# Research on Demand-Based Scheduling Scheme of Urban Low-Altitude Logistics UAVs
## Problem Description and Modeling
### Model Building
#### 2.3.2 Objective Functions  
(1) Time cost  
This paper assumes that the time-cost coefficient \(T_{a}\) is a periodic function, and the period is denoted as \(T_{t}\). Considering the impact of peak times, \(T_{a}\) can be approximately expressed as an exponential function:  
\[T_{a}=\left\{ \begin{array}{ll}T_{a0}&t_{ij}^{m}\leq t_{1}\\ k_{1}\cdot e_{ij}^{m}+b_{1}&t_{1}<t_{ij}^{m}<=t_{2}\\ k_{2}\cdot e_{ij}^{m}+b_{2}&t_{2}<t_{ij}^{m}<=t_{3}\end{array} \right. \tag{8}\]  
where \(T_{a0}\) represents the time-cost coefficient under standard conditions; other variables represent the time-cost coefficient during peak times. Take the time-cost of drone transportation as a factor, which can be expressed as:  
\[Z_{1}=\sum\limits_{m=1}^{M}\sum\limits_{i=1}^{A}\sum\limits_{j=1}^{J}t_{ij}^{m }\cdot T_{\kappa} \tag{9}\]  
where \(t_{ij}^{m}\) refers to the delivery time of the drone, \(T_{\alpha}\) represents the time-cost coefficient of the delivery, and \(Z_{1}\) represents the total time cost of completing all the delivery tasks. \(A\) represents the set of routes for all drone flights, and obviously, the distance of the drone flight route is greater than the total distance of the entire route itself. \(J\) is a collection of requirement points. \(Z_{1}\) represents the time spent. This is the total cost incurred by each drone during each delivery. (2) Risk cost  
Let \(D_{\beta}\) denote the risk-cost coefficient of the drone transportation process, which is related to the transportation distance and peak times. On the one hand, the drone risk-cost coefficient is related to the distribution distance. The longer the distribution distance, the more likely it is that the risk will occur. Therefore, the risk-cost coefficient increases with the distribution distance. Let \(L_{t}\) indicate the total number of routes in the network, then:  
\[l_{a}=\frac{\sum\limits_{i=1}^{A}\sum\limits_{j=1}^{J}l_{ij}}{L_{t}} \tag{10}\]  
where \(l_{a}\) represents the average distance of each delivery route. Let \(c_{a}\) represent the risk-cost coefficient corresponding to the distance \(l_{a}\), then the drone risk-cost coefficient of each delivery route is:  
\[D_{\beta}{}^{ij}=\frac{l_{ij}}{l_{a}}c_{a} \tag{11}\]  
On the other hand, the drone risk-cost coefficient is related to peak times. During peak times of busy distribution, the risk-cost coefficient also increases due to the presence of vehicles, pedestrians, and other influencing factors. This paper assumes that the risk-cost coefficient \(D_{\beta}\) is a periodic function, and the period is recorded as \(T_{t}\). Therefore:  
\[D_{\beta}= \begin{cases}D_{\beta}{}^{ij}&t_{ij}^{m}\leq t_{1}\\ k_{3}\cdot e^{t_{ij}^{m}}+b_{3}&t_{1}<t_{ij}^{m}<=t_{2}\\ k_{4}\cdot e^{t_{ij}^{m}}+b_{4}&t_{2}<t_{ij}^{m}<=t_{3}\end{cases}  \tag{12}\]  
where \(t_{1}\), \(t_{2}\), \(t_{3}\) are the same as those in Formula (8). Take the risk cost of drone transportation as a factor, which can be expressed as:  
\[Z_{2}=\sum_{m=1}^{M}\sum_{i=1}^{A}\sum_{j=1}^{J}\mathrm{x}_{i}\cdot\mathrm{y} _{i}\cdot l_{ij}^{m}\cdot D_{\beta} \tag{13}\]  
where \(l_{ij}^{m}\) is the delivery distance of the drone, \(D_{\beta}\) is the risk-cost coefficient of the delivery, and \(Z_{2}\) represents the total risk cost of completing all the delivery tasks. The farther the distance, the closer the time is to peak hours and the higher the risk. (3)  
Use \(D_{\delta}\) to represent the maneuvering-cost coefficient of the drone transportation process. Every time the drone \(m\) is dispatched to perform the delivery task, a maneuvering cost will be generated. The maneuvering-cost coefficient is related to the load capacity of the drone. The higher the load capacity of the drone, the greater the maneuvering-cost coefficient. The formula for calculating the maneuvering-cost coefficient is as follows:  
\[D_{\delta}=\left\{ \begin{array}{ll}c_{1}&w_{k}=w_{1}\\ c_{2}&w_{k}=w_{2}\\ \ldots&\\ c_{n}&w_{k}=w_{n}\end{array} \right.