The ground state solution of the random dimer model is at a critical point after, which has been shown with random link excitations. In this paper we test the robustness of the random dimer model to the random link excitation by imposing the maximum weight excitation. We numerically compute the scaling exponents of the curves arising in the model as well as the fractal dimension. Although strong finite size corrections are present, the results are compatible with that of the random link excitation. Furthermore, another form of excitation, the ε - coupling excitation is studied. We find that near-optimal configurations belong to the same universality class as the travelling salesman problem. Thus, we confirm a conjecture on the scaling properties of combinatorial optimisation problems, for the specific case of minimum weight perfect matchings on 2-dimensional lattices. This document was submitted as my thesis project for the MSc Complex Systems Modelling course at King's College London in 2021. In particular, I would like to thank my supervisor, Dr Gabriele Sicuro for his insights and guidance.