# Joint Location-Allocation Model for Multi-Level Maintenance Service Network in Agriculture
## 2 Literature Review  
Facility location-allocation problems (FLAPs) have been a research focus for several years and are applied to determine the location of service facilities and the allocation of service resources [14; 15]. FLAPs have been applied in different areas, including emergency service systems [16], supply chain networks [17], transportation and logistics systems [18], urban health and services systems [19], and maintenance service networks [20]. In classical single-level FLAPs, the location of service facilities is selected and the allocation of service resources is also defined. However, there always exist multi-level facilities in service networks, and multi-level facility location-allocation problems (MLFAPs), as a subclass of FLAPs, have attracted more attention in the last two decades [15; 21]. For example, [PERSON] et al. [22] concentrated on the two-echelon humanitarian relief logistics system and developed a Lagrangian relaxation-based heuristic algorithm to determine the location of pre- and post-disaster rescue centres. [PERSON] and [PERSON] et al. [23] conducted research to determine the location of plants and warehouses in reality and combined the Lagrangian relaxation method and simulated annealing algorithm to obtain minimal transportation and production costs. [PERSON] and [PERSON] et al. [24] considered the location problem of warehouses and retailers in a two-echelon supply chain network and developed a novel cutting-plane approach to solve the real case with moderate size. [PERSON] et al. [25] focused on the facility location problem in multi-level delivery systems with retailers, refuel stations, and warehouses, and developed a novel solution method to minimize transportation costs considering demands uncertainty and M/G/K queueing system. [PERSON] et al. [26] considered a multi-objective optimization model to determine the location of intermediate healthcare facilities and two heuristic algorithms, including particle swarm optimization and a non-dominated sorting genetic algorithm, were designed to solve several case studies in healthcare supply chain logistics. As for mathematical models for FLAPs, they are always formulated as mixed-integer program (MIP) models. For example, [PERSON] et al. [27] developed a mixed-integer programming model to solve congested facility location problems considering machine failures. [PERSON] et al. [28] considered the fixed and dynamic demand and developed a mixed-integer linear programming model to determine the location of public facilities. [PERSON] et al. [29] focused on facility location problems in food supply chain networks and formulated a multi-objective model to determine the number and location of service facilities. [PERSON] et al. [30] constructed an MIP-based optimization model to determine the location of collection/inspection centres and minimize the total costs in a closed-loop green supply chain system. [PERSON] et al. [16] developed a multi-objective optimization model to minimize total allocation costs while minimizing the total amount of relief suppliers. [PERSON] et al. [31] considered the service facility location problem in urban service systems and formulated an MILP for minimal costs based on the robust optimization approach. Solution approaches in FLAPs can be classified into two groups: exact approaches and heuristic algorithms [32,33]. As for exact approaches, [PERSON] et al. [34] designed a decomposition-based solution approach to obtain a near-optimal location plan and compared it with another exact method to demonstrate the performance of the developed approach. [PERSON] et al. [35] combined the Benders decomposition method and commercial solver with the brand and cut method to obtain the best service facility location, and computational results were presented to show the method's efficiency compared with exact methods and heuristic algorithms. [PERSON] et al. [36] combined the integer linear programming method and aggregation approach to obtain the optimal location of distribution centres.