Solving Resource Recharging Station Location-routing Problem through a   Resource-space-time Network Representation
  The resource recharging station location routing problem is a generalization of the location routing problem with sophisticated and critical resource consumption and recharging constraints. Based on a representation of discretized acyclic resource-space-time networks, we propose a generic formulation to optimize dynamic infrastructure location and routes decisions. The proposed integer linear programming formulation could greatly simplify the modeling representation of time window, resource change, and sub-tour constraints through a well-structured multi-dimensional network. An approximation solution framework based on the Lagrangian relaxation is developed to decompose the problem to a knapsack sub-problem for selecting recharging stations and a vehicle routing sub-problem in a space-time network. Both sub-problems can be solved through dynamic programming algorithms to obtain optimal solution. A number of experiments are used to demonstrate the Lagrangian multiplier adjustment-based location routing decision making, as well as the effectiveness of the developed algorithm in large-scale networks. 
Gongyuan Lu, Xuesong Zhou, Qiyuan Peng, Bisheng He, Monirehalsadat   Mahmoudi, Jun Zhao
math.OC cs.NI