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Introduction

This is where you will get started solving models like

min⁡∑k∈K∑e∈Ekcexe+∑i∈Nfiyis.t. ∑k∈K∑e∈Ekαjexe+∑i∈Nβjiyi=bjj∈Mxe=∑p∈Pk(∑s∈p:s=e1)λpk∈K,e∈EkLk≤∑p∈Pkλp≤Ukk∈Kλp∈Z+k∈K,p∈Pkxe∈Zk∈K,e∈Ekyi∈Zi∈N \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} && k \in K, e \in E^k \\ & y_i \in \mathbb{Z} && i \in N \\ \end{aligned}

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

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