A variational approach for supply chain networks with environmental interests

Nowadays, the supply chain networks, consisting of different tiers of decision-makers, provide an effective framework for the production, the distribution, and the consumption of goods. In this paper we propose a supply chain network optimization model where manufacturers, retailers and consumers in the demand markets have a degree of interest in environmental sustainability. The manufacturers can improve their energy level (assumed as variables), aim to minimize their environmental emissions (for production and transport) and can also establish the amount of quantity of the production waste to dispose in a eco-sustainable way. The retailers, who are also profit-maximizers, aim to minimize their environmental emissions (which depend on the chosen shipping methods). The consumers at demand markets make their own choices according to the prices and to their degree of aversion to the environmental emissions. We describe the behavior of each decision-maker and we present the mathematical model for each of them, deriving the variational inequality problems. Furthermore, we derive a unique variational inequality formulation for the entire network for whose solution an existence and uniqueness result is obtained. Finally, we illustrate some numerical simulations that highlight how the use of UAVs and the presence of waste sorting centers in the supply chain reduce environmental emissions and related costs.