde (pmk)) have been derived. The Pythagorean ((C) over tilde (pm)) and ((C) over tilde (pmk)) indices have also been applied and tested on some numerical examples based on real case applications from manufacturing industry. The obtained results show that PFPCIs provide wider knowledge about capability of process and to obtain more realistic results. As a result of considering all possibilities about the process, it has been concluded that the process is incapable. In light of this information, the results obtained using different fuzzy set extensions for ((C) over tilde (pm)) and ((C) over tilde (pmk)) indices can be compared.