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The zero-inflated Poisson (ZIP) model is an extension of the standard Poisson distribution. It is often used to describe a near zero-defect process with occasional occurrences of non-conforming products. In the past, research on the control charts for ZIP process has concentrated on univariate ZIP process where there is only one type of defect. However, it is common in some high quality processes that there are several types of defects to be considered and the count variables are correlated. It is not appropriate to monitor the process using independent univariate ZIP based control charts. In this paper, a control charting procedure using a combination of two cumulative sum charts is proposed for monitoring shifts in a bivariate ZIP (BZIP) process, which is a special case of the multivariate ZIP model. We use simulations to obtain the upper control limit of the control charts based on a specified in-control average number of observations to signal. We also use simulations to evaluate the control charting procedure in three situations: shifts only in the p-set parameters; shifts only in the λ-set parameters; and shifts in all the parameters. The simulation results show that the proposed control charts are effective in detecting shifts in the parameters of a BZIP process. Finally, we present an application of our proposed method in the light emitting diode packaging industry.