Consider a random walk on an undirected, connected graph. On each edge we can set a transition probability to connect two adjacent vertices. The mixing rate of the associated Markov chain to the uniform equilib- rium distribution is determined by the second largest eigenvalue in modulus (SLEM) of the transition probability matrix. This problem is called the fastest mixing Markov chain problem (FMMC). This thesis will cover numerical methods for solving the FMMC problem. We will compare different methods for solving the problem, including the subgradient method and primal-dual interior-point methods. We will pro- vide a complexity analysis of implentation of the methods, and make a com- parison with the convex optimization solver CVXOPT written by Andersen, Dahl, and Vandenberghe. Finally, we will also look at small applications of the problem in shuffling.