# Optimization of Optical Networks Based on CDC-ROADM Technology
## 2 Methods
General constraints that should be added to both node-link and link-path formulations to account for WSSs are:  
\[g_{tsnp} \geq\sum_{n^{\prime}\in\mathcal{N}}(y_{tsp}^{(n,n^{\prime})}+y_{ tsp}^{(n^{\prime},n)})\quad\forall t\in\mathcal{T},\forall n\in\mathcal{N}, \forall p\in\mathcal{P},\forall s\in\mathcal{S}\]1 \[g_{tsnp} \geq\sum_{n^{\prime}\in\mathcal{N}}(y_{tsp}^{(n,n^{\prime})}+y_{ tsp}^{(n^{\prime},n)})\quad\forall t\in\mathcal{T},\forall n\in\mathcal{N}, \forall p\in\mathcal{P},\forall s\in\mathcal{S}\]2 (9b) \[g_{tsnp} \geq\sum_{n^{\prime}\in\mathcal{N}}(y_{tsp}^{(n,n^{\prime})}+y_{ tsp}^{(n^{\prime},n)})\quad\forall t\in\mathcal{T},\forall n\in\mathcal{N}, \forall p\in\mathcal{P},\forall s\in\mathcal{S}\]3 (9c) \[g_{tsnp} \geq\sum_{n^{\prime}\in\mathcal{N}}(y_{tsp}^{(n,n^{\prime})}+y_{ tsp}^{(n^{\prime},n)})\quad\forall t\in\mathcal{T},\forall n\in\mathcal{N}, \forall p\in\mathcal{P},\forall s\in\mathcal{S}\]4 (9d) \[g_{tsnp} \geq\sum_{n^{\prime}\in\mathcal{N}}(y_{tsp}^{(n,n^{\prime})}+y_{ tsp}^{(n^{\prime},n)})\quad\forall t\in\mathcal{T},\forall n\in\mathcal{N}, \forall p\in\mathcal{P},\forall s\in\mathcal{S}\]5 (9e) \[g_{tsnp} \geq\sum_{n^{\prime}\in\mathcal{N}}(y_{tsp}^{(n,n^{\prime})}+y_{ tsp}^{(n^{\prime},n)})\quad\forall t\in\mathcal{T},\forall n\in\mathcal{N}, \forall p\in\mathcal{P},\forall s\in\mathcal{S}\]6 (9f) \[g_{tsnp} \geq\sum_{n^{\prime}\in\mathcal{N}}(y_{tsp}^{(n,n^{\prime})}+y_{ tsp}^{(n^{\prime},n)})\quad\forall t\in\mathcal{T},\forall n\in\mathcal{N}, \forall p\in\mathcal{P},\forall s\in\mathcal{S}\]7 (9g) \[g_{tsnp} \geq\sum_{n^{\prime}\in\mathcal{N}}(y_{tsp}^{(n,n^{\prime})}+y_{ tsp}^{(n^{\prime},n)})\quad\forall t\in\mathcal{T},\forall n\in\mathcal{N}, \forall p\in\mathcal{P},\forall s\in\mathcal{S}\]8 (9h) \[g_{tsnp} \geq\sum_{n^{\prime}\in\mathcal{N}}(y_{tsp}^{(n,n^{\prime})}+y_{ tsp}^{(n^{\prime},n)})\quad\forall t\in\mathcal{T},\forall n\in\mathcal{N}, \forall p\in\mathcal{P},\forall s\in\mathcal{S}\]9 (9i) \[x_{tnp}o(t) \geq\sum_{s\in\mathcal{S}}g_{tsnp}\quad\forall t\in\mathcal{T}, \forall n\in\mathcal{N},\forall p\in\mathcal{P}\]0  
In (9a)-(9j) we use the following constants: \(a(n,e)\) is equal to \(1\) if a WSS is installed in node \(n\) on edge \(e\) at time zero and \(0\), otherwise; \(a(n)\) is equal to \(1\) if an access WSS is installed in node \(n\) at time zero and equal to \(0\), otherwise; \(\xi(w)\) is the cost of using WSS in one period and equal to \(0\) otherwise. We use the following variables in the model: \(a_{nep}\) is equal to \(1\) if a WSS is installed in node \(n\) on edge \(e\) in period \(p\) and \(0\) otherwise; \(a_{np}\) is equal to \(1\) if an access WSS is installed in node \(n\) in period \(p\) and \(0\) otherwise; \(r_{np}\) is equal to \(1\) if access is needed in node \(n\) in period \(p\) and \(0\) otherwise; \(u_{ep}\) is equal to \(1\) if edge \(e\) is used in period \(p\). Objective (9a) should be added to objective (1a) if WSSs are considered in a network. Constraint (9b) assures that a WSS is installed on each edge used by a node if routing is needed. Constraints (9c)-(9j) assure that each change in a WSS configuration requires an engineer intervention on site, which increases the OPEX.