## 4 MIP Problem Formulation  
In this section, we formulate the optimization problem that we aim to solve, namely, the OSNR-aware lightpath planning problem in MCF-based SS-FONs. The problem concerns the establishment of lightpath connections for a set of traffic demands in such a way that any spectrum segment of any core of an MCF link is assigned to one lightpath at most and each lightpath satisfies the SNR requirement discussed in Section 3.2. Following the assumption in Section 3, each lightpath has assigned a unique MCF core index, which is the same for each link of its routing path. In addition, the frequency slot assigned to a lightpath does not change in consecutive links of the path. In other words, the spatial and spectral continuity constraints are imposed. The optimization goal is to minimize the maximum spectrum usage, which is defined as the width of spectrum required to serve all demands. Figure 1: Comparison of estimated transmission reaches in a function of crosstalk level for OSNR-based and XT-based QoT models assuming QPSK modulation format. To formulate the optimization problem as an MIP problem, we introduce the notation presented in Table 1. In addition, in Table 1, we define a set of problem variables that we use to model the MIP problem. The MIP formulation is as follows:  
minimize  
\[z=\sum_{s\in\mathcal{S}}y_{s}\]  
, (6) \[\sum_{l\in\mathcal{L}(d)}x_{dl}=1\]  
, \[d\in\mathcal{D}\]  
, (7) \[\sum_{l\in\mathcal{L}(e,\mathcal{S})}x_{d(l)l}=y_{cs} \quad e\in\mathcal{E},c\in\mathcal{C},s\in\mathcal{S}\]  
, (8) \[y_{cs}\leq y_{s} \quad e\in\mathcal{E},c\in\mathcal{C},s\in\mathcal{S}\]  
, (9) \[\beta(e)+\gamma(e)\cdot\sum_{e^{\prime}\in\mathcal{C}(c)}y_{ce^{ \prime}s}=w_{cs} \quad e\in\mathcal{E},c\in\mathcal{C},s\in\mathcal{S}\]  
, (10) \[\sum_{e\in\mathcal{E}(l)}w_{ce(l)s}\leq Q(m(l))+(1-x_{dl})\cdot B \quad d\in\mathcal{D},l\in\mathcal{L}(d),s\in\mathcal{S}(l).\]  
Optimization objective (6) minimizes the number of frequency slices that are actually used in the network (calculated as the sum of variables \(y_{s}\)). Constraint (7) assures that each demand will be realized using exactly one lightpath selected from the set of allowable lightpaths. Constraint (8) assures that the selected lightpaths do not collide with each other; namely, each slice on each core of each link can be used by one lightpath at most since \(y_{cs}\) is a binary variable. Constraint (9) defines variables \(y_{s}\) that indicates whether slice \(s\) is used on at least one link. Eventually, Constraints (10) and (11) implement Model (3) for assuring QoT. Constraint (10) estimates the inverse value of SNR for each slice on each core of each link, taking into account the allocation status in neighbour cores (the sum of variables \(y_{cs}\)). Constraint (11) assures that whenever lightpath \(l\) is selected for demand \(d\) (i.e., if \(x_{dl}=1\)), the sum of inverse SNR values (variables \(w_{cs}\)) corresponding to the spectral-spatial resources used by lightpath \(l\) on its routing path does not exceed the limit allowable for the modulation format used by this lightpath (i.e., value \(Q(m(l))\)), according to Model (3). \(B\) represents a large value that assures that Constraint (11) is always satisfied if variable \(x_{dl}\) is equal to 0. Solving MIP model (4) is difficult even without the presence of OSRN constraints (10)-(11) [30]. Therefore, in the next section, we propose an efficient heuristic method for generating good-quality solutions to larger instances of the optimization problem considered.