Statistical properties of a system of fourth-generation process capability indices C(,psk)(u,v,w)

Process Capability Indices (PCI's) are widely used to provide a unitless measure of process potential and performance and in the evaluation of purchasing decisions. These indices are a function of the process mean ({dollar}mu{dollar}) and standard deviation ({dollar}sigma{dollar}), the process specification limits (USL, upper specification limit; and LSL, lower specification limit) and target value T for the process mean. USL, T and LSL are assigned externally by engineers. Note that, the process must be in statistical control before its capability can be assessed. In other words, a stable, predictable distribution for the output is a prerequisite for capability analysis.; A primary objective of process capability is to determine how well the output from the process meets the preassigned engineering specifications. The criteria commonly adopted in the evaluation of these indices (see, for instance, Nagata (1993)) are: If PCI {dollar}ge{dollar} 1.33, the process capability is satisfactory; if 1.00 {dollar}le{dollar} PCI {dollar}<{dollar} 1.33, the process capability is acceptable; if PCI {dollar}<{dollar} 1.00, the process is unsatisfactory.; The main objectives of this dissertation are to: (1) Propose a fourth generation index C{dollar}sb{lcub}rm psk{rcub}{dollar} for the case where the target value T is not equal to the midpoint of the specification limits (i.e. asymmetric tolerances), and show that this index is more sensitive compared to the standard PCI's in detecting small shifts of the process mean from the target value. (2) Obtain exact formulas for E(C{dollar}sb{lcub}rm psk{rcub}{dollar}) and Var(C{dollar}sb{lcub}rm psk{rcub}{dollar}) under the assumption of a random sample from a process where the variable is normally distributed. (3) Propose a class of fourth generation indices C{dollar}sb{lcub}rm psk{rcub}{dollar}(u,v,w), where u, v and w are positive constants. By varying these constants, this class incorporates all of the PCI's in the literature including the new C{dollar}sb{lcub}rm psk{rcub}{dollar}. An index in this class is defined as follows:{dollar}{dollar}Csb{lcub}psk{rcub}(u,v,w)={lcub}d-uvertmu-Mvert-wvertmu-Tvertover 3sqrt{lcub}sigmasp2+v(mu-T)sp2{rcub}{rcub}{dollar}{dollar}where{dollar}{dollar}d={lcub}USL-LSLover 2{rcub}quad {lcub}rm and{rcub}quad M={lcub}USL+LSLover 2{rcub}{dollar}{dollar}(4) Obtain exact formulas for E(C{dollar}sb{lcub}rm psk{rcub}{dollar}(u,v,w)) and Var(C{dollar}sb{lcub}rm psk{rcub}{dollar}(u,v,w)) assuming normality and independence for the elements of the sample. (5) Provide graphical representation of numerical results.