MP 3 2S-SPLPO mathematical programming formulation.

$$\begin{array}{ccc}\hfill minZ& =& \sum _{j\in J}{f}_j{y}_j+\sum _{\omega \in \Omega }{\alpha }^{\omega }\sum _{i\in I}\sum _{j\in J}{c}_{ij}{x}_{ij}^{\omega }\hfill \end{array}$$ (8) $$\begin{array}{ccc}\hfill \mathrm{subject}\mathrm{to}\sum _{j\in J}{x}_{ij}^{\omega }& =& 1,\forall i\in I,\omega \in \Omega ,\hfill \end{array}$$ (9) $$\begin{array}{ccc}\hfill x_{ij}^{\omega }& \le & y_j,\forall i\in I,j\in J,\omega \in \Omega ,\hfill \end{array}$$ (10) $$\begin{array}{ccc}\hfill \sum _{k\in \overline{W_{ij}^{\omega }}}{x}_{ik}^{\omega }& \ge & y_j,\forall i\in I,j\in J^i,\omega \in \Omega ,\hfill \end{array}$$ (11) $$\begin{array}{ccc}\hfill x_{ij}^{\omega }& \ge & 0,\forall i\in I,j\in J,\omega \in \Omega ,\hfill \end{array}$$ (12) $$\begin{array}{ccc}\hfill y_j& \in & \{0,1\},\forall j\in J.\hfill \end{array}$$ (13)