Economic design of X̄ and R charts under Weibull shock models

Abstract

This paper considers the problem of a continuous production process where both the mean and variance are simultaneously monitored by an X̄ chart and R chart respectively, and generalizes the model of Costa (IIE Transactions 1993; 25(6):27-33). 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 cause changes the process mean and the other changes the process variance. However, the occurrence of one kind of assignable cause does not preclude the occurrence of the other. It is also assumed that the occurrence times of the assignable causes are described by Weibull distributions with increasing failure rates. A cost model is developed and a non-uniform sampling interval scheme is adopted. A two-step search procedure is employed to determine the optimal design parameters. The relative contribution of the paper over the results obtained by Costa is addressed. A sensitivity analysis of the model is conducted and the cost savings associated with the use of non-uniform sampling intervals instead of constant sampling intervals are evaluated. The economic design model is then extended to an economic-statistical design model for achieving desired levels of statistical performance while minimizing the expected cost. Performances of purely economic design and economic-statistical design are compared.