While the joint optimization of production and outbound distribution decisions in a manufacturing context has been intensively studied in the past decade, the integration of production, inventory and inbound transportation from suppliers has received much less attention despite its practical relevance. This paper aims to fill the gap by introducing a general model for the assembly routing problem (ARP), which consists of simultaneously planning the assembly of a finished product at a plant and the routing of vehicles collecting materials from suppliers to meet the inventory requirements imposed by the production. We formulate the problem as a mixed-integer linear program and we propose a three-phase decomposition matheuristic that relies on the iterative solution of different subproblems. The first phase determines a setup schedule while the second phase optimizes production quantities, supplier visit schedules and shipment quantities. The third phase solves a vehicle routing problem for each period in the planning horizon. The algorithm is flexible and we show how it can also be used to solve two well-known problems related to the ARP: the production routing problem (PRP) and the inventory routing problem (IRP). Using the same parameter setting for all problems and instances, we obtained 818 new best known solutions out of 2,628 standard IRP and PRP test instances. In particular, on large-scale multi-vehicle instances, the new algorithm outperforms specialized state-of-the-art heuristics for these two problems.