Article Preview
Top1. Introduction
Process capability indices measure how much of the process' products are in accordance with the requirements of the intended construction engineers or customers. A tolerance interval for a product characteristic, X, consists of lower and upper limits, LSL and USL, together with a target value T somewhere between these limits.
A tolerance interval is symmetric if the target value is the midpoint of the tolerance interval. Although the symmetric cases are by far the more common, there are sufficiently many instances in which the target value is not the midpoint of the tolerance limits. In these cases, the tolerance interval is called asymmetric.
For the processes with symmetric tolerance interval, there are several indices to provide measures of Normally-distributed process potential and performance such as 
and 
(see Chan, Cheng & Spiring, 1988; Kane, 1986; Pearn, Kotz & Johnson, 1992). Vannman (1995) introduced a superstructure index 
that contains these for basic indices.
According to Boyles (1994), it is often assumed that asymmetric tolerances are necessarily related in some cases to a skewed process distribution. In some circumstances the customer may allow the manufacturer to set a specification interval in terms of the process distribution. In some situations, asymmetric tolerances are due to transforming data to achieve approximate Normality. In general, asymmetric tolerances are not related to the shape of the supplier's process distribution; instead they simply reflect the customer's attitude that deviation from target value in one direction is less tolerable than the other.
There have been some indices introduced to handle the processes with asymmetric tolerances. Some superstructure indices are 
and 
. For more information, one can see Chen & Pearn (2001), Chan, Cheng & Spiring (1988), Franklin & Wasserman (1992); Kane (1986); Kushler & Hurley (1992); Pan & Li (2014); Pan & Wendy (2015).
Suppose product characteristic X has Normal distribution with mean 
and standard deviation 
. Some well-known indices which is widely used to measure the capability of the processes with asymmetric tolerances are as the following:
where:
and: