## 3 Mathematical Model of This Study
### Hypothesis  
In this study, we assumed the following conditions. * There is only one UAV involved;
* The operating time of the UAV is inversely proportional to the load weight; the heavier the load weight of the UAV, the shorter the operating time;
* The service time that the UAV spends on each customer will not be a focus;
* The delivery route is fixed;
* The impact of weather, buildings, or UAV malfunctions on the operating route and speed of the unmanned vehicle will not be considered;
* Once the load weight of the UAV is confirmed in the distribution center, the operating time will be confirmed, and will not increase due to the real-time decreasing load weight during delivery;
* The speed is constant; it will not change due to load requests, number of requests, or operating time;
* The UAV can serve many customers; each customer, however, can only be served once;
* The locations of the UAV and customer are known;
* We assume that the UAV is departing from the logistics center;
* Customer requirements will not change and are known;
* The UAV will return to the logistics center after delivery;
* The load weight carried is inversely proportional to the greatest load weight of the UAV;
* The UAV which departs from the logistics center is completely in-tact, everything runs well, and the UAV comes with a full tank of gas/fully charged battery. The route planning of this study is derived from the TSPTW and the period of the time during which the node must satisfy the services, or impassability time of the customer. Based on this condition, the route planning must be able to satisfy the request of the customer's period; the time period specified by the customer assigned is a necessary condition (hard time window). In a real case, some customers' acceptable requirements were delayed. Therefore, the target of the logistics service provider is the maximum profit, instead of obtaining the shortest total delivery time for all the customers. The target of this study is to obtain the maximum profit from an unmanned vehicle. That is the mean of the revenue minus the cost of operating the unmanned vehicle (1). The revenue is the delivery price of the request. The cost includes the variable cost (distance charge minus unmanned vehicle fixed cost), and the penalty cost of missing the time window (penalty cost is divided into delay penalty cost and early arrival penalty cost. The delay penalty coefficient is bigger than the early arrival penalty coefficient). As mentioned above, some customers' acceptable requirements are delayed. Nevertheless, we liberalize the time condition and add a penalty coefficient from the time window in order to deliver on time as often as possible to each node. Finally, we assume the weight of revenue and cost in order to provide different scheduling for the decision-making platform when rush requests occur. (2) After the unmanned vehicle enters one location, it will exit from that location. (3) The unmanned vehicle will avoid transporting within the location. (4) The unmanned vehicle will only serve the same customer once. (5) The time required to complete the total route is the distance divided by the constant speed. (6) The maximum operating time of the unmanned vehicle is such that the operating time of the unmanned vehicle is conversely proportional to the load carried. (7) The required operating time of the unmanned vehicle must be smaller than the maximum operating time. (8) The unmanned vehicle will depart from the current customer to the next customer to avoid looping. (9) There is a penalty cost of the unmanned vehicle missing the time window. There are only three situations: delayed, on time, or early; these cannot occur at the same time, so when the time difference is negative, it will be 0. (10) The rush request condition provides that a rush request is allowed when there is extra capacity for the unmanned vehicle to process a rush request and when the rush request revenue is greater than the rush request cost.