We study theoretically and numerically multiple approaches for approximating the solution of the eigenvalue problem for the reaction-diffusion equation with oscillating coefficients. This equation is especially used to model the neutron flux in a nuclear reactor core in a steady-state regime, where oscillating coefficients describe domain heterogeneity. As for multi-scale problems, numerical approximation of the solution by standard methods is too expensive. Here, we implement a numerical approach using the Multi-scale Finite Element Method ("MsFEM"), which is a Galerkin discretization approach using pre-computed basis functions that are well adapted to the problem of interest. Since these basis functions are solutions of local problems, the intricate task lies in finding the right local problems to solve. The advantage of this method relies upon the fact that once basis functions are pre-computed, any couples of eigenvalue/eigenvector can be obtained in a very short time. We make partial use of theoretical homogenization results in a periodic framework to guide our intuition in order to define appropriate basis functions yielding an efficient approach.