$$\min :y^{*}$$ (9)

Subject to:

$$\begin{array}{}\mathrm{(9a)}& \hfill r_{\mathrm{out}}^c\left(s_c\right)& ={\Gamma }^c\hfill & \hfill \forall c\in C\mathrm{(9b)}& \hfill {\sum }_{c\in C}{r}^c\left(e\right)& \le Z\left(e\right)\hfill & \hfill \forall c\in C,\forall e\in E\mathrm{(9c)}& \hfill r_{in}^c\left(v\right)& =r_{\mathrm{out}}^c\left(v\right)\hfill & \hfill \forall c\in C,\forall v\in V^{\prime }\mathrm{(9d)}& \hfill r_{in}^c\left(s_c\right)& =0\hfill & \hfill \forall c\in C\mathrm{(9e)}& \hfill r_{\mathrm{out}}^c\left(d_c\right)& =0\hfill & \hfill \forall c\in C\end{array}$$

$$y^{*}=\sum _{c\in C}\sum _{e\in E}{x}_e^c\times M\left(e\right).$$