# Optimization of Optical Networks Based on CDC-ROADM Technology
## 2 Methods  
In this section DWDM network optimization problem is formulated using MIP. For this purpose the following sets are defined: \(\mathcal{N}\) the set of all nodes; \(\mathcal{T}\) the set of transponders; \(\mathcal{P}\) the set of time periods; \(\mathcal{S}\) the set of frequency slices; \(\mathcal{E}\) the set of edges; \(\delta^{+}(n)\) the set of edges starting at node \(n\), and \(\delta^{-}(n)\) the set of edges ending at node \(n\). The following objective cost function is optimized using an IP algorithm subject to the listed below constraints:  
\[\min\sum_{p\in\mathcal{P}}\sum_{n\in\mathcal{N}}\left(w_{np}\xi(n)+\sum_{t\in \mathcal{T}}x_{np}\xi(t)\right) \tag{1a}\] \[\sum_{n^{\prime}\in\mathcal{N}}z_{n^{\prime\prime}p}^{(n,n^{ \prime})} =\sum_{n^{\prime}\in\mathcal{N}}z_{n^{\prime\prime}p}^{(n^{\prime },n)}+h(n,n^{\prime\prime},p)\quad\forall n,n^{\prime\prime}\in\mathcal{N}:n \
eq n^{\prime\prime},\forall p\in\mathcal{P}\] (1b) \[\sum_{s\in\mathcal{S}}\sum_{t\in\mathcal{T}}y_{tsp}^{(n,n^{ \prime})}b(t) \geq\sum_{n^{\prime\prime}\in\mathcal{N}}z_{n^{\prime\prime}p}^{(n,n^{ \prime})}\quad\forall n,n^{\prime}\in\mathcal{N},\forall p\in\mathcal{P}\] (1c) \[g_{tsnp} \geq\sum_{n^{\prime}\in\mathcal{N}}(y_{tsp}^{(n,n^{\prime})}+y_{ tsp}^{(n^{\prime},n)})\quad\forall t\in\mathcal{T},\forall n\in\mathcal{N}, \forall p\in\mathcal{P},\forall s\in\mathcal{S}\] (1d) \[x_{tnp}o(t) \geq\sum_{s\in\mathcal{S}}g_{tsnp}\quad\forall t\in\mathcal{T}, \forall n\in\mathcal{N},\forall p\in\mathcal{P}\] (1e) \[w_{np}M \geq\sum_{s\in\mathcal{S}}(g_{tsnp}-g_{tsnp}(p))\quad\forall n\in \mathcal{N},\forall p\in\mathcal{P}:p(p)\
eq\emptyset,\forall t\in\mathcal{T}\] (1f) \[w_{np}M \geq\sum_{s\in\mathcal{S}}(g_{tsnp}(p)-g_{tsnp})\quad\forall n\in \mathcal{N},\forall p\in\mathcal{P}:p(p)\
eq\emptyset,\forall t\in\mathcal{T}\] (1g) \[w_{np}M \geq\sum_{s\in\mathcal{S}}(g_{tsnp}-i(t,s,n))\quad\forall n\in \mathcal{N},\forall p\in\mathcal{P}:p(p)=\emptyset,\forall t\in\mathcal{T}\] (1h) \[w_{np}M \geq\sum_{s\in\mathcal{S}}(i(t,s,n)-g_{tsnp})\quad\forall n\in \mathcal{N},\forall p\in\mathcal{P}:p(p)=\emptyset,\forall t\in\mathcal{T} \tag{1i}\]  
In (1a)-(1i) we are provided with the following constants: \(o(t)\) is the number of outputs of transponder \(t\); \(b(t)\) is the bitrate of one output of transponder \(t\); \(\xi(t)\) is the cost of using transponder \(t\) in one period; \(h(n,n^{\prime},p)\) is the volume of demand from node \(n\) to node \(n^{\prime}\) in period \(p\); \(\xi(n)\) is the cost of intervention in node \(n\); \(p(p)\) is the period before period \(p\); \(p\) is the first period if \(p(p)=\emptyset\); \(i(t,s,n)\) is the number of ports of transponders \(t\) working on slice \(s\) installed in node \(n\) at time zero and \(M\) is a large number. In the formulation we use the following variables: \(x_{tnp}\) is the number of transponders \(t\) installed in node \(n\) in period \(p\); \(z_{n^{\prime}np}^{(n,n^{\prime})}\) is the bitrate in relation \((n,n^{\prime})\) with node \(n^{\prime\prime}\) being the final destination in period \(p\); \(w_{np}\) is a binary variable and equal to 1 if intervention is needed in node \(n\) in period \(p\); \(g_{tsnp}\) is the number of ports of transponders \(t\) working on slice \(s\) installed in node \(n\) in period \(p\), and \(y_{tsp}^{(n,n^{\prime})}\) is the number of outputs of transponder \(t\) working on slice \(s\) installed in relation \((n,n^{\prime})\) in period \(p\). Minimizing the objective function (1a) is equivalent to minimization of the total costs of interventions and transponder purchases. Constraint (1b) imposes the Kirchhoff law for traffic entering and leaving node \(n\). Constraint (1c) assures that flow between nodes \(n\) and \(n^{\prime}\) is realized using only available outputs of transponders. Constraint (1d) assures that all used outputs of transponders have corresponding ports installed. Constraint (1e) assures that all ports are installed in transponders.