Logistics and supply chain planning is one of the vital aspects of modern business. In this regard, location routing plays a prominent role. According to logistics and supply chain system studies, locating depots, regardless of the transportation routes of vehicles, may significantly increase the costs of the logistics systems. Therefore, it is essential to consider depot location and vehicle routing simultaneously in location-routing problems. This study aims to develop a new mathematical model and two meta-heuristic solution methods for the problem. The option of buying or renting vehicles, along with constraints such as vehicle and warehouses capacities, and the maximum allowable vehicle operation time were included in the problem to approach the real-world conditions further. The mathematical model of the problem involved two objectives: minimizing the costs (including warehousing and transportation costs in tours) and maximizing customer service (i.e., maximizing the total amount of demand sent to customers). The efficient ε-constraint method was implemented using the GAMS optimization software to solve the problem. Regarding the complexity and time-consuming nature of the problem, it was impractical to solve large-scale instances using exact methods like the ε-constraint method; thus, the second version of the genetic algorithm was utilized through non-dominated sorting and multi-objective particle swarm optimization algorithm. Both quantitative and qualitative performance indicators were evaluated, and the solutions were analyzed and compared to assess the performance of the proposed method.
