Mathematical modeling for green supply chain considering product recovery capacity and uncertainty for demand.

Competition in today's market is the most important concern of companies and producers in free markets. Buyers are also looking for higher quality and lower prices. Manufacturers should, therefore, reduce production costs and increase budgets for research and product development. On the other hand, the limitation of mineral resources in each country and in the world in general is a very important factor for increasing the price of raw materials which increases the cost of production of a product. In this study, a green aspect of decision-making, concurrent modeling for inventory-routing, and application of maximum entropy (ME) method for overcoming uncertainties of demands are applied to optimize the usage of raw materials and returning of defective products to the production cycle in a closed-looped supply chain under multi-period planning horizon. Also, dynamic modeling is used to balance the inventory level in all stages of the network that leads to optimum usage of the raw materials. For this purpose, the first objective function reduces production, transportation-routing, and inventory costs, and the second objective reduces greenhouse gas emissions through all levels of the network. Finally, this model is solved by using the exact solution method with the help of Gams software as well as the non-dominated sorting genetic algorithm II (NSGAII) and multi-objective particle swarm optimization (MOPSO) algorithm. Sensitivity analysis has been performed on failure rates, greenhouse gas emissions during recycling and production, and the optimistic-pessimistic coefficient of the ME solution method. Solution methods have been compared using several criteria, and the NSGAII method has finally obtained the best result. The results show that the manager should pay more costs in order to prevent backorder demands. Also, collecting the more defective products leads to increasing production amount since the collective products can return to the production line. Finally, it is required for the managers to control products' failure rate to optimize capacity usage in the model.