A production-routing problem with price-dependent demand (PRP-PD) is studied in this paper. Demand follows a general convex, differentiable, continuous and strictly decreasing function in price. The problem is modeled as a mixed integer nonlinear program (MINLP). Two Outer Approximation (OA) based algorithms are developed to solve the PRP-PD. The efficiency of the proposed algorithms in comparison with commercial MINLP solvers is demonstrated. The computational results show that our basic OA algorithm outperforms the commercial solvers both in solution quality and in computational time aspects. On the other hand, our extended (two-phase) OA algorithm provides near-optimal solutions very efficiently, especially for large problem instances. These findings prevail both for linear and for nonlinear demand functions. Additional sensitivity analyses are conducted to investigate the impact of different problem parameters on the optimal solution. The results show that the manufacturer should give higher priority to the retailer who has lower price sensitivity and who is closer to the manufacturer. Another takeaway is that a larger market size and a lower price sensitivity lead to more profit.