Economic Design of and R Charts Under Weibull Shock Models

Abstract

This article considers the problem of a continuous production process, whose mean and variance are simultaneously monitored by and R control charts, respectively. The product variable quality characteristic is assumed to be normally distributed and the process is subject to two independent assignable causes (such as, tool wear-out, overheating, or vibration). One changes the process mean and the other the process variance. The occurrence of one kind of the assignable causes does not preclude the occurrence of the other kind. The occurrence times of the assignable causes are described by Weibull distributions having increasing failure rates. A cost model is developed for determining the economic design parameters. A non uniform decreasing sampling interval scheme is adopted to incorporate the effects of process deterioration. A two-step search procedure is employed to determine the economically optimum design parameters. The relative contribution of this article over the results obtained in Costa (1993) is addressed. This article introduces a few new assumptions and provides some theoretical derivations and results. These results may serve as readily available references for future studies. The article shows through numerical examples that ignoring the true (by assumption) Weibull shock model and incorrectly assuming exponential distributions of times to occurrences of assignable causes (and using constant sampling schemes), results in sizeable cost penalties. A sensitivity analysis of the model with respect to Weibull distribution parameters is performed.

Keywords:

• Increasing hazard rate
• Integrated hazard criterion
• Non uniform sampling interval
• Optimum design parameters

Mathematics Subject Classification:

• 90C99
• 34A34

Acknowledgments

The valuable suggestions and constructive criticisms of the editor and two anonymous reviewers are gratefully acknowledged. The financial support of this article was provided by FAPESP (Fundaço de Amparo a Pesquisa do Estado de So Paulo) and NSERC (Natural Science and Engineering Research Council) of Canada. Their support is gratefully acknowledged. The editorial assistance of Kim Wilson and Krishna Reddy was greatly appreciated.