## 1 Introduction  
Nowadays, global optimization of the supply chain is becoming the main goal of many industrial companies, especially the logistics distribution ones. The objective is to globally reduce the economical cost and to increase the productivity while taking into consideration the social and environmental aspects. Recently, the efficiency and reactivity of the supply chains has become a big challenge for distribution companies to satisfy retailers and customer demands in terms of cost, quality and delivery time [1, 2]. Therefore, with the increase of environmental constraints, supply chain sustainability has emerged as a major approach for logistics firms to enhance their economical, social and environmental sustainability [3; 4; 5]. Since supply chains are composed of different elements (suppliers, production plants, distribution centers, retailers and customers), a robust coordination is necessary to ensure the flexibility of the products flow. In addition to the linkage between all those components of the supply chain, the structure and the configuration of the supply chain has also a major impact on the supply chain sustainability [6; 7]. Cross-docking is a type of distribution centers and it is considered as one of the most efficient distribution techniques used in supply chain management [8; 9]. It consists on transferring products between inbound and outbound trucks, trains or other vehicles. The process of cross-docking starts with the unloading of the products from the inbound vehicles; then, the products may be either stored temporarily in the cross-dock facility or directly transferred and loaded into the outgoing vehicles. In order to overcome the sustainability issue and to consider all its aspects (economical, environmental and social), Physical Internet (PI or \(\pi\)) has been presented as a new paradigm based on a metaphor from the Digital Internet by encapsulating products in standardized modular containers called PI-containers which are handled and moved using standard collaboration protocols inspired from TCP-IP protocols of the Digital Internet [10]. The idea of the PI is to build a worldwide interconnected logistics network through PI-nodes (PI-transits, PI-hubs, etc.) enabling the external and internal routing of PI-containers between the PI-nodes using PI-movers (PI-trucks, PI-wagons, etc.) [11]. The literature of the optimization problems related to the PI concept has considerably grown over the last few years [12; 13; 14]. Most of the papers in the PI literature focus on the global interconnected supply chain rather than PI-nodes, especially cross-docking PI-hubs. In this paper, the focus is on the cross-docking terminals. There are many types of PI-nodes that can be identified in the PI network. For example:  
* Road-Road PI-hubs: Cross-docking terminals used to efficiently transfer the PI-containers between inbound and outbound trucks through PI-sorters. They are composed of PI-sorters connected to the PI-docks using maneuvering areas that arrange the PI-containers after being unloaded from inbound trucks to be routed in the PI-sorters and then grouped and loaded into the outgoing trucks [15]. * Road-Rail PI-hubs: Use the same mechanism as the Road-Road PI-hubs. However, the PI-containers are transferred between trucks and trains and between trains and other trains using PI-sorters for routing and maneuvering areas to arrange containers. The interested reader can refer to [16] for a detailed functional design of the Road-Rail PI-hubs. * Water-Road PI-hubs: Are designed to transfer the PI-containers between boats and trucks in a port terminal [10]. We center our attention in this paper on the Rail-Road cross-docking PI-hubs. The objective is to find a truck schedule that minimizes the total cost which includes the energy cost of the routing of PI-containers through the PI-conveyors and, at the same time, the cost of using outbound trucks for each different destination. The problem is formulated as a Multi-Objective Mixed-Integer Programming model (MO-MIP).