Exact solution approaches for bilevel lot‐sizing

In this paper we propose exact solution methods for a bilevel uncapacitated lot‐sizing problem with backlogs. This is an extension of the classical uncapacitated lot‐sizing problem with backlogs, in which two autonomous and self‐interested decision makers constitute a two‐echelon supply chain. The leader buys items from the follower in order to meet external demand at lowest cost. The follower also tries to minimize its costs. Both parties may backlog. We study the leader’s problem, i.e., how to determine supply requests over time to minimize its costs in view of the possible actions of the follower. We develop two mixed‐integer linear programming reformulations, as well as cutting planes to cut off feasible, but suboptimal solutions. We compare the reformulations on a series of benchmark instances.