# Joint Location-Allocation Model for Multi-Level Maintenance Service Network in Agriculture
## 3 Problem Description and Model Formulation  
### Problem Description  
To supply timely maintenance for failed machinery in the busy farming seasons, manufacturers always construct a multi-layer maintenance service network. There exist three layers in the service network, which include spare part centres, service stations, and service units. Service units are a set of connected production areas including agricultural machinery, just like a county in China, and a connected service area always means a similar climate and ripening time, which represents maintenance demands. Service stations with several service vehicles are built by manufacturers to supply maintenance for the demand for service units.  
Thus, the multi-layer facility location-allocation problem (MLFLAP) can be described as follows. There are \(n\) service units with defined maintenance demands in agriculture. In order to supply maintenance, \(l\) service stations are selected from the potential location set of service stations to supply maintenance service, and \(m\) service vehicles with defined service capacity are allocated to the selected service stations. In addition, several spare part centres are built by the manufacturer to supply spare parts for service stations.  
Assuming that there are 20 service units, four service stations are selected to supply service for these service units, and nine service vehicles are allocated to the selected service stations. In Figure 1, service stations are located in service units 4, 8, 14, and 17, and the service units served by each service station are distinguished by bold lines. As for service station 4, there exists one service vehicle that can provide maintenance service for four service units, e.g., 3, 4, 5, and 10. As for service station 17, two service vehicles are allocated to this service station, and five service units, such as 11, 12, 16, 17, and 18, are served by service station 17. In addition, an indefinite number of spare part centres are built by the manufacturer to supply spare parts for service stations, which are served by the nearest spare part centres. As shown in Figure 1, in order to supply spare parts for service stations, two spare part centres are selected in service units 12 and 9, and the maximal distribution distance for these spare part centres is shown with black dotted lines. Thus, the spare part centre in 9 can supply spare parts for service stations 4, 8, and 14, while the spare part centre in 12 can supply spare parts for service stations 8 and 17.  
Thus, the proposed MLFLAP can be divided into several subproblems: (1) Service station location and allocation problem in agriculture, which is to decide the location of service stations while allocating service vehicles to these stations. (2) Spare part centre location problem, which is to determine the location of spare part centres. (3) Service unit allocation problem, which is to assign these service units to selected service stations. The objective of the proposed MLFLAP is to minimize total costs, which include total service mileage costs from service stations to service units, total usage costs of service vehicles, and total construction costs for spare part centres.  
Several assumptions are presented in this research:  
(1) The maintenance requirement in service units is known.  
(2) One service unit can only be served by one service station.  
(3) Several service units served by one service station are continuous.  
(4) There is a defined service capacity for service vehicles, and there exists a maximal distribution distance for spare part centres to service stations.