# Dynamic Planning of Mobile Service Teams' Mission Subject to Orders Uncertainty Constraints
## 4 Problem Description
### Declarative Model  
Following the assumptions stated above, the proposed reference model consists of:  
Parameters:  
* Crisp parameters:
* graph of a transportation network \(G=(N,E)\), where \(N=\{N_{1},\ldots,N_{\lambda},\ldots,N_{n}\}\) is a set of nodes and \(E=\left\{\left(N_{i},N_{j}\right)\middle|i,\ j\in N\,i\
eq j\right\}\) is a set of edges, \(n\)--the number of nodes;
* set of MSTs: \(\mathcal{U}=\{U_{1},\ldots,U_{k},\ldots,U_{k}\}\), where \(U_{k}\) is the \(k\)-th MST;
* size of the fleet;
* family of required sets of service qualifications: \(\Psi=\{\psi_{1},\ldots,\psi_{\lambda},\ldots,\psi_{n}\}\), where \(\psi_{\lambda}\) is a set of qualifications required by customer \(N_{\lambda}\) (see example in Figure 3);
* family of sets of offered qualifications: \(\Phi=\{\Phi_{1},\ldots,\Phi_{k},\ldots,\Phi_{k}\}\), where \(\Phi_{k}\) is a set of qualifications offered by \(U_{k}\) (see example in Figure 3);
* set of service deadlines: \(\Delta=\{\Delta_{1},\ldots,\Delta_{\lambda},\ldots,\Delta_{n}\}\), where \(\Delta_{\lambda}=[ld_{\lambda};ud_{\lambda}]\) is a deadline for service at the customer \(N_{\lambda}\) (see example in Figure 5);
* disturbance \(IS=(M,\Delta^{*})\) where: \(M\) is a state of fleet mission at the moment \(t^{*}\): \(M=((\mu_{1},\ldots,\mu_{k},\ldots,\mu_{K}),t^{*})\), where \(\mu_{k}\in N\) is the node occupied by \(U_{k}\) (or the node the \(U_{k}\) is headed to) at time \(t^{*}\), the information about the disturbance is received. For example, in the situation shown in Figure 5, the information about the disturbance \(IS\) was received at moment \(t^{*}=2500\) where the mission state is equal to: \(M=((N_{9},N_{3},N_{5}),2500)\);
* is a set of changed service deadlines (caused by the appearance of disturbances): \(\Delta^{*}=\left\{\Delta_{1}^{*},\ldots,\Delta_{M}^{*},\ldots,\Delta_{n}^{*}\right\}\), where \(\Delta_{\lambda}^{*}=\left[ld_{\lambda}^{*};\ ud_{\lambda}^{*}\right]\) is a new deadline (after the occurrence of disturbance) for providing a service to customer \(N_{\lambda}\);
* window width, understood as a period, repeated at regular intervals, in which all nodes should be serviced (see Figure 5-\(T=2000\));
* set of routes \(\pi_{k}\) before the occurrence of the disturbance \(IS\), where \(\pi_{k}\) is a route of \(U_{k}\): \[\pi_{k}=\left(N_{k_{1}},\ldots,N_{k_{l}},N_{k_{l+1}},\ldots,N_{k_{l}}\right)\], where \(x_{k_{k},k_{l+1}}^{k}=1\) for \(i=1,\ldots,\ \mu-1\) and \(x_{k_{l},k_{1}}^{k}=1\)\[x^{k}_{\beta,\lambda}= \begin{cases}1&\text{if $U_{k}$ travels from node $N_{\beta}$ to node $N_{\lambda}$}\\ 0&\text{otherwise}\end{cases} \]  
Imprecise parameters: (defined as positive-oriented OFNs and marked by \"\({}^{\sim}\)\"):  
\(\widehat{d_{\beta,\lambda}}\): traveling time along edge \(\left(N_{\beta},N_{\lambda}\right)\);  
\(\widehat{t_{\lambda}}\): time of node \(N_{\lambda}\) occupation;  
\(\widehat{\mathbb{Y}}\): fuzzy schedule of fleet \(\mathcal{U}\), \(\widehat{\mathbb{Y}}=\left(\widehat{Y},\widetilde{W}\right)\) before the disturbance \(IS\):  
\(\widehat{Y}\): family of \(\widehat{\mathbb{Y}}\), where \(\widehat{Y}^{k}\) is a sequence of moments \(\widehat{y}^{k}_{\lambda}\): \(\widehat{Y}^{k}=\left(\widehat{y}^{k}_{1},\ldots,\ \widehat{y}^{k}_{\lambda},\ldots,\widehat{y}^{k}_{n}\right)\), \(\widehat{y}^{k}_{\lambda}\) is fuzzy time at which \(U_{k}\) arrives at node \(N_{\lambda}\);  
\(\widehat{W}\): family of \(\widehat{W^{k}}\), where \(\widehat{W^{k}}\) is a sequence of laytimes \(\widehat{w^{k}_{\lambda}}\): \(\widehat{W^{k}}=\left(\widehat{w}^{k}_{1},\ldots,\ \widehat{w}^{k}_{\lambda},\ldots, \widehat{w}^{k}_{n}\right)\), \(\widehat{w}^{k}_{\lambda}\) is daytime at node \(N_{\lambda}\) for \(U_{k}\). Variables:  
Crisp variables:  
\({}^{*}x^{k}_{\beta,\lambda}\): binary variable indicating the travel of \(U_{k}\) between nodes \(N_{\beta}\), \(N_{\lambda}\) after disturbance \(IS\):  
\[{}^{*}x^{k}_{\beta,\lambda}= \begin{cases}1&\text{if $U_{k}$ travels from node $N_{\beta}$ to node $N_{\lambda}$}\\ 0&\text{otherwise}\end{cases}