A classic production routing problem (PRP), in which a plant produces and distributes a single product to a set of customers over a finite time horizon, consists of planning simultaneously the production, inventory and routing activities to minimize the total cost. The last few decades have witnessed the increasing efforts made to solve such a complex problem. In this paper, we investigate a generalized PRP by considering multiple products and outsourcing (MPRP-OS). The newly studied problem is first formulated into a mixed integer linear program. Then a three-level mathematical-programming-based heuristic called TLH is developed to solve it. TLH combines a two-phase iterative method, a repairing strategy and a fix-and-optimize procedure to find near-optimal solutions. In addition, it is adaptable to solve the classic PRP. Computational experiments on 225 newly generated MPRP-OS instances with up to 200 customers, 20 vehicles, 6 periods and 12 products show the effectiveness and efficiency of the proposed heuristic. The performance of TLH is further demonstrated by testing 1530 classic PRP benchmark instances with up to 200 customers, 13 vehicles and 20 periods. Experimental results indicate that TLH is able to solve large-sized MPRP-OS instances and generally outperforms the state-of-the-art heuristics for the classic PRP. In addition, TLH provides new best solutions for 364 out of 1530 benchmark instances. Particularly, it finds 88 new best solutions for 90 very large-sized benchmark instances.