# Dynamic Planning of Mobile Service Teams' Mission Subject to Orders Uncertainty Constraints
## 3 Preliminaries
### Illustrative Example  
Let us consider the graph \(G=(N,E)\) modelling a transportation network composed of \(|N|=\omega=11\) delivery points (hereinafter referred to as nodes), i.e., customers and the service base, as shown in Figure 2. The points include 1 node representing the service point \(N_{1}\) and 10 nodes representing customers \(N_{2}\)-\(N_{11}\). The customers \(N_{2}\)-\(N_{11}\) are cyclically serviced (with period \(T=2000\) u.t.) by the mobile service teams (MSTs) traveling form node \(N_{1}\). The beginning moment of the node \(N_{\lambda}\) occupation (service) by team \(U_{k}\) is described by variable \(y_{\lambda}^{k}\). The service is executed in intervals determined by the service deadline \(\Delta_{\lambda}=[ld_{\lambda};ud_{\lambda}]\in\Delta\) (see Table 1), i.e., \(y_{\lambda}^{k}\geq ld_{\lambda}\) and \(y_{\lambda}^{k}+t_{\lambda}\leq ud_{\lambda}\) (where \(t_{\lambda}\) is time of node \(N_{\lambda}\) occupation). Moreover, each node \(N_{\lambda}\) can be serviced by MSTs offering required qualifications and confirmed with the appropriate certificates. The considered sets of qualifications \(\psi_{\lambda}\in\Psi\) which are required by customers \(N_{\lambda}\) are shown in Table 2.  
Figure 1: (**a**) Ordered fuzzy number (OFN) \(\hat{A}\) represented as a convex fuzzy number; (**b**) functions \(f_{A},\ g_{A}\) determining \(\hat{A}\) (positive orientation); and (**c**) the discrete representation of \(\hat{A}\) (\(dx=0.5\)). Figure 2: Graph \(G\) modeling transportation network. For example, customer \(N_{4}\) should be serviced within the interval time [900; 1500] by MSTs offering qualifications \(A\) and \(B\) (one MST offering set \(\{A,B\}\) or two MSTs: the first offering \(A\) and the second offering \(B\)). Each edge \(\left(N_{\beta},N_{\lambda}\right)\in E\) linking nodes \(N_{\beta}\) and \(N_{\lambda}\) is labelled with a fuzzy variable (in the OFN representation) representing the uncertainty of the traveling time \(d_{\beta,\lambda}\) between the nodes \(N_{\beta}\) and \(N_{\lambda}\) (see Figure 3). Given is a set of MSTs \(\mathcal{U}=\{U_{1},\ldots,U_{k},\ldots,U_{K}\}\) servicing customers spatially dispersed in network \(G\). For each \(U_{k}\) the set \(\Phi_{k}\) of the offered qualifications is assigned. For example, the available set \(\mathcal{U}=\{U_{1},U_{2},U_{3}\}\) in a network \(G\) (Figure 2) contains three MSTs offering the following qualifications: \(\Phi_{1}=\{A,B\}\); \(\Phi_{2}=\{C,A\}\); \(\Phi_{3}=\{B,C\}\). This means that:  
* The team \(U_{1}\) can completely satisfy the expectations of the nodes: \(N_{2}\), \(N_{3}\), \(N_{4}\), \(N_{5}\), \(N_{9}\), \(N_{10}\), and partially those of the nodes: \(N_{6}\), \(N_{11}\);
* The team \(U_{2}\) can completely satisfy the expectation of nodes: \(N_{3},N_{7},N_{8},N_{9},N_{10},N_{11}\), and partially of nodes: \(N_{4}\), \(N_{6}\); \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & \(N_{2}\) & \(N_{3}\) & \(N_{4}\) & \(N_{5}\) & \(N_{6}\) & \(N_{7}\) & \(N_{8}\) & \(N_{9}\) & \(N_{10}\) & \(N_{11}\) \\ \hline \(ld_{\lambda}\) & 1200 & 300 & 900 & 300 & 450 & 700 & 900 & 300 & 600 & 700 \\ \(ud_{\lambda}\) & 1900 & 600 & 1500 & 600 & 750 & 1200 & 1300 & 800 & 1200 & 1100 \\ \hline \end{tabular}
\end{table} 
Table 1: Service deadlines for customers \(N_{2}\)–\(N_{11}\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & \(N_{2}\) & \(N_{3}\) & \(N_{4}\) & \(N_{5}\) & \(N_{6}\) & \(N_{7}\) & \(N_{8}\) & \(N_{9}\) & \(N_{10}\) & \(N_{11}\) \\ \hline \(\psi_{\lambda}\) & \(\{B\}\) & \(\{A\}\) & \(\{A,B\}\) & \(\{B\}\) & \(\{B,C\}\) & \(\{C\}\) & \(\{C\}\) & \(\{A\}\) & \(\{A\}\) & \(\{C,A\}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Sets of required qualifications \(N_{2}\)–\(N_{11}\). Figure 3: Traveling times between nodes for the networks from Figure 2 (an OFN representation). * The team \(U_{3}\) can completely satisfy the expectations of the nodes: \(N_{2}\), \(N_{5}\), \(N_{6}\), \(N_{7}\), \(N_{8}\), and partially those of the nodes: \(N_{4}\), \(N_{5}\) and \(N_{11}\).