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# How to understand the mathematical constraint of potentials?

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Problem Detail:

Good evening! I am studying scheduling problems and I have some difficulties understanding constraints of potentials:

Let be \$t_j\$ the time when a task \$j\$ starts, and \$t_i\$ the time when a task \$i\$ starts. I'm assuming that \$a_{ij}\$ is the length of the task \$i\$ but I'm not sure. Why is a "constraint of potential" mathematically expressed by:

\$\$t_j-t_i \le a_{ij}\$\$

Shouldn't it be the reverse, \$t_j-t_i \ge a_{ij}\$? If we know that the length of task \$i\$ is \$a_{ij}\$, isn't it impossible to do something in less that the necessary allocated time?

I suspect there's some misunderstanding about the definition/meaning of \$a_{ij}\$. The time to complete task \$i\$ depends only on \$i\$ (not on \$j\$), so it wouldn't make sense to use notation like \$a_{ij}\$ for the length of task \$i\$: we'd instead expect to see only the index \$i\$, but not \$j\$, appear in that notation.

I suspect you'll probably need to go back to your textbook or other source on scheduling and look for a precise definition of the notation that it uses. If your textbook doesn't define its notation, look for a better textbook.

As far as the constraint \$t_j - t_i \le a_{ij}\$, that is expressing that \$t_j\$ should start at most \$a_{ij}\$ seconds after \$t_i\$ starts. So, it'd make more sense for \$a_{ij}\$ to be a permissible delay that expresses the maximum time you can wait to start task \$j\$, once task \$i\$ has been started.