Vännman has earlier studied a class of capability indices, containing the indices Cp, Cpk, Cpm, and Cpmk, when the tolerances are symmetric. We study the properties of this class when the tolerances are asymmetric and suggest a new enlargened class of indices. Under the assumption of normality an explicit form of the distribution of the new class of the estimated indices is provided. Numerical investigations are made to explore the behavior of the estimators of the indices for different values of the parameters. Based on the estimator a decision rule that can be used to determine whether the process can be considered capable or not is provided and suitable criteria for choosing an index from the family are suggested.