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today I came across a paper that uses a type of model notation I have never come across before. These are the objective function and constraints I don't quite understand. I am specifically interested in what the dots in front of the capital letters mean. For example, for $i.NC=1$, does this mean that it only applies to all $NC$ of index $i$?

Likewise, how do I understand the sum signs with the $\wedge$ symbol?

\begin{align} I)~\min\sum_{j,b}^{}j.W\cdot l_{jb} \end{align} \begin{align} II)~\sum_{i:i.B}^{}p_{jit}\le l_{jb}~~~\forall j,t,b \end{align} \begin{align} III)~\sum_{i,t:i.NC=1\wedge i.B=b}^{}p_{jit}\le 3~~~\forall j,b \end{align} \begin{align} VI)~\sum_{i,t:i.SUN=1\wedge i.B=b}^{}p_{jit}\le 1~~~\forall j,b\end{align} \begin{align} V)~\sum_{i,t:i.B=b}^{}i.SAS\cdot p_{jit}\le j.AS\cdot l_{jb}~~~\forall j,b\end{align} \begin{align} VI)~p_{jit}+\sum_{k:k.B=b\wedge k.NC\neq1}^{}p_{jk(t+1)}\le 1~~~\forall j,t,i:i.NC=1\end{align} \end{align}

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The $\wedge$ simply means "and". So for example, constraint III is telling you to sum over all pair of $i$ and $t$ such that $i.NC = 1$ and $i.B=b$.

As for what the authors mean by the "." notation itself (such as $i.NC$), that is impossible to say for certain without context; it is not a standard notation as far as I know. But I would be willing to wager that each $i$ comes from set, and for each $i$ there are a bunch of related parameters, including $NC$ and $B$. For example, in a minimum cost flow problem, each edge $e$ has the parameters "cost" and "capacity". Normally, we think of these as vectors/functions and write them as $c_e$ and $u_e$ (or $c(e)$ and $u(e)$. However, if you wanted to emphasize that these are properties of the edge (NB: this has no mathematical meaning but could improve pedagogy) you could choose to write them as $e.c$ and $e.u$ (or even $e.cost$ and $e.capacity$). The reason I think this is what the authors mean is that this is the syntax that many popular programming languages use if you want to access a property of a specific object.

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    $\begingroup$ The ^ character is called a caret, not a carat. Also $\wedge$ is known as the logical AND symbol. This has nothing to do with the caret symbol so I suggest updating your answer accordingly. $\endgroup$ Commented Nov 17, 2023 at 6:04
  • $\begingroup$ Thank you for the correction. I have removed the incorrect information. $\endgroup$ Commented Nov 17, 2023 at 20:21

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