Exact Relationship of C″pk with Proportion of Nonconformance and Some Other Properties of C″p(u,v)

Establishing relationship between two parallel measures of process performance, namely, process capability indices and proportion of nonconformance (PNC), is a very challenging problem and has been studied by many researchers. The situation worsens for asymmetric specifications as in that case, the PNC differs depending on which one of the specification limits is under consideration. Although some formulation of the upper bound of PNC as a function of C″pk = C″p(1,0) is available, very little has been carried out for similar formulation of an exact relationship between C″pk and expected PNC. In the present article, this problem has been addressed, and it has been shown that the proposed formulation is analogous to that of the symmetric specification. Moreover, the expected PNC values are also less than the already available upper bounds. Some other useful properties of C″p(u,v) such as the interrelationship between member indices of the superstructure, threshold value of C″p = C″p(0,0), and optimality on target of C″p(u,v) have also been studied. A numerical example has been discussed to strengthen the theoretical findings.