Modeling dependent assignable causes via Marshall–Olkin copula in the economic–statistical design of T2 control charts

Economic–statistical design of control charts has been developed under the assumption that assignable causes occur independently, which simplifies the quality cycle analysis but does not allow simultaneous or common origin occurrences to be represented explicitly. To address this limitation, a modeling framework is proposed in which occurrence times follow proportional hazard marginal distributions and are coupled through the Marshall–Olkin copula. Both individual and simultaneous occurrences of assignable causes are embedded within the quality cycle, and analytical expressions for the performance measures and cost function are derived. The proposed framework can be applied to Shewhart-type control charts without additional theoretical development. As an illustration, the framework is implemented for the multivariate T2 chart with Weibull marginals under a shock model. Numerical results for this shock model show that ignoring dependence overestimates the in–control duration and leads to economically optimistic sampling policies. Therefore, for processes in which assignable causes may share common physical sources, incorporating a dependence structure in economic–statistical design is recommended.