A cross docking facility is a type of warehouse in supply chain management that allows orders to be prepared with or without going through the phase of storing products in the warehouse and subsequently selecting them for delivery. The goods are unloaded from incoming trucks called origins on inbound doors of a cross-docking facility platform and, using a handling device inside the platform such as a forklift, immediately transferred to outbound doors to be loaded into outgoing trucks named destinations or delivery trucks for distribution to customers. Contrary to a traditional warehouse, goods are unloaded and loaded without placing them in temporary storage inside the cross-docking facility. The goal of the cross-docking assignment problem (CDAP) is to assign origins to inbound doors and destinations to outbound doors so that the total cost inside the cross-dock platform is minimized. To the best of our knowledge, there are only three mixed integer programming (MIP) formulations of the CDAP in the literature. We propose eight new MIP models and demonstrate the mathematical equivalence of all 11 models, together with rigorously proving some of their properties. In order to detect which of these 11 models is best, we conduct an extensive comparative analysis on benchmark instances from the literature, which discloses that the best model is one proposed in this paper for the first time.