*3.1. Problem Description*

Assume that multiple demand nodes are distributed in a two-dimensional service area. Demand node *x* has demand rate *λ<sup>x</sup>* and is independent from the other demand nodes in the area. *K* number of UAVs are available for operation, and a UAV will serve a single demand (a demand instance at a demand node) per trip.

Given the setting, we seek a set of zones **Z** and base locations *μ* such that (1) all the demand nodes are divided into *<sup>K</sup>* number of zones **<sup>Z</sup>** <sup>=</sup> {*Z*1, ... *ZK*}, (2) a flight path between the base of a zone and a demand node within the zone is guaranteed, and (3) the total service level for the demand nodes by the zoning solution is maximized. The zoning problem of this study can be formally described by

$$P\_{\text{zoting}} = \underset{\mathbf{Z}, \ \mu}{\text{arg}\max} \sum\_{k=1}^{K} \sum\_{\mathbf{x} \in \mathcal{Z}\_k} \mathcal{L}(\mathbf{x} | \mu\_{k\prime} \mathbf{Z}),\tag{1}$$

where <sup>L</sup>(*x*|*μk*,**Z**) is the service level for demand node *<sup>x</sup>* given a base location *<sup>μ</sup><sup>k</sup>* and a set of zones **Z**.

From the safety perspective, the most relevant measure for the service level would be the distance between operating UAVs in different zones. Considering this idea, a zoning solution that maximizes the separation between demand nodes in different zones would be desired. This concept is well-aligned with the *k*-means clustering algorithm, which partitions observations into *k* clusters such that observations are assigned to the clusters with the nearest mean. Based on this, the original zoning problem can be rewritten as

$$P\_{\text{zoting}}^{\*} = \underset{\mathbf{Z}, \ \mu}{\text{arg min}} \sum\_{k=1}^{K} \sum\_{\mathbf{x} \in \mathbf{Z}\_{k}} \lambda\_{\mathbf{x}} \cdot ||\mathbf{x} - \mu\_{k}||^{2} \tag{2}$$

which is intended to minimize the weighted within-cluster variances (which is the same as maximizing between-cluster variances). It should be clear that this formulation is one of the options to approximate the original zoning problem (Equation (1)); thus, a different formulation can be designed based on the use case's specific constraints, for example, the minimum distance between demands and the center of a zone or no-fly zone.
