## 4 MILP Optimization Model
Constraint (4) assures that the TP established for some demand \(\bar{d}\) is used to serve demand \(d\) if a direct TP is not established for \(d\) (i.e., when \(y_{d}=1\)). Constraint (5) imposes that a TP established for demand \(\bar{d}\), which also serves demand \(d\), should be routed through the remote site associated with \(d\). Constraint (6) determines variable \(x_{d}\), which indicates if the TP established for demand \(d\) is also used to serve some other demand. Constraints (7)-(12) are responsible for the assignment of wavelengths to particular demands. Namely, Constraint (7) assigns \(W(d)\) wavelengths requested by demand \(d\) by setting to one an adequate number of variables \(x_{dl}\). Constraint (8) assures that two demands \(d\) and \(\bar{d}\) may use the same wavelength only if demand \(d\) is not served by a TP established for demand \(\bar{d}\). Constraint (9) sets variable \(y_{d\bar{d}}\) equal to one if wavelength \(l\) is assigned to demand \(d\) and demand \(d\) is served by the TP established for demand \(\bar{d}\in\mathcal{D}\). Constraint (10) assures that at most one demand \(d\) may have assigned wavelength \(l\) among all demands served be the TP established for demand \(\bar{d}\). Whenever a TP is established for demand \(d\) (i.e., when \(y_{d}=0\)), Constraint (11) determines the number of wavelengths utilized on the TP as the sum of wavelengths requested by \(d\) and all other demands served by the TP. Constraint (12) assures that the number of utilized wavelengths does not exceed the WDM transmission capacity (\(W\)). Constraints (13)-(17) are used to determine the type of MUXs deployed at the ends of TPs. The MUXs are installed only for the demands with established TPs (\(y_{d}=0\)), which is imposed by Constraints (13) and (14), respectively, for the hub site and the remote site. Constraint (15) assures that the number of channels of the MUX installed at the hub is not lower than the number of wavelengths utilized on the TP. Constraint (16) assures that the number of channels of the MUX installed at the end remote site is sufficient to serve the wavelengths originated at the site. Moreover, Constraint (17) assures the use of a MUX at the end remote site whenever the TP serves also other demands. Constraints (18)-(20) implement the transmission distance model described in Section 3.3 for each demand that has established a TP. Constraint (18) calculates the attenuation of MUXs used at the ends of the TP, taking into account the type of the MUXs applied. Constraint (19) calculates the attenuation of OADMs installed on the TP at intermediate remote sites. Constraint (20) estimates the optical power budget, accounting for the components of the optical path, to determine the maximum transmission distance allowable for the TP selected. Constraints (21)-(23) determine the cost elements. Constraint (21) estimates the cost of the optical fiber leased, taking into account the overall length of TPs. Constraint (22) calculates the cost of MUXs deployed at the ends of the TPs, namely, at the hub site and the remote sites. Constraint (22) calculates the cost of OADMs installed at the remote sites that are served by the TPs established for other demands. To reflect both transmission directions, coefficient 2 appears in Constraints (21)-(23). Finally, Constraints (24) and (25) determine the type of variables. MILP model (2)-(25) corresponds to the Flex-O network deployment scenario, in which different types of OADM and MUX devices are available and can be applied in a flexible way on the TRs, according to the wavelength demands of particular remote sites. The reference scenarios described in Section 3 can be implemented by introducing the following modifications in the MILP model. * In scenario Fix-O, sets \(\mathcal{M}\) and \(\mathcal{O}\) are limited to the single types of MUX and OADM devices, respectively, such that they support the maximum number of channels available in the WDM system; in addition, set \(\mathcal{M}\) includes a fictitious type of MUX that supports one channel.