A maximum double exponentially weighted moving average (EWMA) chart, called the MaxDEWMA chart, is used for fast detection of small and moderate shifts in the process mean and/or variance. A loss model that combines Lorenzen and Vance's cost model with linear, quadratic and exponential loss functions is applied in this work to develop the statistical constraints of the MaxDEWMA chart. The optimal decision variables, namely sample size n, sampling interval time h, control limit width L and smoothing constant lambda, are obtained by minimizing the expected cost function. Via simulations, the MaxDEWMA charts have smaller expected cost than the MaxEWMA charts based on different loss functions. In addition, adopting different loss functions impacts the optimal expected cost and decision variables. Sensitivity analyses reveal that a larger magnitude of the mean and/or variance shifts leads to a smaller sample size and a more frequent sampling interval.