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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?

Asked By : Marine1
Answered By : D.W.

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.

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