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In this paper we examine a consolidation and dispatching problem motivated by a multinational chemical company which has to decide routinely the best way of delivering a set of orders to its customers over a multi-day planning horizon. Every day the decision to be made includes order consolidation, vehicle dispatching as well as load packing into the vehicles. We develop a heuristic based on a cutting plane framework, in which a simplified Integer Linear Program (ILP) is solved to optimality. Since the ILP solution may correspond to a infeasible loading plan, a feasibility check is performed through a tailored heuristic for a three-dimensional bin packing problem with side constraints. If this test fails, a cut able to remove the infeasible solution is generated and added to the simplified ILP. Then the procedure is iterated. Computational results show that our procedure allows achieving remarkable cost savings.