In quality control, such as other statistical problems, we may confront imprecise concepts. One case is a situation in which specification limits are two fuzzy sets. In such a fuzzy environment, the product is not qualified with a two valued Boolean view, but to some degree depending on the quality level of the product and the strictness of the decision maker. This matter can be cause to a justified judgment in decision making on manufacturing processes. The generalized process capability indices $C_{\widetilde{p}}$, $C_{\widetilde{pk}}$ and $C_{\widetilde{pm}}$ can be helpful and necessary for measuring the fuzzy quality in an in-control process. In this paper, the generalized Taguchi index $C_{\widetilde{pm}}$ is proposed to provide an assessment of the ability of the fuzzy process to be clustered around the target value. Considering fuzzy specification limits we present four approximate 100(1 − γ)% confidence intervals for the generalized process capability index $C_{\widetilde{pm}}$ in this paper. Then, we obtain several fast computable 100(1 − γ)% confidence intervals for $C_{\widetilde{pm}}$, where the fuzzy specification limits are linear, normal or elliptic. An industrial example is given to show the performance of the proposed method.