## 3 Methodology
### Constraints  
The mass balance for SNF was formulated independently for reactors since they generate waste. Further, for all years but the first year, the amount stored in the year before must be taken into account. \[\sum_{nn}\big{(}SNF\_t\_nn,y\big{)}-\sum_{nn}\big{(}SNF\_t\_r,nn,y \big{)}+gen\_SNF(r,y)\\ +\big{(}SNF\_s\_r,y-1\big{)}\,if\,(y>1)-SNF\_s\_r,y=0\qquad \forall\,r\in R,y\in Y. \tag{2}\]  
For all nodes but reactors and hot cells, the mass balance for \(SNF\) was defined. \[\sum_{nn}\big{(}SNF\_t\_nn,n,y\big{)}-\sum_{nn}\big{(}SNF\_t\_n, nn,y\big{)}+\big{(}SNF\_s\_n,y-1\big{)}\,if\,(y>1)-SNF\_s\_n,y=0\\ \forall\,n\in N,y\in Y. \tag{3}\]  
As refilling waste into new casks was only done in hot cells, the waste \(WAST\_NC\) can be considered--model-wise--to be generated in hot cells. For all other nodes therefore the standard mass balance holds. \[\sum_{nn}\big{(}WAST\_NC\_t\_nn,y\big{)}-\sum_{nn}\big{(}W AST\_NC\_t\_n,nn,y\big{)}\\ +\big{(}WAST\_NC\_s,y-1\big{)}\,if\,(y>1)-WAST\_NC\_s,y=0\qquad \forall\,n\in N\
otin HC,y\in Y \tag{4}\]  
In analogy to \(WAST\_NC\), vitrified waste \(WAST\_VI\) was only generated in hot cells and the standard mass balance therefore holds for all other nodes. \[\sum_{nn}\left(WASTE\_VI\_t_{nn,n,y}\right)-\sum_{nn}\left(WASTE\_VI\_t_ {n,nn,y}\right)\\ +\left(WASTE\_VI\_s_{n,y-1}\right)if\left(y>1\right)-WASTE\_VI\_s_{n, y}=0\qquad\forall\,n\in N\
otin HC,y\in Y. \tag{5}\]  
Each node n can be described by its storage capacity \(s\_cap_{n,y}\). In all years the amount of waste stored in the node--independent of the nature of the waste--must not exceed the local storage capacity. \[ \begin{split} SNF\_s_{n,y}+WASTE\_NC\_s_{n,y}+WASTE\_VI\_s_{n, y}\leq s\_cap_{n,y}\\ \forall\,n\in N,y\in Y.\end{split}  \tag{6}\]  
This constraint expresses whether a transport between two nodes is possible. For all non existing connections between two nodes the \(dist_{n,nn}\) was zero, meaning transportation was not possible. For all other connections the transportation capacity was unlimited, due to our assumptions. \[ \begin{split} SNF\_t_{n,nn,y}+WASTE\_NC\_t_{n,nn,y}+WASTE\_VI\_t_ {n,nn,y}\leq dist_{n,nn}\cdot BIG\\ \forall\,n,nn\in N,y\in Y.\end{split}  \tag{7}\]  
An overall reprocessing/vitrification capacity \(v\_cap\) was defined per year, which resulted from the capacity of the hot cells. \[\sum_{n}\left(v\_effect\cdot WASTE\_VI\_t_{hc,n,y}\right)\leq v\_cap\qquad \forall\,hc\in HC,y\in Y. \tag{8}\]  
This specific mass balance for hot cells took into account that the amount of waste can be reduced through vitrification by the factor \(v\_effect\). Therefore, all incoming waste was either transformed to the same amount of newly casked waste \(WASTE\_NC\) or into a reduced amount of vitrified waste \(WASTE\_VI\). \[\sum_{n}\left(SNF\_t_{n,hc,y}-WASTE\_NC\_t_{hc,n,y}-v\_effect \cdot WASTE\_VI\_t_{hc,n,y}\right)=0\\ \forall\,hc\in HC,y\in Y. \tag{9}\]  
Neither the waste in new casks, nor vitrified waste can be processed in a hot cell. Therefore the amount of units transported to all hot cells in each year of both these kinds of waste must be zero. \[WASTE\_NC\_t_{n,hc,y}+WASTE\_VI\_t_{n,hc,y}=0\qquad\forall\,hc\in HC,n\in N,y \in Y. \tag{10}\]  
Each interim storage facility can only be built only once, therefore the binary decision over all years must be smaller or equal to one. \[\sum_{y}\left(B_{i,y}\right)\leq 1\qquad\forall\,i\in I. \tag{11}\]  
Neither can waste be stored in, nor transported to or from an interim storage facility in each year, unless it has been built before the current year, taking into account that the building takes five years. \[\sum_{y\leq yy}\left(B_{i,y}\right)\cdot BIG\geq\sum_{y,i,fy\leq yy }\left(\left(SNF\_s_{i,y}+WASTE\_NC\_s_{i,y}\right.\right.\\ +WASTE\_VI\_s_{i,y})+\sum_{n}\left(SNF\_t_{n,i,y}+SNF\_t_{i,n,y}+ WASTE\_NC\_t_{n,i,y}\right.\\ +WASTE\_NC\_t_{i,n,y}+WASTE\_VI\_t_{i,n,y}+WASTE\_VI\_t_{n,i,y}) \right)\qquad\forall\,i\in I,yy\in Y. \tag{12}\]  
Cask decay is a result of the security requirements regarding storage of nuclear waste.