Closed-loop supply chain network design with multiple transportation modes under stochastic demand and uncertain carbon tax

We optimize the design of a closed-loop supply chain network that encompasses flows in both forward and reverse directions and is subject to uncertainty in demands for both new and returned products. The model also accommodates a carbon tax with tax rate uncertainty. The proposed model is a three-stage hybrid robust/stochastic program that combines probabilistic scenarios for the demands and return quantities with uncertainty sets for the carbon tax rates. The first stage decisions are facility investments, the second stage concerns the plan for distributing new and collecting returned products after realization of demands and returns, and the numbers of transportation units of various modes are the third stage decisions. The second- and third-stage decisions may adjust to the realization of the carbon tax rate. For computational tractability, we restrict them to be affine functions of the carbon tax rate. Benders cuts are generated using recent duality developments for robust linear programs. Computational results show that adjusting product flows to the tax rate provides negligible benefit, but the ability to adjust transportation mode capacities can substitute for building additional facilities as a way to respond to carbon tax uncertainty.