An exact method for the two-echelon split-delivery vehicle routing problem for liquefied natural gas delivery with the boil-off phenomenon

In this paper we investigate a two-echelon vehicle routing problem for liquefied natural gas (LNG) delivery to determine how to transport LNG from an overseas production terminal to a set of import terminals by vessels, and transport the LNG from the import terminals to a set of filling stations either by tanker trucks or bunker barges. Some important features of this problem are that part of LNG will be evaporated during delivery and split deliveries are allowed at both import terminals and filling stations, which render the problem more intractable than those considered in most of the existing two-echelon vehicle routing studies. The objective is to find the optimal first-echelon and second-echelon delivery schemes to minimize the sum of the routing cost and boil-off cost. To solve the problem, we develop a customized branch-and-price-and-cut (BPC) algorithm incorporating a specialized labeling algorithm tailored to address the issues of LNG evaporation and split deliveries in solving the challenging pricing subproblems. To speed up the solution algorithm, we introduce some heuristic pricing strategies to quickly solve the pricing subproblems, and explore the (strong) $k$-path inequalities and subset-row inequalities to tighten the lower bound obtained by column generation. We conduct extensive numerical studies on simulation instances and a case study of LNG delivery in region along the Yangtze river, China to verify the effectiveness of the model and proposed algorithm. The numerical results demonstrate that our algorithm significantly outperforms CPLEX and the existing BPC algorithm on related topic, confirm the superiority of our integrated two-echelon solution method over its sequential solution counterpart, and illustrate that the locations of the production terminal and import terminals are highly related to the solution performance.