A Lagrangian Relaxation Heuristic for a Bi-Objective Multimodal Transportation Planning Problem

We study a realistic Bi-objective Multimodal Transportation Planning Problem (BMTPP) faced by logistics companies when trying to obtain cost advantages and improve the customer satisfaction in a competitive market. The two objectives considered are: the minimization of total transportation cost and the maximization of service quality. Given a set of transportation orders described by an origin, a destination and a time window, solving BMTPP involves determining the delivery path for each order in a capacitated network as well as selecting the carrier with the best service quality for each edge of the path. The BMTPP is formulated as a novel bi-objective mixed integer linear programming model and an iterative $\epsilon $ -constraint method is applied to solve it. As the NP-hardness of the single-objective problems derived from BMTPP, a Lagrangian Relaxation (LR) heuristic which can not only provide a near-optimal solution but also a lower bound for each of the single-objective problems is developed. 100 randomly generated instances are tested and the computational results demonstrate the effectiveness of the heuristic in obtaining a tight lower bound and a high-quality near-optimal solution for the derived single-objective problem. Various performance indicators show the high-quality of the Pareto front of the bi-objective problem obtained by the heuristic. We also provide a case study for the proposed LR heuristic in a logistics network in China.