For a graph G= (V,E), a double Roman dominating function (DRDF) is a function f : V —>{0,1,2,3} having the property that if f (v) = 0, then vertex v must have at least two neighbors assigned 2 under f or one neighbor u with f (u) = 3, and if f (v) = 1, then vertex v must have at least one neighbor u with f (u) \ge 2. In this paper, we consider the double Roman domination problem (DRDP), which is an optimization problem of finding the DRDF f such that $\sum_{v\in V} f (v)$ is minimum. We propose several integer linear programming (ILP) formulations with a polynomial number of constraints for this combinatorial optimization problem, and present computational results for graphs of different types and sizes. To the best of our knowledge, this is the first time that ILP formulations are proposed for DRDP. Further, we prove that some of our ILP formulations are equivalent to the others regardless of the variables relaxation or usage of less number of constraints and variables.