The location routing problem (LRP) entails two problems: the Facility Location Problem (FLP) and the Vehicle Routing Problem with Time Windows (VRPTW). Both are NP-hard problems and thus have been studied individually. The simple approach to solving the LRP is to solve each problem sequentially. The sequential method solves the FLP first and then solves the VRPTW. This method is relatively easy.
  There is only one transportation mode for the original LRP. The transportation cost of the mode is calculated by travel distance. In the real world, however, we combine transportation mode options so as to reduce the cost. Our objective is to extend the original LRP, so as to handle transportation mode options. We propose an extended model of the LRP, along with a solving algorithm. Our method is based on Mixed Integer Programming (MIP), which has a pre-process and post-process to combine the VRPTW. Computational experiments showed that our method reduced delivery cost by 12% compared to the sequential method.