The diversification of customers’ geographic locations forces companies’ delivery systems to travel long distances increasing distribution costs. In this regard, customers in an inconvenient location for one distribution company could be ideal for another company. Therefore, a set of distribution companies may shorten the delivery distance by combining urban distribution networks. The feeder vehicle routing problem (FVRP) is a new type of VRP to provide fast services in urban transportation. Unlike VRP, FVRP includes a fleet of heterogeneous vehicles (i.e., trucks and motorcycles) in which trucks and motorcycles move from the depot to serve customers. In this problem, motorcycles pass easily in crowded areas, and the traffic of urban logistics is distributed easily. Since returning to the depot and re-shipment to the customers increases cost and the distance traveled, one strategy to deal with this problem is applying the “joint” mechanism. In this mechanism, motorcycles visit the trucks rather than return to the depot at the joint points. Also, the feeder approach lowers the number of times the vehicle returns to the central depot for loading, resulting in cost and time savings. This study introduces a collaborative feeder vehicle routing problem with flexible time windows (CFVRPFlexTW) as a bi-objective model that simultaneously minimizes routing costs and maximizes customer satisfaction with a flexible time window. After modeling CFVRPFlexTW through mixed-integer linear programming (MILP), the augmented epsilon constraint (AEC) approach is applied in the CPLEX solver to solve the problem. Also, multi-objective particle swarm optimization with dynamic inertia weigh (WMOPSO) and MOPSO with adaptive learning strategy (LAMOPSO) were developed regarding the complexity of the problem. Then, their performance is compared with that of the Pareto solutions produced by the non-dominated sorting genetic algorithm-II (NSGA-II). The computational outcomes indicate the outperformance of the WMOPSO based on some related metrics. Eventually, the AHP-TOPSIS method is applied to prioritize and analyze the algorithms. The results indicate the proposed LAMOPSO algorithm is more efficient in small-size instances. Furthermore, in large-size instances, the WMOPSO algorithm outperforms the MOPSO, LAMOPSO, LAWMOPSO, and NSGA-II algorithms.
