The Minimized Aggregated Wasserstein (MAW) distance for Gaussian mixture models (GMM) has been used as a computationally efficient approximation to the Wasserstein metric. Recently, significant theoretical advance on MAW has been made, providing deep insight about its optimality. In this talk, we develop a new algorithm for computing the barycenter of GMMs under MAW, and prove that this barycenter has the same expectation as the Wasserstein barycenter. In addition, we illustrate its practical use with examples of solving label switching problem for Bayesian analyses of GMMs based on Markov chain Monte Carlo methods.