Fuzzy nonparametric estimation of capability index $$\textit{C}_{pk}$$Cpk

Process capability indices have been widely used in the manufacturing industry to measure the potential performance. This paper proposes a nonparametric approach for estimating the proportion of non-conforming items and capability index $$C_{pk}$$Cpk, when sample observations and specification limits of a process are reported as imprecise numbers. In this approach, first the $$\alpha $$α-pessimistic values of the imprecise observations were first applied to determine an unbiased estimator for population variance and optimal bandwidth. Thereafter, the fuzzy proportion of non-conforming items based on kernel distribution function was obtained. Finally, the fuzzy proportion of non-conforming items was applied to obtain the membership function of fuzzy nonparametric capability index $${\widetilde{C}}_{pk}$$C~pk . Moreover, the proposed nonparametric methods are examined to compare with some other existing parametric methods and their performance will be cleared via some numerical examples and some comparison studies.