# Optimization of Optical Networks Based on CDC-ROADM Technology
## 2 Methods
If a node-link formulation is considered the following constraints need to be added:  
\[\sum_{s\in\mathcal{S}}\sum_{t\in\mathcal{T}}y_{tsp}^{(n,n^{ \prime})}b(t) \geq\sum_{n^{\prime\prime}\in\mathcal{N}}z_{n^{\prime\prime}p}^{(n,n^{ \prime})}\quad\forall n,n^{\prime}\in\mathcal{N},\forall p\in\mathcal{P}\]2 \[\sum_{s\in\mathcal{S}}\sum_{t\in\mathcal{T}}y_{tsp}^{(n,n^{ \prime})}b(t) \geq\sum_{n^{\prime\prime}\in\mathcal{N}}z_{n^{\prime\prime}p}^{(n,n^{ \prime})}\quad\forall n,n^{\prime}\in\mathcal{N},\forall p\in\mathcal{P}\]3 (5b) \[\sum_{s\in\mathcal{S}}\sum_{t\in\mathcal{T}}y_{tsp}^{(n,n^{ \prime})}b(t) \geq\sum_{n^{\prime\prime}\in\mathcal{N}}z_{n^{\prime\prime}p}^{(n,n^{ \prime})}\quad\forall n,n^{\prime}\in\mathcal{N},\forall p\in\mathcal{P}\]4  
In (5a)-(5c) \(r(e)\) is the edge opposite to edge \(e\), \(k_{sept}^{(n,n^{\prime})}\) is a binary variable and equal to 1 if slice \(s\) is occupied on edge \(e\) in period \(p\) by a transmission between nodes \(n\) and \(n^{\prime}\) using transponder \(t\) and 0, otherwise. Constraint (5a) expresses the Kirchhoff law for all nodes that are neither source nor sink nodes of a demand. Constraint (5b) expresses the Kirchhoff law for source nodes of demands. Constraint (5c) assures that the bandwidth is reserved for both directions. Link-path formulation is well known and often used in the literature [11; 18]. For this method sets \(\mathcal{G}_{(n,n^{\prime})}\) of all paths between nodes \(n\) and \(n^{\prime}\) should be defined and the following constraints added:  
\[\sum_{s\in\mathcal{S}}\sum_{t\in\mathcal{T}}y_{tsp}^{(n,n^{ \prime})}b(t) \geq\sum_{n^{\prime\prime}\in\mathcal{N}}z_{n^{\prime\prime}p}^{(n,n^{ \prime})}\quad\forall n,n^{\prime}\in\mathcal{N},\forall p\in\mathcal{P}\]5 \[\sum_{s\in\mathcal{S}}\sum_{t\in\mathcal{T}}y_{tsp}^{(n,n^{ \prime})}b(t) \geq\sum_{n^{\prime\prime}\in\mathcal{N}}z_{n^{\prime\prime}p}^{(n,n^{ \prime})}\quad\forall n,n^{\prime}\in\mathcal{N},\forall p\in\mathcal{P}\]6  
In (6a) and (6b) \(\mathcal{G}_{(n,n^{\prime})}\) is the paths set between nodes \(n\) and \(n^{\prime}\); \(u(g,e)\) is a binary constant and equal to 1 if path \(g\) uses edges \(e\) or \(r(e)\) and equal to 0, otherwise; \(k_{sgpt}\) is a binary variable and equal to 1 if slice \(s\) is occupied on path \(g\) in period \(p\) by transponder \(t\). Constraint (6a) assures that each demand is served. Constraint (6b) assures that the bandwidth is are reserved for both directions. Additional constraints that should be added to the node-link model to have directioned transponders are:  
\[\sum_{s\in\mathcal{S}}\sum_{t\in\mathcal{T}}y_{tsp}^{(n,n^{ \prime})}b(t) \geq\sum_{n^{\prime\prime}\in\mathcal{N}}z_{n^{\prime\prime}p}^{(n,n^{ \prime})}\quad\forall n,n^{\prime}\in\mathcal{N},\forall p\in\mathcal{P}\]7 \[\sum_{s\in\mathcal{S}}\sum_{t\in\mathcal{T}}y_{tsp}^{(n,n^{ \prime})}b(t) \geq\sum_{n^{\prime\prime}\in\mathcal{N}}z_{n^{\prime\prime}p}^{(n,n^{ \prime})}\quad\forall n,n^{\prime}\in\mathcal{N},\forall p\in\mathcal{P}\]8  
Constraints (7a) and (7b) assure that the first edge and the last edge, respectively, are used. Additional constraints that should be added to the link-path model to have directioned transponders are:  
\[\sum_{s\in\mathcal{S}}\sum_{t\in\mathcal{T}}y_{tsp}^{(n,n^{ \prime})}b(t) \geq\sum_{n^{\prime\prime}\in\mathcal{N}}z_{n^{\prime\prime}p}^{(n,n^{ \prime})}\quad\forall n,n^{\prime}\in\mathcal{N},\forall p\in\mathcal{P}\]9 \[g_{tsnp} \geq\sum_{n^{\prime}\in\mathcal{N}}(y_{tsp}^{(n,n^{\prime})}+y_{ tsp}^{(n^{\prime},n)})\quad\forall t\in\mathcal{T},\forall n\in\mathcal{N}, \forall p\in\mathcal{P},\forall s\in\mathcal{S}\]0  
The constraints assure that the first and last edges specified by variables \(f_{tsp}^{(n,n^{\prime})}\) and \(l_{tsp}^{(n,n^{\prime})}\) are consistent with the routing specified by variables \(k_{sgpt}\). Wavelength Selective Switches (WSS) are additional WDM equipment, except transponders, which are taken into account in the nodal model.