## Appendix A Proof of Property 4:  
**Proof of only if part. ** Let \(h_{nkt}\), \(w_{nkt}\) and \(l_{nt}\) for \(G_{n}(V_{nt},E_{n})\) such that \(\underset{t\in\{1,2,\ldots,\theta\}}{\Sigma}l_{n\theta}=Q\) be a solution to Problem 2. We need to prove that there exists a solution to Problem 3 with \(l_{nt}=0\) for all \(t>\theta\). As \(h_{nkt}\), \(w_{nkt}\) and \(l_{nt}\) is a solution to Problem 2, it must be a feasible solution to Problem 2. \(\square\)  
Due to conservation of flows in the network of \(G_{n}(V_{nt},E_{n})\), \(\underset{t\in\{1,2,\ldots,T\}}{\Sigma}l_{nt}=Q\). As \(h_{nkt}\), \(w_{nkt}\) and \(l_{nt}\) is a feasible solution to Problem 2, \(\underset{t\in\{1,2,\ldots,\theta\}}{\Sigma}l_{n\theta}=Q\). Therefore, \(l_{nt}=0\) for all \(t>\theta\). Therefore, \(h_{nkt}\),\(w_{nkt}\) and \(l_{nt}\) is a feasible solution to Problem 2 with \(l_{nt}=0\) for all \(t>\theta\). As the constraints (1)-(9) in Problem 2 and Problem 3 are the same and \(h_{nkt}\), \(w_{nkt}\) and \(l_{nt}\) is a feasible solution to Problem 2, \(h_{nkt}\), \(w_{nkt}\) and \(l_{nt}\) must be a feasible solution to Problem 3. Next, we show that \(h_{nkt}\), \(w_{nkt}\) and \(l_{nt}\) is an optimal solution to Problem 3. To prove this, we show that the objective function of \(h_{nkt}\), \(w_{nkt}\) and \(l_{nt}\) is less than or equal to an optimal solution \(h_{nkt}^{*}\), \(w_{nkt}^{*}\) and \(l_{nt}^{*}\) to Problem 3. As \(h_{nkt}^{*}\), \(w_{nkt}^{*}\) and \(l_{nt}^{*}\) is an optimal solution to Problem 3, \(h_{nkt}^{*}\), \(w_{nkt}^{*}\) and \(l_{nt}^{*}\) must be a feasible solution that satisfies constraints (1)-(9). Let \(\Gamma=\{t\,|\,l_{nt}^{*}>0,t\in\{\theta+1,\theta+2,\ldots,T\}\}\). The objective function value for \(h_{nkt}^{*}\), \(w_{nkt}^{*}\) and \(l_{nt}^{*}\) is  
\[\underset{t=1}{\overset{T}{\sum}}\underset{k=1}{\overset{K_{n }+1}{\sum}}\left(c_{nkt}^{h}h_{nkt}^{*}\right)+\underset{t=1}{\overset{T}{ \sum}}\underset{k=1}{\overset{K_{n}+1}{\sum}}\left(c_{nkt}^{w}w_{nkt}^{*}\right) +\underset{t=1}{\overset{T}{\sum}}\left(c_{nt}^{l}l_{nt}^{*}\right)=\underset {t=1}{\overset{T}{\sum}}\left(c_{nt}^{l}l_{nt}^{*}\right)=\] \[\underset{t\in\{1,2,\ldots,T\}\backslash\Gamma}{\sum}\left(c_{ nt}^{l}l_{nt}^{*}\right)+\underset{t\in\Gamma}{\overset{T}{\sum}}\left(c_{ nt}^{l}l_{nt}^{*}\right)=\underset{t\in\Gamma}{\overset{T}{\sum}}\left(c_{ nt}^{l}l_{nt}^{*}\right)=\underset{t\in\Gamma}{\overset{T}{\sum}}\left( \Phi l_{nt}^{*}\right)=\Phi(\underset{t\in\Gamma}{\overset{T}{\sum}}l_{nt}^{*} )=\Phi\mathbb{Q}.\]  
The objective function value for \(h_{nkt}\), \(w_{nkt}\) and \(l_{nt}\) is  
\[\underset{t=1}{\overset{T}{\sum}}\underset{k=1}{\overset{K_{n }+1}{\sum}}\left(c_{nkt}^{h}h_{nkt}\right)+\underset{t=1}{\overset{T}{\sum}} \underset{k=1}{\overset{K_{n}+1}{\sum}}\left(c_{nkt}^{w}w_{nkt}\right)+ \underset{t=1}{\overset{T}{\sum}}\left(c_{nt}^{l}l_{nt}\right)=\underset{t=1}{ \overset{T}{\sum}}\left(c_{nt}^{l}l_{nt}\right)=\] \[\underset{t\in\{1,2,\ldots,T\}\backslash\Gamma}{\sum}\left(c_{ nt}^{l}l_{nt}\right)+\underset{t\in\Gamma}{\overset{T}{\sum}}\left(c_{ nt}^{l}l_{nt}\right)=\underset{t\in\Gamma}{\overset{T}{\sum}}\left(c_{ nt}^{l}l_{nt}\right)=0\leq\Phi\mathbb{Q}.\]The objective function of \(h_{nkt}\), \(w_{nkt}\) and \(l_{nt}\) is less than or equal to an optimal solution \(h^{*}_{nkt}\);\(w^{*}_{nkt}\) and \(l^{*}_{nt}\) to Problem 3. Therefore, \(h_{nkt}\), \(w_{nkt}\) and \(l_{nt}\) is an optimal solution to Problem 3. **Proof of if part. ** In this part, we need to prove that if there exists a solution \(h_{nkt}\), \(w_{nkt}\) and \(l_{nt}\) to Problem 3 with \(l_{nt}=0\) for all \(t>\theta\), there exists a solution to Problem 2 with \(\Sigma\limits_{t\in\{1,2,\ldots,\theta\}}l_{n\theta}=Q\). \(\square\)  
Suppose \(h_{nkt}\), \(w_{nkt}\) and \(l_{nt}\) is a solution to Problem 3 with \(l_{nt}=0\) for all \(t>\theta\). The solution \(h_{nkt}\), \(w_{nkt}\) and \(l_{nt}\) must be a feasible solution that satisfy constraints (1)-(9) in Problem 2 and Problem 3.