In this work, a novel multi-objective design optimization procedure is presented for the Minkowski Reflectarray RA s using a complete 3-D CST Microwave Studio MWS- based Multilayer Perceptron Neural Network MLP NN model including the substrate constant Er with a hybrid Genetic GA and Nelder-Mead NM algorithm. The MLP NN model provides an accurate and fast model and establishes the reflection phase of a unit Minkowski RA element as a continuous function within the input domain including the substrate 1 <= epsilon(r) <= 6; 0.5 mm <= h <= 3 mm in the frequency between 8 GHz <= f <= 12 GHz. This design procedure enables a designer to obtain not only the most optimum Minkowski RA design all throughout the X-band, at the same time the optimum Minkowski RAs on the selected substrates. Moreover a design of a fully optimized X-band 15 x 15 Minkowski RA antenna is given as a worked example together with the tolerance analysis and its performance is also compared with those of the optimized RA s on the selected traditional substrates. Finally it may be concluded that the presented robust and systematic multi-objective design procedure is conveniently applied to the Microstrip Reflectarray RAs constructed from the advanced patches.