## Appendix  
### Problem Solving Steps  
The solution approach to the problem formulation is based on Lagrangian Relaxation. Constraints (IP 1), (IP 3) and (IP 5) are relaxed and multiplied by nonnegative Lagrangian multipliers, respectively. They are added to the objective functions as follows:  
\[ \begin{array}{l}\hline\begin{array}{l}Z_{xx}(\mu_{v}^{1},\mu_{v}^{2},\mu_{v}^{3 })=\\ \min\sum_{x_{in}}D_{i}(g_{1})g_{1}\\ +\sum_{v_{in}}\mu_{v}^{1}(\sum_{u=1}^{L}D_{i}(g_{1})y_{u}-D_{v})\\ +\sum_{u,l}\mu_{v}^{2}(\sum_{u=1}^{L}\sum_{v=1}x_{v}\delta_{v}y_{u}-g_{1})\\ +\sum_{v_{in}}\sum_{u,l}\mu_{v_{in}}^{3}(\sum_{v=1}x_{v}\delta_{v}-y_{u})\\ \end{array}&\begin{array}{l}\text{(LR)}\\ \forall w\in W\\ \forall l\in L\\ \forall p\in P_{v,w}\in W\\ \forall w\in W,\forall l\in L\end{array}&\begin{array}{l}\text{(IP 2)}\\ \forall l\in L\\ \end{array}\\ \begin{array}{l}x_{p}=0\,\text{or}\,1\\ y_{u}=0\,\text{or}\,1\end{array}&\begin{array}{l}\text{(IP 6)}\\ \forall w\in W,\forall l\in L\end{array}&\begin{array}{l}\text{(IP 7)}\\ \end{array}\\ \hline\end{array} \]  
The constraints are relaxed in such a way that the corresponding Lagrangian multipliers \(\mu_{v}^{l}\), \(\mu_{v}^{2}\), and \(\mu_{v}^{3}\) are non-negative. This approach is not guaranteed to have a feasible solution due to accessing a big value of \(\gamma_{v}\) into the network which will result in unsatisfactory QoS requirement to the O-D pair \(w\). However, LR can be decomposed into the following two independent and solvable optimization sub-problems in a sufficient way. Step 1: Sub-problem 1 (related to \(x_{p}\))  
\[\begin{array}{l}\hline\hline\begin{array}{l}\hline\hline\omit\span \omit\span\omit\span\omit\  
Step 2: Sub-problem 2 (related to \(y_{wl}\) and \(g_{l}\)) \begin{tabular}{l l l} \hline \hline
**Objective Function:** & \\ \(Z_{s_{\text{sub}}}(\mu_{w}^{1},\mu_{u}^{2},\mu_{u}^{3})\) & \\ \(=\min[\sum\limits_{s\in L}(D_{s}(g_{l})g_{l}+\sum\limits_{w\in W}\mu_{w}^{1}Y_{ wt}D_{s}(g_{l})-\mu_{t}^{2}g_{l}-\sum\limits_{w\in W}\mu_{wt}^{3}Y_{wt})-D_{w}\sum \limits_{w\in W}\mu_{w}^{1}]\) & (SP 2) \\
**Subject to:** & \\ \(0\leq g_{l}\leq\frac{C_{l}}{I(l)}\) & \(\forall l\in L\) & (IP 4) \\ \(y_{wl}=0\,\text{or}\,1\) & \(\forall w\in W,\forall l\in L\). & (IP 7) \\ \hline \hline \end{tabular}   
In objective function (SP 2), the last term \(-D_{w}\sum\limits_{w\in W}\mu_{w}^{1}\) is not changed to determine the optimal solution. It can be disregarded first and then added back to the objective value. Therefore, (SP 2) can be reformulated and further decomposed into \(|L|\) independent sub-problems. The objective function (SP 2) can be determined to function (SP 2.1) for each link \(l\). \begin{tabular}{l l l} \hline \hline \(\min[D_{s}(g_{l})g_{l}+\sum\limits_{w\in W}\mu_{w}^{1}Y_{wt}D_{l}(g_{l})-\mu_{l }^{2}g_{l}-\sum\limits_{w\in W}\mu_{wt}^{3}Y_{wt}]\) & (SP 2.1) \\
**Subject to:** & \\ \(0\leq g_{l}\leq\frac{C_{l}}{I(l)}\) & \(\forall l\in L\) & (IP 4) \\ \(y_{wl}=0\,\text{or}\,1\) & \(\forall w\in W,\forall l\in L\). & (IP 7) \\ \hline \hline \end{tabular}   
In problem (SP 2.1), the first term \(D_{l}(g_{l})g_{l}\) is a monotonically increasing and nonnegative function. If the minimum value of the other terms in (SP 2.1) can be determined, the first term \(D_{l}(g_{l})g_{l}\) can be eliminated in the first step. (SP 2.1) can be expressed as (SP 2.1') for each link \(l\), \begin{tabular}{l l l} \hline \hline \(\min[\sum\limits_{w\in W}\mu_{w}^{1}Y_{wt}D_{l}(g_{l})-\mu_{l}^{2}g_{l}-\sum \limits_{w\in W}\mu_{wt}^{3}Y_{wt}]\) & (SP 2.1') \\
**Subject to:** & \\ \(0\leq g_{l}\leq\frac{C_{l}}{I(l)}\) & \(\forall l\in L\) & (IP 4) \\ \(y_{wl}=0\,\text{or}\,1\) & \(\forall w\in W,\forall l\in L\). & (IP 7) \\ \hline \hline \end{tabular}   
**Table A1. ** Algorithm for solving problem (SP 2.1'). \begin{tabular}{l l l} \hline \hline Step 1: & Solve \(y_{w}^{*}(g_{l})=\mu_{w}^{1}D_{l}(g_{l})-\mu_{wl}^{3}=0\) for each O-D pair \(w\), and define the results as a set of break points of \(g_{l}\).