# Reactive Planning-Driven Approach to Online UAVs Mission Rerouting and Rescheduling
## 3 Approach to UAV Fleet Online Routing  
We consider a distribution network, that is modelled by the graph \(G=(N,E)\) where \(N=\{N_{1},\ldots,N_{\lambda},\ldots,N_{n}\}\) signifies the set of \(n=|N|\) nodes and \(E=\big{\{}\big{(}N_{\beta},\ N_{\lambda}\big{)}\big{|}\beta,\lambda\in\{1,\ldots,n\},\beta\
eq\lambda\big{\}}\) signifies the set of edges determining the possible connections between nodes. Set \(NB\subset N\) contains the nodes representing the bases and a set of \(ND\subset N\) contains the nodes representing delivery points: \(NB\cup\ ND=N\), \(NB\cap\ ND=\varnothing\). Given a fleet of UAVs \(\mathcal{U}=\{U_{1},\ldots,U_{k},\ldots,U_{K}\}\) that delivers goods to the points \(ND\), the fleet \(\mathcal{U}\) is spread between the bases \(NB\), the subset \(\mathcal{U}^{\lambda}\subseteq\ \mathcal{U}\) determines the UAVs allocated at base \(N_{\lambda}\in NB\). An ordered quantity of goods \(z_{\lambda}\in\mathbb{N}\) (taken from the base \(N_{1}\)) should be transported to each delivery point \(N_{\lambda}\in ND\). Deliveries are made as part of the mission \(S\), which consists of sub-missions \({}^{l}S\) (i.e., delivery plans that include a single course of UAVs: base-delivery points-base). \(Z\) denotes a sequence consisting of variables \(z_{\lambda}\): \(Z=(z_{1},\ldots,z_{n})\) where \(z_{\lambda}=0\) for each \(N_{\lambda}\in NB\). It is assumed that all required goods should be delivered in the given horizon time \(H\). The amount of goods delivered during one sub-mission \({}^{l}S\) by the \(U_{k}\) to the delivery point \(N_{\lambda}\in ND\) is determined by the variable \({}^{l}c_{\lambda}^{k}\in\mathbb{N}\). \({}^{l}C\) is a sequence: \({}^{l}C=(^{l}c_{1}^{1},\ldots,^{l}c_{1}^{K},\ldots,^{l}c_{n}^{k},\ldots,^{l}c_ {n}^{k})\) determining the payload weight delivered by fleet \(\mathcal{U}\) (\({}^{l}c_{\lambda}^{k}=0\) for each \(N_{\lambda}\in NB\)). Variable \(Q_{k}\) denotes the payload capacity of \(U_{k}\) (amount of goods transported by \(U_{k}\) cannot exceed \(Q_{k}\)). Moreover, each \(U_{k}\) is described by the following technical parameters: the battery capacity \(CAP\), airspeed \(va\), drag coefficient \(C_{D}\), front surface \(A\), and UAV width \(b\). The time spent on take-off and landing \(U_{k}\) on the delivery point \(N_{\lambda}\) is indicated by variable \(w_{\lambda}\in\mathbb{N}\). Note that \({}^{l}\mathcal{U}\subseteq\mathcal{U}\) denotes a set of UAVs used during sub-mission \({}^{l}S\). The time when the \(U_{k}\in{}^{l}\mathcal{U}\) arrives at the delivery point \(N_{\lambda}\) during a sub-mission \({}^{l}S\) is indicated by the variable \({}^{l}y_{\lambda}^{k}\in\mathbb{N}\) [s]. In this context, the sequence \({}^{l}Y\) consisting of moments \({}^{l}y_{\lambda}^{k}\) is called the schedule of the fleet \({}^{l}\mathcal{U}\): \({}^{l}Y=(^{l}y_{1}^{1},\ldots,^{l}y_{1}^{k},\ldots,^{l}y_{n}^{1},\ldots,^{l}y _{n}^{k})\). We assume that the variable \(t_{\beta,\lambda}\in\mathbb{N}\) determines travelling time between nodes \(N_{\beta}\), \(N_{\lambda}\), where \(\big{(}N_{\beta},N_{\lambda}\big{)}\in E\), and routes of \(U_{k}\in{}^{l}\mathcal{U}\) during sub-mission \({}^{l}S\) are represented by the sequences \({}^{l}\pi_{k}=\big{(}N_{k_{1}},\ldots,N_{k_{l}},N_{k_{i+1}},\ldots,N_{k_{p}} \big{)}\), where \(k_{i}\in\{1,\ldots,n\}\), \((N_{k_{i}},N_{k_{i+1}})\in E\). \({}^{l}\Pi\) denotes a sequence of UAV routes executed during the sub-mission \({}^{l}S\): \({}^{l}\Pi=(^{l}\pi_{1},\ldots,^{l}\pi_{k},\ldots,^{l}\pi_{K})\) (in case when \(U_{k}\
otin{}^{l}\mathcal{U}\) then \({}^{l}\pi_{k}=\bigtriangleup\)). The delivery plan of one UAV's sub-mission \({}^{l}S\) is defined as a sequence: \({}^{l}S=(^{l}\mathcal{U},^{l}\Pi,^{l}Y,^{l}C)\). It is assumed that a plan of the UAVs \({}^{l}S\) sub-missions \({}^{l}S\) is implemented under specific weather conditions, i.e., the weather forecast is known for each sub-mission \({}^{l}S\).