A new aggregation–disaggregation algorithm

This paper proposes a new algorithm for solving the Kolmogoroff equations for large Markov chains. The algorithm consists of two stages. At the first stage, it finds the optimal partition of states into blocks which makes the Markov chain as completely lumpable or decomposable as possible. We define an objective function for evaluating the lumpability or decomposability of a Markov chain with respect to a partition. A lower bound of the objective function is presented and the Branch-and-Bound method is proposed for obtaining the optimal partition. At the second stage, the Kolmogoroff equations are solved by the iterative aggregation-disaggregation method with the optimal partition. Numerical results are presented in order to demonstrate the effectiveness of the proposed algorithm.