A multi-path routing-inventory problem for a closed-loop supply chain considering the heterogeneous fleet of vehicles

In this paper, the inventory-routing problem is studied for a closed-loop supply chain. This closed-loop supply chain considers suppliers, manufacturers, whole-sellers, and disposal centers. To formulate this problem, a mixed integer linear programming model is proposed. This mathematical model minimizes the total costs of the supply chain, including the fixed and variable costs of vehicles, and holding inventory costs of final products and scraps. The proposed model considers the road roughness degree, multi-path setting and the heterogeneous fleet of vehicles, which increases its flexibility and the quality of solutions. Then, two symmetry-breaking constraints are proposed to reduce the complexity of the mathematical model. In order to evaluate the integrity of the proposed model, 20 instances of different sizes are randomly generated and solved. Finally, a comprehensive sensitivity analysis is conducted with respect to five key features of the problem, such as the impact of the symmetry-breaking constraints on the CPU time, multi-path setting, fixed cost of vehicles, heterogeneous fleet of vehicles, and lost sales. The results indicate that the consideration of multi-path setting and the heterogeneous fleet of vehicles improves the quality of solutions significantly.