Capability adjustment for gamma processes with mean shift consideration in implementing Six Sigma program

In the 1980s, Motorola, Inc. introduced its Six Sigma quality program to the world. Some quality practitioners questioned why the Six Sigma advocates claim it is necessary to add a 1.5[sigma] shift to the process mean when estimating process capability. Bothe [Bothe, D.R., 2002. Statistical reason for the 1.5[sigma] shift. Quality Engineering 14 (3), 479-487] provided a statistical reason for considering such a shift in the process mean for normal processes. In this paper, we consider gamma processes which cover a wide class of applications. For fixed sample size n, the detection power of the control chart can be computed. For small process mean shifts, it is beyond the control chart detection power, which results in overestimating process capability. To resolve the problem, we first examine Bothe's approach and find the detection power is less than 0.5 when data comes from gamma distribution, showing that Bothe's adjustments are inadequate when we have gamma processes. We then calculate adjustments under various sample sizes n and gamma parameter N, with power fixed to 0.5. At the end, we adjust the formula of process capability to accommodate those shifts which can not be detected. Consequently, our adjustments provide much more accurate capability calculation for gamma processes. For illustration purpose, an application example is presented.

[1]  Surajit Pal Evaluation of Nonnormal Process Capability Indices using Generalized Lambda Distribution , 2004 .

[2]  Dong Ho Park,et al.  Estimation of capability index based on bootstrap method , 1996 .

[3]  Alberto Luceño,et al.  Computing the Run Length Probability Distribution for CUSUM Charts , 2002 .

[4]  Douglas C. Montgomery,et al.  PROCESS CAPABILITY INDICES AND NON-NORMAL DISTRIBUTIONS , 1996 .

[5]  Alan M. Polansky,et al.  A smooth nonparametric approach to process capability , 1998 .

[6]  N. L. Johnson,et al.  Distributional and Inferential Properties of Process Capability Indices , 1992 .

[7]  James M. Lucas,et al.  Fast Initial Response for CUSUM Quality-Control Schemes: Give Your CUSUM A Head Start , 2000, Technometrics.

[8]  Lora S. Zimmer,et al.  Process Capability Indices in Theory and Practice , 2000, Technometrics.

[9]  Victor E. Kane,et al.  Process Capability Indices , 1986 .

[10]  Jianmin Ding,et al.  A Method of Estimating the Process Capability Index from the First Four Moments of Non‐normal Data , 2004 .

[11]  L. Franklin,et al.  Bootstrap Lower Confidence Limits for Capability Indices , 1992 .

[12]  Peter R. Nelson,et al.  The Effect of Non-Normality on the Control Limits of X-Bar Charts , 1976 .

[13]  Haim Shore A new approach to analysing non-normal quality data with application to process capability analysis , 1998 .

[14]  John Gilson A new approach to engineering tolerances : a critical presentation of the considerations necessary for the allocation and maintenance of realistic tolerances in modern economic production , 1951 .

[15]  W. Pearn,et al.  Capability indices for non-normal distributions with an application in electrolytic capacitor manufacturing , 1997 .

[16]  Davis R. Bothe,et al.  Statistical Reason for the 1.5σ Shift , 2002 .

[17]  Fred A. Spiring,et al.  A New Measure of Process Capability: Cpm , 1988 .

[18]  Arthur Bender Statistical Tolerancing as it Relates to Quality Control and the Designer (6 times 2.5 = 9) , 1968 .

[19]  Edward J. Dudewicz,et al.  Modern Mathematical Statistics , 1988 .

[20]  A. Singh Exponential Distribution: Theory, Methods and Applications , 1996 .

[21]  Thomas Pyzdek,et al.  PROCESS CAPABILITY ANALYSIS USING PERSONAL COMPUTERS , 1992 .

[22]  James M. Lucas,et al.  The Design and Use of V-Mask Control Schemes , 1976 .

[23]  David H. Evans,et al.  Statistical Tolerancing: The State of the Art, Part III. Shifts and Drifts , 1975 .