# Proactive Mission Planning of Unmanned Aerial Vehicle Fleets Used in Offshore Wind Farm Maintenance
## 3 Model Formulation
### Problem Statement  
The introduced model allows for defining two kinds of problems:  
* The analysis problem comes down to searching for answers to the following questions: Do the available resources (vessels, service teams, and UAV fleets) allow for the servicing of a given set of \(T\) WTs in the assumed time horizon \(H\)? And, if so, what is the service plan that guarantees minimal cost \(C\)?
* The synthesis problem comes down to the search for an answer to the following question: What is/are the structure/parameters of the used resources (vessels, service teams, and UAV fleets) that guarantee the service of a given set of \(T\) WTs in the assumed time horizon \(H\) and the given cost \(C\)?  
The ability to answer the above questions comes down to a solution dedicated to the constraint satisfaction problem (CSP). Depending on the planning stage, this problem is described in (66) and (67):  
* For proactive planning: \[CP_{AP}=(\mathcal{V}_{AP},\mathcal{D}_{AP},\mathcal{C}_{AP})\] (66) where \[\mathcal{V}_{AP}=\left\{\Pi_{AP},Y_{AP},A_{AP},WY_{AP}\right\}\text{--a set of decision variables that determine the plan of mission }S:\] \[\odot\qquad\Pi_{AP}\text{--a set containing the variables of the vessel route:}\ \Pi_{AP}=\left\{dx_{i,j},px_{i,j}\middle|i,j=1\dots NI\right\}\text{;}\] \[\odot\qquad Y_{AP}\text{--the schedule of a vessel:}\ Y_{AP}=\left\{\overline{dy_{i}}, \underline{dy_{i}},\overline{py_{i}},\underline{py_{i}}\middle|i=1\dots NI\right\} \text{;}\] \[\odot\qquad A_{AP}\text{--an assignment for the service teams:}\ A_{AP}=\left\{aw_{\omega,i}, \left|\omega=1\dots NW,i=1\dots NI\right\}\text{;}\] \[\odot\qquad WY_{AP}\text{--a schedule for the service teams:}\ WY_{AP}=\left\{\overline{sy_{i}}, \underline{sy_{i}}\middle|i=1\dots NI\right\}\text{;}\] \[\mathcal{D}_{AP}\text{--a finite set of decision variable domain descriptions:}\ \overline{dy_{i}},\underline{dy_{i}},\overline{py_{i}},\underline{py_{i}}, \overline{sy_{i}},\underline{sy_{i}}\in\mathbb{N}\text{,}\] \[dx_{i,j},\ px_{i,j},aw_{\omega,i}\in\{0,1\}\text{;}\] \[\mathcal{C}_{AP}\text{'a set of constraints specifying the relationships between the vessels, service teams and routes, and turbine networks (1)-(42).
* For reactive planning: \[CP_{AR}=(\mathcal{V}_{AR},\mathcal{D}_{AR},\mathcal{C}_{AR}(\mathcal{V}_{AP}))\] (67) where \[\mathcal{V}_{AR}=\left\{\Pi_{AR},Y_{AR}\right\}\text{--a set of decision variables that determine the plan of mission }S:\] \[\odot\qquad\Pi_{AR}\text{'a set containing the variables of the UAV route:}\ \Pi_{AR}=\left\{ux_{i,j}\middle|i,j=1\dots NI\right\}\text{;}\] \[\odot\qquad Y_{AR}\text{--the schedule of a UAV:}\ Y_{AR}=\left\{s,\overline{uy_{i}}, \underline{uy_{j}}\middle|i=1\dots NI\right\}\text{;}\] \[\odot\qquad C_{AR}\text{--the weight of the delivered equipment:}\ C_{AR}=\left\{c_{i}\middle|i=1\dots NI\right\}\text{;}\] \[\mathcal{D}_{AR}\text{--a finite set of decision variable domain descriptions:}\ s,\overline{uy_{i}},\underline{uy_{j}},c_{i}\in \mathbb{N}\text{,}\ ux_{i,j}\in\{0,1\}\text{;}\] \[\mathcal{C}_{AR}(\mathcal{V}_{AP})\text{--a set of constraints specifying the relationships between the UAVs, transported equipment, and proactive plans }\mathcal{V}_{AP}\text{ (\ref{eq:P1})-(\ref{eq:P1}).}\]  
To solve the \(CP_{AP}\) and \(CP_{AR}\) defined by Formulas (66) and (67), the values of the decision variables for which all the constraints are satisfied need to be found. For relevant examples of how to implement these problems in IBM ILOG, see the following case studies.