A branch-and-price algorithm for location-routing problems with pick-up stations in the last-mile distribution system

In catering to the needs of the growing e-commerce demand and environmentally friendly urban delivery solutions, package delivery companies are increasingly focusing on user experience and sustainable operations, such as alternative delivery methods (e.g., pick-up stations) and green delivery vehicles (e.g., electric vehicles). We consider designing a location-routing problem with pick-up stations (LRP-PS) where the location of pick-up stations to open and the delivery plan of green vehicles (GVs) are optimized simultaneously. The purpose is to satisfy the total demand while minimizing the sum of opening cost and handling cost of pick-up stations and the fixed cost and routing cost of GVs. Then, we present a compact mixed-integer programming formulation to define the LRP-PS and reveal some interesting properties of the optimal solution. We propose an effective branch-and-price (B&P) algorithm to solve the LRP-PS, which is demonstrated to outperform greatly commercial branch-and-cut solvers such as CPLEX in the computational study. Finally, through a comprehensive analysis of several key parameters (e.g., coverage ranges of pick-up stations and battery driving ranges of GVs), we assess the impact of pick-up stations on this last-mile distribution system and provide some helpful business insights for package delivery companies.