# Problems understanding model notation in LPs

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}

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.
• 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. Nov 17 at 6:04