## 3 Algorithms
### Parcel-Receiving Station and Public Route Planning Based on GA-MMAS
#### 3.2.2 Public Air-Route Planning Based on MMAS  
The process of determining the public air-route configuration is the process of generating the minimum tree for all nodes. The greedy algorithm is used when planning the route of each station with a certain planning sequence to form the public air route. When planning the route for station i, based on the public air-route planning results of the first i-1 stations, we select the node-connection mode with the smallest increment to Function (17). According to Function (17), if the route of station i is a direct route, Function (17) should increase the air-range cost and flight-risk cost of the direct route on the basis of its original value. If the route of station i is a transit route, Function (17) should increase the cost of the air range passed by all edges and the cost of the flight risk from the transit point to station i on the basis of its original value. As shown in Figure 6a, the dotted line in Step 0 is the set of edges that can be selected between nodes after single-air-route planning. In Step 0, since there is no station that has a planned route with warehouse O, only the direct route OA can be planned for station A. In Step 1, because there is an air route between stations A and O, we can choose to plan the OC direct route and OA-AC transit route when planning the air route for station C. According to Function (17), the increased cost of the OC direct route includes the air-range cost and flight-risk cost of edge OC. The increased cost of the OA-AC transit route includes the cost of the air range of edge OA and edge AC and the cost of the flight risk of edge AC. We compare the increased cost of OC and OA-AC and choose the smaller increased cost-planning method (in Figure 6a, it is assumed that the increased cost of OC is smaller). In Step 2, the route that can be planned for station B includes the direct route OB and the transit routes OA-AB and OC-CB, and the planning method with the minimum increased cost is selected (OA-AB is assumed to have the minimum increased cost in the figure). In the order of the planning sequence of A-C-B, the route of station A is the direct route OA, the route of station B is the transit route OA-AB, and the route of station C is the direct route OC. The public air-route contains three edges, which are OA, AB, and OC. In Figure 6b, we find that different route configurations can be obtained with the planning sequence of B-A-C using the same greedy algorithm. The route of station A is the transit route OB-BA, the route of station B is the direct route OB, and the route of station C is the transit route OB-BC. The public air-route contains three edges, which are OB, BA, and BC. Therefore, after the parcel-receiving station location is determined, it is also necessary to determine the optimal public air-route planning sequence of stations, and then plan the public route according to the greedy algorithm above to obtain the optimal public route configuration. In this section, the max-min ant system (MMAS) is used to solve the route-planning sequence of each parcel-receiving station and then determine the configuration of the public air route. Compared with the ordinary ant colony algorithm, the max-min ant system used in this section is mainly different in the following ways:  
1. The max-min ant system strengthens the feedback of the optimal path information, only allowing the pheromone updates on the optimal path of each generation: \[\tau_{ij}(t+1)=(1-\rho)\tau_{ij}(t)+\Delta\tau_{ij}^{best}\] (28) where \(\Delta\tau_{ij}^{best}=1/c_{best}\), and \(c_{best}\) refers to the optimal ant's total cost of public air-route operating in this iteration. The cost of each ant can be calculate by Function (17). 2.