## 2 Problem Statement
### Mathematical Model  
The corridor planning problem is concerning finding the shortest path joining two specific nodes in an undirected weighted graph [6]. The mathematical model of the corridor routes is described by relations (1)-(5). The topographical network (rectangular mesh or grid) representing the study area is encoded using a graph (\(N\), \(A\)), where \(N=\{u_{1},\,\ldots,\,u_{n}\}\) is the set of \(n\) nodes, and \(A=\{(u_{1},\,v_{1}),\,\ldots,\,(u_{m},\,v_{m})\}\) is the set of \(m\) edges connecting the nodes. A node represents the center of a cell in the rectangular mesh (matrix of visited nodes), and each edge is associated with a viability cost (penalty) \(c_{u,v}^{p}\in R\) of passing from node \(u\) to node \(v\) in the graph. Furthermore, each edge has associated an elevation value \(c_{u,v}^{DEM}\), based on a digital elevation model (DEM). In this model, \(s\) and \(t\) are the source and destination nodes connected by the corridor, respectively, and a binary variable \(x_{uv}\) is used to indicate if the edge (\(u\),\(v\)) is or not part of the route.  
\[\textit{min}z(x)=\textit{min}\sum_{(u,v)\in A}x_{u,v}\left(\sum_{l=1}^{l=Nl}pl_ {u,v}+c_{u,v}^{DEM}\right)u\
eq v \tag{1}\]  
s.f.  
\[\sum_{(u,v)\in A}x_{uv}-\sum_{(v,u)\in A}x_{vu}=\begin{cases}1&\text{if }u=s \\ -1&\text{if }u=t\\ 0&\text{if }u\
eq s,t\end{cases} \tag{2}\]  
\[x_{u,v}=\{0,1\},\forall(u,v)\in A \tag{3}\]  
\[(u,v)\in\textit{corridor},\forall(u,v)\in A \tag{4}\]  
\[c_{u,v}^{DEM}\leq e_{max} \tag{5}\]  
The goal, shown in Equation (1), is to find the lowest cost route from \(s\) to \(t\), accomplishing the constraints expressed in (2)-(5). In this work, the route cost is a combination of the viability costs of the edges (\(u\),\(v\)) belonging to the proposed route and the elevation of the visited nodes in the corridor. The route between nodes \(s=0\) and \(t=k\) is a finite sequence of nodes \(n=\{u_{0},\,\cdots,u_{k}\}\). The accumulated costs matrix represents the feasibility of passing between two nodes, using a feasibility scale of 1 to 10, where 1 represents the lowest penalty cost, i.e., there exists a priority to pass on the path, while value 10 means that it is prohibited. For creating this matrix, a feasibility cost \(p\) (penalty) for each topographic \(l\) (layer) is assigned according to its relevance. The values of the \(Nl\) layers involved in the corridor design are added and weighted in Equation (1). Equation (6) shows the weighting operation of a raster cell \(c_{uv}^{p}\) in the accumulated cost matrix. Elevation on the ground is also penalized with higher values of \(c_{uv}^{DEM}\) in Equation (1). Constraints (2) guarantees that \(s\) and \(t\) are only used as the initial and final node in the route, respectively, as well as no cycles are in the solution, fulfilling the constraints (3). Furthermore, constraints (4) indicates that edge (\(u\), \(v\)) must be located within the corridor. The corridor is given in units of length and has as its midpoint the shortest route found by an exact algorithm such as the Dijkstra algorithm. Finally, constraint (5) establishes that \(e_{max}\) is the maximum elevation allowed on the ground.  
\[c_{u,v}^{p}=\sum_{l=1}^{l=Nl}pl_{u,v} \tag{6}\]The optimization model presented in (1) to (5) is used to find cost-optimized routes located in the corridor generated by the shortest route. These optimized routes are obtained by the simulated annealing algorithm, which creates neighboring routes to the shortest route.