Aerodynamic knowledge for practical flapping-wing micro air vehicle (MAV) design is obtained by application of the design exploration framework coupled with CFD to a multiobjective aerodynamic design optimization problem of two-dimensional flapping motion of an airfoil. Lift and thrust are maximized while required power is minimized in the design problem. Pareto-optimal solutions are obtained by a multiobjective evolutionary optimization and analyzed with the self-organizing map. Aerodynamic performance of each flapping motion is evaluated by a two-dimensional Navier-Stokes solver. The result reveals tradeoff information between each objective and effect of each design parameters on them. Analysis of the time histories of lift, thrust, and required power coefficients and corresponding pressure coefficient distribution of the extreme Pareto-optimal solutions leads to useful guidelines for the lift maximization, thrust maximization, and required power minimization.