Two-Dimensional Uncertainty Analysis for Cp and Cpk Process Capability Indices

Abstract

Process capability analysis (PCA) is an efficient statistical technique for calculating of process’ ability to meet predetermined specification limits (SLs) that defined by customer, engineers or designers. Measurements and evaluations for PCA may be vague, incomplete or inaccurate in the real-case problems. In that cases, the process capability should be successfully measured by using fuzzy set extensions to model uncertainties of the process. One of fuzzy set extensions named Pythagorean fuzzy Sets (PFSs) that also contains the non-membership function can be employed as an effective tool to model uncertainty better than traditional fuzzy sets (TFSs). In this paper, a novel approach based on PFSs is suggested to increase flexibility and sensitivity of the PCA and to successfully model the uncertainties. For this aim, two of frequently used process capability indices (PCIs) C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> and C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pk</inf> , are analyzed based on PFSs. Then, the Pythagorean fuzzy process capability indices (PFPCIs) have been derivate respectively for the indices ${\tilde C_p}$ and ${\tilde C_{pk}}$ and the mathematical backgrounds of these indices have been developed for the first time in the literature. Additionally, the proposed indices ${\tilde C_p}$ and ${\tilde C_{pk}}$ have been applied to a real case problem from manufacturing industry. The obtained PCIs based on PFSs provide some additional flexibility and information about the process since they better modeled process uncertainty. Moreover, it is demonstrated that the proposed PFCPIs can be effectively applied on process to manage PCA.