This paper is devoted to the computation of the maximum likelihood estimates of the Mallows-Bradley-Terry ranking model parameters. The maximum likelihood method is avoid because of the normalizing constant that may involve an untractable sum with a very large number of terms. We show how to implement a Monte Carlo Maximization-Minimization algorithm to estimate the model parameters: the evaluation of the mathematical expectations involved in the log-likelihood equation is obtained by generating samples of Monte Carlo Markov chain from the stationary distribution. In addition, a simulation study for asymptotic properties assessment has been made. The proposed method is applied to analyze real life data set of the literature. The present paper is restricted to the Mallows-Bradley-Terry ranking model that does not allow for possibility of ties. This case has been studied elsewhere.