A thin elastic multilayered plate consisting of alternating hard and soft isotropic layers is studied. One of the face planes is subject to a normal pressure and the other face plane is free. A formula for the deflection of an infinite plate under a doubly periodic external force is delivered using asymptotic expansions. This formula is also applied for a rectangular plate with Navier boundary conditions on its edges. The maximal deflection is accepted as a measure of the plate stiffness. The purpose of the present paper is to obtain an expression for the plate deflection and find an optimal distribution of hard and soft layers assuming that their total thicknesses are given. The Monte Carlo method is used for finding the optimal distribution of layers.