# Dynamic Planning of Mobile Service Teams' Mission Subject to Orders Uncertainty Constraints
## 3 Preliminaries
### Illustrative Example
The routes traveled by team \(U_{k}\) are denoted by sequences of nodes: \(\pi_{k}=(N_{k_{1}},\ldots,N_{k_{l}},N_{k_{i+1}},\ldots,N_{k_{k}})\), where \(k_{i}\in\{1,\ldots,K\}\), \(\forall_{k_{i}\
eq k_{j}}\,N_{k_{i}}\
eq N_{k_{j}}\), \(\left(N_{k_{i}},N_{k_{i+1}}\right)\in E\). Nodes representing the service point (e.g., \(N_{1}\)) appear along every route. Moreover, each route \(\pi_{k}\) consists of nodes in which customers \(N_{\lambda}\) assigned to them expect services that require qualifications \(\psi_{\lambda}\), i.e., for each team \(U_{k}\) offering qualifications \(\Phi_{k}\) the following condition holds \(\Phi_{k}\cap\ \psi_{\lambda}\
eq\varnothing\). In this context, the problem of the proactive planning of service team trips boils down to the question: do the schedule and routings of MSTs guarantee the timely execution of the ordered services? Given a set \(\mathcal{U}\) of MSTs providing services (according to given qualifications \(\Phi_{k}\)) to the customers allocated in a network \(G\) (ordering an assumed kind of services \(\Psi\)). Does there exist a set of routes \(\Pi\) guaranteeing the timely execution of the ordered services (according to given service deadlines \(\Delta_{\lambda}\))? The examples of such routes \(\Pi\) and the associated fuzzy schedule for the network \(G\) for Figure 2 are illustrated in Figures 4 and 5. The routes are specified by the sequences of nodes: \(\pi_{1}=(N_{1},N_{9},N_{10},N_{4},N_{1})\), \(\pi_{2}=(N_{1},N_{3},N_{11},N_{1}),\pi_{3}=(N_{1},N_{5},N_{6},N_{7},N_{8},N_{2},N_{1})\). It should be noted that in the presented solution, customer service is provided only by the necessary MSTs. Moreover, despite the uncertain (fuzzy) traveling times \(d_{\beta,\lambda}\), it is also assumed that all customers are serviced cyclically (with period \(T=2000\)) due to given service deadlines \(\Delta_{\lambda}\)--see Figure 5.  
Due to the occurrence of unforeseen disturbances, the implementation of proactively designated customers service plans becomes practically impossible. An example of such a disturbance are the unforeseen changes of service deadlines. Such a kind of disturbance is presented in Figure 5 where the dispatcher receives information about changing the date of the customer service being located at the node \(N_{6}\) (from \(\Delta_{6}=[450;750]\) to \(\Delta_{6}^{*}=[650;950]\)), see the second window (moment \(t^{*}=2500\) when \(U_{1}\) occupies \(N_{9}\), \(U_{2}\) occupies \(N_{3}\) and \(U_{5}\) occupies \(N_{5}\)). Due to this change, the adopted routes do not guarantee the implementation of maintenance services on the set dates--the handling of \(N_{6}\) according to the new service deadline \(\Delta_{6}^{*}=[650;950]\) prevents the timely handling of the client \(N_{8}\) and vice versa. In such a situation, it becomes necessary to answer the following question:_Given a set \(\mathcal{U}\) of MSTs providing services (according to given qualifications \(\Phi_{k}\)) to the customers allocated in a network (ordering assumed kind of services \(\Psi\)), MSTs move along a given set of routes \(\Pi\) according to a cyclic fuzzy schedule \(\hat{\mathbb{Y}}\). Given is a disturbance changing \(\Delta_{\lambda}\) to \(\Delta_{\lambda}^{\star}\) at the moment \(t^{\star}\). Does there exist a rerouting '\(\Pi\) and rescheduling '\(\overline{\Upsilon}\): of MSTs, which guarantee the timely execution of the ordered services?_  
The possibility of the reactive (dynamic) planning of MST missions in the event of the disruption of service deadlines is the subject of the following chapters.