A methodology for the cross-dock door platforms design under uncertainty

The cross dock door design problem consists of deciding on the number and capacity of inbound and outbound doors for receiving product pallets from origin nodes and exiting them to destination nodes. The uncertainty, realized in scenarios, lies in the occurrence of these nodes, the number and cost of the pallets, and the disruption of the capacity of the doors. It is represented using a stochastic two stage binary quadratic model. The first stage decisions are related to the cross dock infrastructure design, and the second stage decisions are related to the node to assignments of the doors. This is the first time, as far as we know, that a stochastic two stage binary quadratic model has been presented for minimizing the construction cost of the infrastructure and its exploitation expected cost in the scenarios. Given the difficulty of solving this combinatorial problem, a mathematically equivalent mixed integer linear formulation is introduced. However, searching an optimal solution is still impractical for commercial solvers. Thus, a scenario cluster decomposition based matheuristic algorithm is introduced to obtain feasible solutions with small optimality gap and reasonable computational effort. A broad study to validate the proposal gives solutions with a much smaller gap than the ones provided by a state of the art general solver. In fact, the proposal provides solutions with a 1 to 5% optimality gap, while the solver does it with up to a 12% gap, if any, and requires a wall time two orders of magnitude higher.