Optimal Control and Value of Reverse Logistics in Supply Chains with Multiple Flows of Product

Abstract

Reverse logistics has been gaining recognition in practice (and theory) for helping companies better match supply with demand, and thus reduce costs in their supply chains. In this paper, we study reverse logistics from the perspective of a supply chain in which each location can initiate multiple flows of product. Our first objective is to jointly optimize ordering decisions pertaining to regular, reverse and expedited flows of product in a stochastic, multi-stage inventory model of a logistics supply chain, in which the physical transformation of the product is completed at the most upstream location in the system. Due to those multiple flows of product, the feasible region for the problem acquires multi-dimensional boundaries that lead to the curse of dimensionality. To address this challenge, we develop a different solution method that allows us to reduce the dimensionality of the feasible region and, subsequently, identify the structure of the optimal policy. We refer to this policy as a nested echelon basestock policy, as decisions for different product flows are sequentially nested within each other. We show that this policy renders the model analytically and numerically tractable. Our results provide actionable policies for firms to jointly manage three different product flows in their supply chains, and allow us arrive at insights regarding the main drivers of the value of reverse logistics. One of our key findings is that, when it comes to the value generated by reverse logistics, demand variability (i.e., demand uncertainty across periods) matters more than demand volatility (i.e., demand uncertainty within each period). This is because, in the absence of demand variability, it is effectively never optimal to return product upstream, regardless of the level of inherent demand volatility. Our second objective is to extend our analysis to product transforming-supply chains, in which product transformation is allowed to occur at each location. In such a system, it becomes necessary to keep track of both the location and stage of completion of each unit of inventory, so that the number of state and decisions variables increases with the square of the number of locations in the system. To analyze such a supply chain, we first identify a policy that provides a lower bound on the total cost. Then, we establish a special decomposition of the objective cost function that allows us to propose a novel heuristic policy. We find that the performance gap of our heuristic policy relative to the lower-bounding policy averages less than 5% across a range of parameters and supply chain lengths.