## 7 Conclusions  
This paper addressed the multi-objective sustainable truck scheduling in the Road-Rail PI-hub cross-dock. The problem was formulated as a Multi-Objective Mixed Integer Programming model (MO-MIP) considering two different objectives. The first objective minimizes the energy consumption cost for the routing of the PI-containers using PI-conveyors. The second one minimizes the cost of using the outgoing trucks for each destination. The model was then solved using Lexicographic Goal Programming in CPLEX solver. Due to the long computational times, two Multi-Objective hybrid meta-heuristics were proposed: MO-VNSSA and MO-VNSTS. The MO-MIP and the two meta-heuristics were evaluated on 27 small instances and 5 additional large instances. CPLEX found optimal solutions for only 23 instances. The obtained results showed that the two meta-heuristics were able to generate near optimal and optimal solutions within short computational times. The results were validated through an ANOVA analysis.  
In this study we used a lexicographic based method, which requires a priori knowledge from the decision maker. As an important direction of this work, we intend to extend our study by developing other Pareto based approaches that do not require a priori preferences about the relative importance of the objective functions. Those Pareto based approaches can be developed using population based  
Figure 6: Convergence behavior of MO-VNSSA and MO-VNSTS for the second objective \(F_{2}\).  
meta-heuristics such as NSGA, MOPSO and MOGA [75]. As another possible direction of this work, integrating external simulators could be interesting for taking into account the possible perturbations that can occur in the PI-hub facility such as trucks delays, customers changing orders at the last minute, etc.  
Conceptualization, T.C., A.B., M.R. and D.T.; data curation, T.C.; formal analysis, T.C.; funding acquisition, A.B., M.R. and D.T.; investigation, T.C.; methodology, T.C., A.B., M.R. and D.T.; project administration, A.B., M.R. and D.T.; software, T.C.; supervision, A.B., M.R. and D.T.; validation, T.C.; writing-original draft, T.C.; writing-review and editing, T.C., A.B., M.R. and D.T.  
This research was supported by the ELSAT2020 project of CPER sponsored by the French ministry of sciences, the Hauts de France region and the FEDER. This work was supported also by the ANR PI-NUTS Project (grant ANR-14-CE27-0015).  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} ACO & Ant Colony Optimization \\ GA & Genetic Algorithm \\ GLS & Guided Local Search \\ GRASP & Greedy Randomized Adaptive Search Procedure \\ LS & Local Search \\ MOGA & Multi-Objective Genetic Algorithm \\ MO-MIP & Multi-Objective Mixed Integer Programming Model \\ MOPSO & Multi-Objective Particle Swarm Optimization \\ MO-VNSSA & Multi-Objective Variable Neighborhood Search - Simulated Annealing \\ MO-VNSTS & Multi-Objective Variable Neighborhood Search - Tabu Search \\ NSGA & Non-dominated Sorting Genetic Algorithm \\ PI & Physical Internet \\ PMA & Pareto Memetic Algorithm \\ PSO & Particle Swarm Optimization \\ SA & Simulated Annealing \\ SCBM & Simulated Constraint Boundary Method \\ TL & Tabu List \\ TS & Tabu Search \\ VNS & Variable Neighborhood Search \\ \end{tabular}