Application of Additive Exponential Distribution in Design of \(\overline{{\varvec{X}} }\) Control Chart-Statistical and Economic Perspective

Economic control charts are broadly utilized in the quality control and quality assurance of several products. The assignable cause of occurrence in these charts is generally presumed to follow either an Exponential or a Weibull distribution. But in some products, the time to happening of a known cause may be due to the sum of the occurrence times of two events, such as (i) due to mechanical failure and (ii) due to human failure. Each failure time has an Exponential distribution with different parameters. In such a case, the time taken for a known cause to occur may follow an Additive Exponential distribution. The focus of this paper is to optimize the design of the \(\overline{X }\) $\overline{X}$ control chart with Additive Exponential in-control times, and the quality feature follows a Normal distribution. The statistical parameters \(\alpha\) $\alpha$ and \(\beta\) $\beta$ are fixed in order to identify out of control times easily and effectively. By minimizing the average cost per unit time with respect to the constraints on type I and type II error probabilities, the parameters of optimal design, such as sample size, time span between consecutive samples and the control limits are derived. Sensitivity analysis of the design reveals that the distribution parameters of in-control times significantly impact the parameters of optimal design and cost per unit time. These charts are extremely valuable for quality control in manufacturing processes such as chemicals, glassware, paints, etc., where the in-control times of the process follow an Additive Exponential Distribution.