We propose a new binary formulation of the Travelling Salesman Problem (TSP), with which we overcame the best formulation of the Vehicle Routing Problem (VRP) in terms of the minimum number of necessary variables. Furthermore, we present a detailed study of the constraints used and compare our model (GPS) with other frequent formulations (MTZ and native formulation). Finally, we have carried out a coherence and efficiency check of the proposed formulation by running it on a quantum annealing computer, D-Wave\_2000Q6.

The Vehicle Routing Problem (VRP) and its variants are found in many fields, especially logistics. In this study, we introduced an adaptive method to a complex VRP. It combines multi-objective optimization and several forms of VRPs with practical requirements for an urban shipment system. The optimizer needs to consider terrain and traffic conditions. The proposed model also considers customers' expectations, shipper considerations as goals, and the common goal like transportation cost. We offered compromise programming to approach the multi-objective problem by decomposing the original multi-objective problem into a minimized distance-based problem. We designed a hybrid version of the Genetic algorithm with the Local Search algorithm to solve the proposed problem. We evaluate the effectiveness of the proposed algorithm with the Tabu Search algorithm, the original Genetic algorithm on the tested dataset. The results show that our method is an effective decision-making tool for the multi-objective VRP and an effective solver for the new variation of VRP.