The subject matter of the present paper is Model Predictive Control (MPC). MPC is a well known approach to optimal control that tackles a long time-horizon control problem by sequentially optimizing small portions of it. MPC is generally regarded as a powerful tool for efficiently finding approximate solutions to complex optimal control problems. However, in the context of mixed-integer non-linear optimal control, MPC suffers from the high computational cost of mixed-integer non-linear programming. As a consequence, in real-time applications, the suitability of mixed-integer nonlinear MPC is limited. In this paper we present the Mixed-Integer Real Time Optimal Control (MIRT-OC) algorithm: a novel MPC technique that reduces the cost of each MPC iteration by reusing the information generated during past iterations. MIRT-OC extends the basic ideas behind LP/NLP-Branch&Bound to MPC. In LP/NLP-B&B, a mixed-integer convex optimization problem is tackled as a linear one where new linear constraints are added on the run in order to enforce the original nonlinear constraints on the solution. MIRT-OC in addition to using a LP/NLP-B&B procedure at each MPC iteration, introduces two forward shifting techniques. The first technique transforms the linearizations generated during one MPC iteration into linearizations for the sub-problem to solve in the subsequent MPC iteration. The second one extrapolates from the B&B tree built during the solution of one MPC sub-problem into a partially explored B&B tree for the next MPC sub-problem. Consequently, the non-linear MPC problem is tackled as a single mixed-integer linearly-constrained optimal control problem in which new linearizations are added on the run and, at each given moment, only a portion of the problem is optimized.The collected empirical data shows how the proposed algorithm is capable of providing sizeable computational savings, representing a first step towards a true real-time mixed-integer convex MPC scheme.