# Drone-Based Parcel Delivery Using the Rooftops of City Buildings: Model and Solution
## 3 Model Development
### Mathematical Model  
Considering the above assumptions and the current drone technologies, we propose a new drone operation planning model for effectively delivering parcels in urban areas. The proposed model is based on MILP. To formulate the model, the following parameters and variables are required. _Indices_  
\[i\]  
drone index ( \[i=1,\ldots,l\] ). \[j\]  
destination (rooftops) index ( \[j=1,\ldots,l\] ). \[k\]  
drone operation index ( \[k=1,\ldots,K\] ). The upper bound of the drone operation index, \[K\], can be set by adding the estimated number of recharging operations and the estimated number of delivery operations. Of course, \[K\] can be adjusted during the pre-experiments. _Parameters_  
\[D_{j}\]  
demand at destination \[j\] (unit: number of parcels). \[i=1,\ldots,l\]0  
distance between the drone operation center and rooftop \[i=1,\ldots,l\]1 (unit: km). \[i=1,\ldots,l\]2  
worst-case battery consumption (unit: kWh). \[i=1,\ldots,l\]3  
battery capacity of the drone (unit: kWh). \[i=1,\ldots,l\]4  
average speed of the drone (unit: km/h). \[i=1,\ldots,l\]5  
required time to load, unload, take-off, and land the drone (unit: hour). \[i=1,\ldots,l\]6  
daily available hours for drone operations (unit: hour). \[i=1,\ldots,l\]7  
minimum recharging ratio to prevent full discharge. The battery capacity of the drone should not fall below this ratio. \[i=1,\ldots,l\]8  
battery recharging time from the minimum charging ratio ( \[i=1,\ldots,l\]9 ) to full charge (unit: hour). _Decision Variables_  
\[j\]0  
1 if drone \[j\]1 delivers a parcel to destination \[j\]2 as the \[j\]3th operation, and 0 otherwise. \[j\]4  
1 if drone \[j\]5 recharges its battery as the \[j\]6th operation, and 0 otherwise. \[j\]7  
battery recharging ratio (%) when drone \[j\]8 recharges its battery as the \[j\]9th operation. \[j=1,\ldots,l\]0. \[j=1,\ldots,l\]1  
the battery charge after drone \[j=1,\ldots,l\]2 finishes its \[j=1,\ldots,l\]3th operation. The initial battery charge is set to \[j=1,\ldots,l\]4. \[j=1,\ldots,l\]5  
time after drone \[j=1,\ldots,l\]6 finishes its \[j=1,\ldots,l\]7th operation. The initial time is set to zero, and it increases as the operations are performed. Using the parameters and the decision variables defined above, the drone operation planning model can be formulated as follows. _Drone Operation Planning Model (DROP-Model)_  
\[j=1,\ldots,l\]8  
Subject to:  
\[j=1,\ldots,l\]9  
\[k\]0  
\[k\]1  
\[k\]2\[k\]3  
\[k\]4  
\[k\]5  
\[k\]6  
\[k\]7  
\[k\]8  
\[k\]9  
\[k=1,\ldots,K\]0  
\[k=1,\ldots,K\]1  
The objective function in Equation (1) works to maximize the number of parcels delivered. In many past related studies, the authors pursued the minimization of the total transportation cost. However, it is impossible for drones to take all the portion of delivery demand in any building because drones are not used to replace the conventional transport modes but to supplement them, as mentioned in the assumptions. Thus, in this study, drones need to be operated continuously during the operating hours. In short, drones are required to deliver as many parcels as possible during the given hours. Additionally, the related costs (mainly battery recharging) to operate several drones are not high and can be disregarded in comparison with delivery van operations, which require relatively substantial fuel costs. The constraint in Equation (2) indicates that the number of deliveries to any rooftop should not be more than its demand. The constraint in Equation (3) ensures that drone \(i\) conducts either delivery or recharging, but it is not allowed to do both concurrently. The constraint in Equation (4) guarantees that drone \(i\) delivers a parcel as the next task if the current task is recharging the battery. The constraint in Equation (5) ensures that the mission of the drone is terminated when both tasks (delivery and recharging) are not assigned and conducted in the previous step. This constraint prohibits any unnecessary breaks for drones during operating hours.