Traditional control charts often assume that the mean and variance of an in-control process remain constant. However, in real-world production or research scenarios, this assumption is often difficult to satisfy. For example, the levels of blood chemical components in a patient's body may change over time. When their mean and variance have shifted over time, the direction and magnitude of shifts are often unknown. Faced with such complex situations, traditional control charts, due to their inherent limitations, fail to effectively monitor the actual process. To address these, the work applies a log-normal transformation to the square of the coefficient of variation, and designs a two-sided adaptive exponentially weighted moving average (AEWMA) control chart for monitoring the coefficient of variation. Subsequently, it investigates the effect of parameters on its performance? and compares the out-of-control performance between the proposed and existing charts. Comparison results show that the proposed chart only outperforms some existing charts. To enhance detection capability of the proposed chart, the work introduces variable sample sizes (VSS) into the design of the chart, and proposes a VSS AEWMA chart for monitoring the efficient of variation, and compares its out-of-control performance with existing charts. The results demonstrate that the proposed VSS AEWMA chart achieves the best overall performance in monitoring the coefficient of variation and can effectively identify process abnormalities.