Fuzzy process capability analyses with fuzzy normal distribution

Process capability indices (PCIs) are very a useful statistical analysis tool to summarize process’ dispersion and location by process capability analysis (PCA). Additionally PCA produces another two more important results that show us the process ability satisfying specification limits (SLs) and the ratios of conforming (CIs) and nonconforming (NCIs) items which are the probabilities of producing within and out of SLs. However, there are some limitations which prevent a deep and flexible analysis because of the crisp definition of SLs. In this paper, the fuzzy set theory is used to add more sensitiveness to PCA including more information and flexibility. For this aim, fuzzy normal distribution with crisp SLs is first used to calculate the fuzzy percentages of conforming (FCIs) and nonconforming (FNCIs) items by taking into account fuzzy process mean, μ˜ and fuzzy variance, σ˜2 which are obtained by using fuzzy extension principles. The calculation of the percentages of CIs and NCIs items in fuzzy numbers gains more flexible evaluation ability for the process engineer. Then fuzzy SLs are used together with μ˜ and σ˜2 to produce fuzzy PCIs (FPCIs) and fuzzy normal distribution. The fuzzy formulation of the indices CpCp and CpkCpk, most used two traditional PCIs, are produced when SLs are either triangular (TFN) or trapezoidal fuzzy numbers (TrFN). The proposed methodologies are applied in a piston manufacturer in Konya’s Industrial Area, Turkey. FPCIs, FCIs, and FNCIs ratios are determined for piston diameter measurements. The results show that fuzzy estimations of PCIs, CIs, and NCIs have much more treasure to evaluate the process when it is compared with the crisp case.