Cumulative quantity control chart for the mixture of inverse Rayleigh process

• Proposed control chart is based on model having increasing/decreasing failure rate.
• Simple and mixture distribution is considered to develop a control chart.
• Likelihood structure is very simple as compared to Majeed, Aslam, and Riaz (2013).
• Asymptotic properties of ARLs, different loss functions to compute ALI and EQL are considered.
• Simulation study as well as a real life case study is also discussed to help engineers.

The engineering processes are made up of a number of the phenomenons working together that may lead to defects with multiple causes. In order to model such types of multiple cause defect systems we may not rely on simple probability models and hence, the need arises for mixture models. The commonly used control charts are based on simple models with the assumption that the process is working under the single cause defect system. This study proposes a control chart for the two component mixture of inverse Rayleigh distribution. The proposed chart namely IRMQC chart is based on mixture cumulative quantity using the quantity of product inspected until specified numbers of defects are observed. The single cause chart is also discussed as a special case of the proposed mixture cumulative quantity chart. The control structure of the proposed chart is designed, and its performance is evaluated in terms of some useful measures, including average run length (ARL), expected quality loss (EQL) and relative ARL (RARL). An illustrative example along a case study, is also given to highlight the practical aspects of the proposal.