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minkKeEkcexe+iNfiyis.t. kKeEkαjexe+iNβjiyi=bjjMxe=pPk(sp:s=e1)λpkK,eEkLkpPkλpUkkKλpZ+kK,pPkxeZ,yiZeE,iN \begin{aligned} \min{ } & \sum_{k \in K} \sum_{e \in E^k} c_e x_e + \sum_{i \in N} f_i y_i \\ \text{s.t. } & \sum_{k \in K} \sum_{e \in E^k} \alpha_{je} x_e + \sum_{i \in N} \beta_{ji} y_i = b_j & & j \in M \\ & x_e = \sum_{p \in P^k} \Big ( \sum_{ s \in p: s = e} \mathbf{1} \Big ) \lambda_p & & k \in K, e \in E^k \\ & L^k \leq \sum_{p \in P^k} \lambda_p \leq U^k && k \in K \\ & \lambda_p \in \mathbb{Z}^+ && k \in K, p \in P^k \\ & x_e \in \mathbb{Z}, y_i \in \mathbb{Z} && e \in E, i \in N \end{aligned}

given graphs G(Vk,Ek),  kKG(V^k, E^k),\; k \in K with paths pPkp \in P^k subject to resource constraints RkR^k.

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