Magnonic crystals with spin waves as information carriers are the magnetic counterparts of photonic and phononic crystals. The studies of spin waves or magnons in magnonic crystals have attracted increasing attention, especially for the characteristics of band gaps. However, most of the previous work has paid attention to the magnonic crystals with simple lattices. In this paper, the model of magnonic crystals with complex lattices which is composed of two different scatterers of ferromagnetic materials periodically embedded in another kind of ferromagnetic matrix material is proposed for the first time. Then, the plane-wave expansion method is developed by using the idea of super cells, in which the Fourior coefficient of exchange constant in the space of reciprocal lattice vector is analytically derived, and this method can be used to numerically investigate the eigen-properties of spin waves in magnonic crystals with complex lattices. Of course, it can be applied to the fields of other artificial crystals with complex lattices after the corresponding process, such as photonic crystals and phononic crystals. Band structures of two-dimensional magnonic crystal with complex lattices consisting of two different sizes of Fe cylinders alternately arranged in Euo matrix, are numerically calculated by using the above plane-wave expansion method. The behaviors of band gaps of spin waves changing with the total filling fraction of volume f and also with the mismatch of the filling fraction of volume of two Fe cylinders in EuO matrix are numerically studied. The results of magnonic crystals with complex lattices are compared with those of magnonic crystal with simple latticeic. Some conclusions are summarized as follows. In the same filling fraction of volume f, the width of band gap B4, 5 in the magnonic crystal with complex lattice is always larger than that with the simple lattice, but the width of band gap B8, 9 in the complex lattice is less than that in the simple lattice. When f = (fA + fB)/2 = 0.5, the width of band gap B4, 5 increases as the mismatch between fA and fB increases, but the behavior of the gap B8, 9 is opposite. Moreover, some new spin-wave gaps can be generated by changing the mismatch between fA and fB. This is because the gaps in our studied systems result from the mechanism of Bragg scattering of spin wave in periodic ferromagnetic materials. When the mismatch between fA and fB increases, the multiple scattering effects become stronger. All of these results show that the width or the frequency of band gap can be optimized or tuned by using the complex lattice. Such an approach through fabricating complex lattices may open a new scope for engineering and designing the band gaps of magnonic crystals.