Representation of conditions in integer linear programming

Question

How to represent in PLI the fact that inequality (1) or inequality (2) must be satisfied but not both?

• $j$ is executed before $k \rightarrow t_{ij} + p_{ij} \leq t_{ik}$ (1)
• $j$ is executed after $k \rightarrow t_{ij} \geq t_{ik} + p_{ik}$ (2)

Answer 1

Let binary decision variable $x_{jk}$ indicate whether $j$ is executed before $k$, and impose linear constraints \begin{align} x_{jk} + x_{kj} &= 1 &&\text{for $j < k$} \\ t_{ij} + p_{ij} - t_{ik} &\le M_{ijk} (1 - x_{jk}) &&\text{for all $i$ and $j \not= k$} \end{align}