Economic Design of VSI x-Bar Control Charts for Non Normally Distributed Data under Gamma (λ, 2) Failure Models

Abstract

This investigation develops the economic design of x-bar control charts with variable sampling interval (VSI) for non normal data under Gamma (λ, 2) shock models. In the past, most of the economic design of control charts follows the Poisson process. However, pragmatically speaking, this process is usually not appropriate. Banerjee and Rahim (1988) presented a cost model which uses variable sampling intervals. This can be contrasted with sampling intervals of fixed length under a process-failure mechanism, which follows a Gamma (λ, 2) model with an increased hazard rate. In a separate study by Al-Oraini and Rahim (2003), an economic statistical design of x-bar control charts with the assumption of the Gamma (λ, 2) failure mechanism was proposed. When the x bar chart is designed to monitor a manufacturing process, three parameters need to be determined via engineers or participators. These parameters are sample size, sampling interval between successive samples, and control limits. However, measurements in the subgroup are assumed to be normally distributed when designing control charts. Said assumption may be untenable. This article employs a numerical example to indicate the solution procedure and to implement sensitivity analysis. Also, the situation of non normal and normal in the Gamma (λ, 2) model was compared. Results in non normal assumption revealed the following: smaller sample size (n) is needed; the initial sampling interval (h 1) can become longer; the sampling interval (h 2) can become shorter; the control limit width (L) can become narrower; and the expected cost per hour (ECT) can likewise be reduced.