# Planning of Optical Connections in 5G Packet-Optical xHaul Access Network
## 4 Optimization Problem
### ILP Formulation  
To model the optimization problem, we introduce a set of decision variables. Namely, binary variable \(x_{dp},d\in\mathcal{D},p\in\mathcal{P}(d)\) indicates whether transmission path \(p\) is established for demand \(d\). Binary variable \(y_{d},d\in\mathcal{D}\) indicates whether demand \(d\) is served by a transmission path established for some other demand, which is achieved by means of an OADM device introduced into the path. Eventually, binary variable \(y_{d\bar{d}},d\in\mathcal{D},\bar{d}\in\mathcal{D}\) indicates whether demand \(d\) is served by a transmission path established for demand \(\bar{d}\). The meaning of variables is also summarized in Table 3.  
The ILP formulation of the connection planning problem considered is the following:  
\[\text{minimize}\ z=A\cdot\sum_{d\in\mathcal{D}}x_{dp}+\sum_{d\in \mathcal{D}}\sum_{p\in\mathcal{P}(d)}D(p)\cdot x_{dp}, \tag{2}\] \[\sum_{p\in\mathcal{P}(d)}x_{dp}+y_{d}=1,\qquad\forall d\in \mathcal{D},\] (3) \[\sum_{d\in\mathcal{D}\bar{d}\
eq d}y_{d\bar{d}}=y_{d},\qquad \forall d\in\mathcal{D},\] (4) \[\sum_{p\in\mathcal{Q}(d)}x_{dp}\geq y_{d\bar{d}},\qquad\forall d, \bar{d}\in\mathcal{D},\bar{d}\
eq d,\] (5) \[W(d)\cdot(1-y_{d})+\sum_{\bar{d}\in\mathcal{D}\bar{d}\
eq d}W(d) \cdot y_{\bar{d}d}\leq W,\qquad\forall d\in\mathcal{D},\] (6) \[\sum_{\bar{d}\in\mathcal{D}\bar{d}\
eq d}y_{d\bar{d}}\leq\sum_{p \in\mathcal{P}(d)}N(p)\cdot x_{dp},\qquad\forall d\in\mathcal{D}. \tag{7}\]  
Optimization objective (2) aims at minimizing the number of transmission paths, which is a primary goal imposed by coefficient \(A\) (we consider \(A=1000\)), and the overall length of used fibers (a secondary optimization goal). Constraints (3) assure that either a transmission path, selected from the set of candidate paths, is established for a demand (\(y_{d}=0\)) or the demand is served by the transmission path established for some other demand by means of an OADM device (\(y_{d}=1\)). Constraints (4) determine which other demand serves on its transmission path the demand that has not established its own path (i.e., when \(y_{d}=1\)). Constraints (5) assure that, for demands supporting some other demands, their own transmission paths should exist. Constraints (6) assure that the sum of requested wavelengths of a demand for which a transmission path is established (when \(y_{d}=0\)) and the requests wavelengths of other demands served by this transmission path (when \(y_{\bar{d}d}=1\)) must not exceed the fiber capacity (expressed by \(W\)). Finally, Constraints (7)  
\begin{table}
\begin{tabular}{l l} \hline \hline \multicolumn{2}{l}{**Sets and Parameters**} \\ \hline \(\mathcal{V}\) & set of remote and hub nodes \\ \(\mathcal{E}\) & set of links connecting the nodes \\ \(\mathcal{D}\) & set of demands \\ \(\mathcal{P}(d)\) & set of candidate transmission paths of demand \(\bar{d}\in\mathcal{D}\) \\ \(\mathcal{Q}(d,\bar{d})\) & set of candidate transmission paths of demand \(\bar{d}\) which can serve demand \(\bar{d}\) \\ \(D(p)\) & length of transmission path \(p\) \\ \(N(p)\) & allowable number of OADMs on transmission path \(p\) \\ \(W(d)\) & number of requested wavelengths of demand \(\bar{d}\) \\ \(W\) & capacity of WDM system (maximum number of wavelengths carried in a fiber) \\ \hline \multicolumn{2}{l}{**Variables**} \\ \hline \(x_{dp}\) & binary, \(x_{dp}=1\) when path \(p\) is used by demand \(\bar{d}\); \(x_{dp}=0\) otherwise \\ \(y_{d}\) & binary, \(y_{d}=1\) when demand \(\bar{d}\) is served by a path belonging to other demand; \(y_{d}=0\) \\ \(y_{d\bar{d}}\) & otherwise \\ \hline \hline \end{tabular}
\end{table}
Table 3: Notation.  
assume that the number of other demands served by an established transmission path does not exceed the allowable number of OADMs traversed by the path (expressed by \(N(p)\)).