Order-preserving submatrices are an important tool for the analysis of gene expression data. As finding large order-preserving submatrices is a computationally hard problem, previous work has investigated both exact but exponential-time as well as polynomial-time but inexact algorithms for finding large order-preserving submatrices. In this paper, we propose a novel exact algorithm to find maximum order preserving submatrices which is fixed parameter tractable with respect to the number of columns of the provided gene expression data. In particular, our algorithm is based on solving a sequence of mixed integer linear programs and it exhibits better guarantees as well as better runtime performance as compared to the state-of-the-art exact algorithms. Our empirical study in benchmark datasets shows large improvement in terms of computational speed.